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International Symposium on Fundamentals of Computation Theory

FCT 2017: Fundamentals of Computation Theory pp 418–430Cite as

Automatic Kolmogorov Complexity and Normality Revisited

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10472))

Abstract

It is well known that normality (all factors of a given length appear in an infinite sequence with the same frequency) 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. Other known notions (Shallit and Wang [15], Calude et al. [8]) of description complexity related to finite automata are discussed (see the last section).

As a byproduct, this approach provides simple proofs of classical results about normality (equivalence of definitions with aligned occurrences and all occurrences, Wall’s theorem saying that a normal number remains normal when multiplied by a rational number, and Agafonov’s result saying that normality is preserved by automatic selection rules).

A. Shen—Supported by ANR-15-CE40-0016-01 RaCAF grant.

On leave from IITP RAS, Moscow, Russia.

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References

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Acknowledgements

I am grateful to Veronica Becher, Olivier Carton and Paul Heiber for many discussions of their paper [3] and the relations between incompressibility and normality, and for the permission to use observations made during these discussions in the current paper. I am also grateful to my colleagues in LIRMM (ESCAPE team) and Moscow (Kolmogorov seminar, Computer Science Department of the HSE). I am thankful to the anonymous referees of an earlier version of this paper submitted to ICALP (and rejected) and to the anonymous referees of the final version.

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  1. LIRMM CNRS, University of Montpellier, Montpellier, France

    Alexander Shen

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  1. Alexander Shen
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Correspondence to Alexander Shen .

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Editors and Affiliations

  1. CNRS, University of Bordeaux, Talence cedex, France

    Ralf Klasing

  2. LaBRI, University of Bordeaux, Talence cedex, France

    Marc Zeitoun

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Shen, A. (2017). Automatic Kolmogorov Complexity and Normality Revisited. In: Klasing, R., Zeitoun, M. (eds) Fundamentals of Computation Theory. FCT 2017. Lecture Notes in Computer Science(), vol 10472. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-55751-8_33

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  • DOI: https://doi.org/10.1007/978-3-662-55751-8_33

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