## Limit complexities revisited

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Bienvenu, Laurent

Muchnik, Andrej

Shen, Alexander

Vereshchagin, Nikolay

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##### Description

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 $\limsup_n\KS(x|n)$ (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.

##### Keywords

Computer Science - Computational Complexity