Limit complexities revisited

TL;DR

This paper simplifies and unifies known results in Kolmogorov complexity and randomness, providing clearer proofs and new criteria for 2-randomness using the low-basis theorem.

Contribution

It offers simplified proofs of key results in Kolmogorov complexity and randomness, introduces a new criterion for 2-randomness, and applies the low-basis theorem for improved results.

Findings

Unified perspective on limit complexities and their proofs.

New criterion for 2-randomness based on prefix complexity.

Enhanced results on effectively open sets and randomness criteria.

Abstract

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.