Limit Complexities Revisited

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.