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Limit Complexities Revisited

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Abstract

The main goal of this article is to put some known results in a common perspective and to simplify their proofs.

We start with a simple proof of a result from Vereshchagin (Theor. Comput. Sci. 271(1–2):59–67, 2002) saying that lim sup  n C(x|n) (here C(x|n) is conditional (plain) Kolmogorov complexity of x when n is known) equals C 0′(x), the plain Kolmogorov complexity with 0′-oracle.

Then we use the same argument to prove similar results for prefix complexity, a priori probability on binary tree and measure of effectively open sets, and also to improve results of Muchnik (Theory Probab. Appl. 32:513–514, 1987) about limit frequencies. As a by-product, we get a criterion of 0′ Martin-Löf randomness (called also 2-randomness) proved in Miller (J. Symb. Log. 69(2):555-584, 2004): a sequence ω is 2-random if and only if there exists c such that any prefix x of ω is a prefix of some string y such that C(y)≥|y|−c. (In the 1960ies this property was suggested in Kolmogorov, IEEE Trans. Inf. Theory IT-14(5):662–664, 1968, as one of possible randomness definitions; its equivalence to 2-randomness was shown in Miller, J. Symb. Log. 69(2):555-584, 2004.) Miller (J. Symb. Log. 69(2):555-584, 2004) and Nies et al. (J. Symb. Log. 70(2):515–535, 2005) proved another 2-randomness criterion: ω is 2-random if and only if C(x)≥|x|−c for some c and infinitely many prefixes x of ω.

We show that the low-basis theorem can be used to get alternative proofs of our results on Kolmogorov complexity and to improve the result about effectively open sets; this stronger version implies the 2-randomness criterion mentioned in the previous sentence.

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Authors and Affiliations

  1. Laboratoire d’Informatique Fondamentale, CNRS & Université Aix-Marseille, Marseille, France

    Laurent Bienvenu & Alexander Shen

  2. Moscow State Lomonosov University, Moscow, Russia

    Nikolay Vereshchagin

  3. CNRS Poncelet Laboratory, Moscow, Russia

    Alexander Shen & Nikolay Vereshchagin

  4. IITP RAS, Moscow, Russia

    Alexander Shen

Authors
  1. Laurent Bienvenu
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  2. Andrej Muchnik
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  3. Alexander Shen
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  4. Nikolay Vereshchagin
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Corresponding author

Correspondence to Alexander Shen.

Additional information

The research of L. Bienvenu and A. Shen was supported in part by ANR Sycomore and NAFIT ANR-08-EMER-008-01 grants.

The research of N. Vereshchagin was supported in part by RFBR 05-01-02803-CNRS-a, 06-01-00122-a.

Andrej Muchnik (24.02.1958–18.03.2007) worked in the Institute of New Technologies in Education (Moscow). For many years he participated in Kolmogorov seminar at the Moscow State (Lomonosov) University. N. Vereshchagin and A. Shen (also participants of that seminar) had the privilege to know Andrej for more than two decades and are deeply indebted to him both as a great thinker and noble personality. The text of this paper was written after Andrej’s untimely death but it (like many other papers written by the participants of the seminar) develops his ideas.

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Bienvenu, L., Muchnik, A., Shen, A. et al. Limit Complexities Revisited. Theory Comput Syst 47, 720–736 (2010). https://doi.org/10.1007/s00224-009-9203-9

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  • DOI: https://doi.org/10.1007/s00224-009-9203-9

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