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FO logic on cellular automata orbits equals MSO logic

Published 25 Apr 2024 in cs.DM, cs.LO, and math.DS | (2404.16430v1)

Abstract: We introduce an extension of classical cellular automata (CA) to arbitrary labeled graphs, and show that FO logic on CA orbits is equivalent to MSO logic. We deduce various results from that equivalence, including a characterization of finitely generated groups on which FO model checking for CA orbits is undecidable, and undecidability of satisfiability of a fixed FO property for CA over finite graphs. We also show concrete examples of FO formulas for CA orbits whose model checking problem is equivalent to the domino problem, or its seeded or recurring variants respectively, on any finitely generated group. For the recurring domino problem, we use an extension of the FO signature by a relation found in the well-known Garden of Eden theorem, but we also show a concrete FO formula without the extension and with one quantifier alternation whose model checking problem does not belong to the arithmetical hierarchy on group Z2.

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References (44)
  1. S. Amoroso and Y.N. Patt. Decision procedures for surjectivity and injectivity of parallel maps for tessellation structures. Journal of Computer and System Sciences, 6(5):448–464, October 1972. doi:10.1016/s0022-0000(72)80013-8.
  2. Causal graph dynamics. Information and Computation, 223:78–93, February 2013. doi:10.1016/j.ic.2012.10.019.
  3. Cellular automata over generalized cayley graphs. Mathematical Structures in Computer Science, 28(3):340–383, May 2017. doi:10.1017/s0960129517000044.
  4. About the Domino Problem for Subshifts on Groups, pages 331–389. Springer International Publishing, 2018. doi:10.1007/978-3-319-69152-7_9.
  5. The domino problem is undecidable on surface groups. Schloss Dagstuhl, Leibniz-Zentrum für Informatik, 2019. doi:10.4230/LIPICS.MFCS.2019.46.
  6. The domino problem on groups of polynomial growth. Groups, Geometry, and Dynamics, 12(1):93–105, March 2018. doi:10.4171/ggd/439.
  7. Laurent Bartholdi. The domino problem for hyperbolic groups, 2023. arXiv:2305.06952.
  8. Simulations and the lamplighter group. Groups, Geometry, and Dynamics, 16(4):1461–1514, November 2022. doi:10.4171/ggd/692.
  9. R. Berger. The undecidability of the domino problem. Mem. Amer. Math Soc., 66, 1966.
  10. Winning Ways for your Mathematical Plays, volume 2. Academic Press, 1982. chapter 25.
  11. Complexity of fixed point counting problems in boolean networks. Journal of Computer and System Sciences, 126:138–164, June 2022. URL: http://dx.doi.org/10.1016/j.jcss.2022.01.004, doi:10.1016/j.jcss.2022.01.004.
  12. T. Ceccherini-Silberstein and M. Coornaert. Cellular automata and groups. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi:10.1007/978-3-642-14034-1.
  13. Cellular Automata Modeling of Physical Systems. Cambridge University Press, December 1998. doi:10.1017/cbo9780511549755.
  14. Bruno Courcelle. The monadic second-order logic of graphs, ii: Infinite graphs of bounded width. Mathematical Systems Theory, 21(1):187–221, December 1988. doi:10.1007/bf02088013.
  15. Bruno Courcelle. The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Information and Computation, 85(1):12–75, March 1990. URL: http://dx.doi.org/10.1016/0890-5401(90)90043-H, doi:10.1016/0890-5401(90)90043-h.
  16. Bruno Courcelle. The monadic second order logic of graphs vi: on several representations of graphs by relational structures. Discrete Applied Mathematics, 54(2-3):117–149, October 1994. URL: http://dx.doi.org/10.1016/0166-218X(94)90019-1, doi:10.1016/0166-218x(94)90019-1.
  17. Graph structure and monadic second-order logic. A language-theoretic approach. Encyclopedia of Mathematics and its applications, Vol. 138. Cambridge University Press, June 2012. Collection Encyclopedia of Mathematics and Applications, Vol. 138.
  18. Non-uniform cellular automata: Classes, dynamics, and decidability. Information and Computation, 215:32–46, June 2012. URL: http://dx.doi.org/10.1016/j.ic.2012.02.008, doi:10.1016/j.ic.2012.02.008.
