Automatic Kolmogorov complexity, normality and finite state dimension revisited

TL;DR

This paper revisits the concepts of Kolmogorov complexity and finite state dimension, providing simplified proofs of classical results on normal sequences and extending the theory with new characterizations and definitions.

Contribution

It introduces an explicit notion of automatic Kolmogorov complexity aligned with traditional theory and extends finite state dimension definitions with new equivalent formulations.

Findings

Simplified proofs of classical normality results

New characterizations of finite state dimension

Extension of normality preservation by automatic rules

Abstract

It is well known that normality can be described as incompressibility via finite automata. Still the statement and the proof of this result as given by Becher and Heiber (2013) in terms of "lossless finite-state compressors" do not follow the standard scheme of Kolmogorov complexity definition (an automaton is used for compression, not decompression). We modify this approach to make it more similar to the traditional Kolmogorov complexity theory (and simpler) by explicitly defining the notion of automatic Kolmogorov complexity and using its simple properties. Using this characterization and a sufficient condition for normality in terms of Kolmogorov complexity derived from it, we provide easy proofs for classical results about normal sequences (Champernown, Wall, Piatetski-Shapiro, Besicovitch, Copeland, Erdos et al.) Then we extend this approach to finite state dimension. We show that the block entropy definition of the finite state dimension remains the same if non-aligned blocks are used. Then we provide equivalent definitions in terms of automatic complexity, superadditive bounds for Kolmogorov complexity, calibrated superadditive functions and finite state a priori probability and use them to give simple proofs for known results about finite state dimension, and for Agafonov's result saying that normality is preserved by automatic selection rules as well as the results of Schnorr and Stimm that relate normality to finite state martingales. Some results of this paper were presented at the Fundamentals in Computing Theory conferences in 2017 and 2019. Preliminary version of this paper (that does not mention the finite state dimension) was published in arxiv in~2017 (see the previous version of this submission).