Finite-State Dimension and Lossy Decompressors

Abstract: This paper examines information-theoretic questions regarding the difficulty of compressing data versus the difficulty of decompressing data and the role that information loss plays in this interaction. Finite-state compression and decompression are shown to be of equivalent difficulty, even when the decompressors are allowed to be lossy. Inspired by Kolmogorov complexity, this paper defines the optimal decompression *ratio achievable on an infinite sequence by finite-state decompressors (that is, finite-state transducers outputting the sequence in question). It is shown that the optimal compression ratio achievable on a sequence S by any *information lossless finite state compressor, known as the finite-state dimension of S, is equal to the optimal decompression ratio achievable on S by any finite-state decompressor. This result implies a new decompression characterization of finite-state dimension in terms of lossy finite-state transducers.