  19. Finite Model Theory. Springer Berlin Heidelberg, 1995. doi:10.1007/978-3-662-03182-7.
  20. The structure of quasi-transitive graphs avoiding a minor with applications to the domino problem, 2023. doi:10.48550/ARXIV.2304.01823.
  21. Rice-like theorems for automata networks. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. doi:10.4230/LIPICS.STACS.2021.32.
  22. Martin Gardner. Mathematical games. Scientific American, 223(4):120–123, October 1970. doi:10.1038/scientificamerican1070-120.
  23. Walter Gottschalk. Some general dynamical notions, pages 120–125. Springer Berlin Heidelberg, 1973. doi:10.1007/bfb0061728.
  24. R. Halin. Über unendliche wege in graphen. Mathematische Annalen, 157(2):125–137, April 1964. doi:10.1007/bf01362670.
  25. David Harel. Recurring Dominoes: Making the Highly Undecidable Highly Understandable, pages 51–71. Elsevier, 1985. doi:10.1016/s0304-0208(08)73075-5.
  26. G. A. Hedlund. Endomorphisms and Automorphisms of the Shift Dynamical Systems. Mathematical Systems Theory, 3(4):320–375, 1969.
  27. The Domino Problem Is Undecidable on Every Rhombus Subshift, pages 100–112. Springer Nature Switzerland, 2023. URL: http://dx.doi.org/10.1007/978-3-031-33264-7_9, doi:10.1007/978-3-031-33264-7_9.
  28. A characterization of the entropies of multidimensional shifts of finite type. Annals of Mathematics, 171(3):2011–2038, 2010. doi:10.4007/annals.2010.171.2011.
  29. Jarkko Kari. Reversibility and surjectivity problems of cellular automata. Journal of Computer and System Sciences, 48(1):149–182, February 1994. doi:10.1016/s0022-0000(05)80025-x.
  30. Jarkko Kari. Rice’s theorem for the limit sets of cellular automata. Theoretical Computer Science, 127(2):229–254, May 1994. doi:10.1016/0304-3975(94)90041-8.
  31. S.A. Kauffman. Metabolic stability and epigenesis in randomly constructed genetic nets. Journal of Theoretical Biology, 22(3):437–467, March 1969. URL: http://dx.doi.org/10.1016/0022-5193(69)90015-0, doi:10.1016/0022-5193(69)90015-0.
  32. On brambles, grid-like minors, and parameterized intractability of monadic second-order logic. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2010, Austin, Texas, USA, January 17-19, 2010, pages 354–364. SIAM, 2010. doi:10.1137/1.9781611973075.30.
  33. Logical aspects of cayley-graphs: the group case. Annals of Pure and Applied Logic, 131(1-3):263–286, January 2005. doi:10.1016/j.apal.2004.06.002.
  34. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, December 2020. doi:10.1017/9781108899727.
  35. Symbolic dynamics. Amer. J. Math., 3:286–303, 1936.
  36. The theory of ends, pushdown automata, and second-order logic. Theoretical Computer Science, 37:51–75, 1985. doi:10.1016/0304-3975(85)90087-8.
  37. Fabian Reiter. Distributed graph automata. In 2015 30th Annual ACM/IEEE Symposium on Logic in Computer Science. IEEE, July 2015. doi:10.1109/lics.2015.27.
  38. Adrien Richard. Fixed points and connections between positive and negative cycles in boolean networks. Discrete Applied Mathematics, 243:1–10, July 2018. doi:10.1016/j.dam.2017.12.037.
  39. Graph minors. ii. algorithmic aspects of tree-width. Journal of Algorithms, 7(3):309–322, September 1986. doi:10.1016/0196-6774(86)90023-4.
  40. Graph minors. v. excluding a planar graph. Journal of Combinatorial Theory, Series B, 41(1):92–114, August 1986. doi:10.1016/0095-8956(86)90030-4.
  41. H. Rogers. Theory of Recursive Functions and Effective Computability. MIT Press, 1987.
  42. Local normal forms for first-order logic with applications to games and automata, pages 444–454. Springer Berlin Heidelberg, 1998. doi:10.1007/bfb0028580.
  43. René Thomas. Boolean formalization of genetic control circuits. Journal of Theoretical Biology, 42(3):563–585, December 1973. URL: http://dx.doi.org/10.1016/0022-5193(73)90247-6, doi:10.1016/0022-5193(73)90247-6.
  44. Wolfgang Thomas. On logics, tilings, and automata, pages 441–454. Springer Berlin Heidelberg, 1991. doi:10.1007/3-540-54233-7_154.
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Authors (1)

  1. Guillaume Theyssier 

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