Limit complexities revisited

Authors Laurent Bienvenu, Andrej Muchnik, Alexander Shen, Nikolay Veraschagin

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Laurent Bienvenu
Andrej Muchnik
Alexander Shen
Nikolay Veraschagin

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Laurent Bienvenu, Andrej Muchnik, Alexander Shen, and Nikolay Veraschagin. Limit complexities revisited. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 73-84, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)


The main goal of this paper 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, 2002) saying that $limsup_{nKS(x|n)$ (here $KS(x|n)$ is conditional (plain) Kolmogorov complexity of $x$ when $n$ is known) equals $KS^{mathbf{0'(x)$, the plain Kolmogorov complexity with $mathbf{0'$-oracle. Then we use the same argument to prove similar results for prefix complexity (and also improve results of (Muchnik, 1987) about limit frequencies), a priori probability on binary tree and measure of effectively open sets. As a by-product, we get a criterion of $mathbf{0'$ Martin-L"of randomness (called also $2$-randomness) proved in (Miller, 2004): a sequence $omega$ is $2$-random if and only if there exists $c$ such that any prefix $x$ of $omega$ is a prefix of some string $y$ such that $KS(y)ge |y|-c$. (In the 1960ies this property was suggested in (Kolmogorov, 1968) as one of possible randomness definitions; its equivalence to $2$-randomness was shown in (Miller, 2004) while proving another $2$-randomness criterion (see also (Nies et al. 2005)): $omega$ is $2$-random if and only if $KS(x)ge |x|-c$ for some $c$ and infinitely many prefixes $x$ of $omega$. Finally, we show that the low-basis theorem can be used to get alternative proofs for these results and to improve the result about effectively open sets; this stronger version implies the $2$-randomness criterion mentioned in the previous sentence.
  • Kolmogorov complexity
  • limit complexities
  • limit frequencies
  • 2-randomness
  • low basis


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