Automatic Kolmogorov complexity, normality and finite state dimension revisited
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 finitestate 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, PiatetskiShapiro, 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 nonaligned 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).
 Publication:

arXiv eprints
 Pub Date:
 January 2017
 arXiv:
 arXiv:1701.09060
 Bibcode:
 2017arXiv170109060K
 Keywords:

 Computer Science  Information Theory;
 Computer Science  Formal Languages and Automata Theory;
 68Q45;
 F.1.1
 EPrint:
 Revised version (2019): finite state dimension, criterion of normality in terms of complexity implying results of Champernowne et al,, superadditive calibrated functions. Covers FCT 2017 and FCT2019 conference papers. Sept. 16, 2019: spelling of author's name. August 2020: corrections and additions due to reviewer's report for the submitted version (including strong dimensions)