Limit complexities revisited [once more]

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 of Vereshchagin [13] saying that limsupnC(x|n) (here C(x|n) is conditional (plain) Kolmogorov complexity of x when n is known) equals C0 ′ (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, to prove Conidis’ theorem [3] about limits of effectively open sets, and also to improve the results of Muchnik [8] about limit frequencies. As a byproduct, we get a criterion of 0′ Martin-Löf randomness (called also 2-randomness) proved in Miller [7]: 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 [5] as one of possible randomness definitions; its equivalence to 2-randomness was shown in Miller [7]). Miller [7] and Nies et al. [9] proved another 2randomness criterion: ω is 2-random if and only if C(x)> |x|− c for some c and infinitely many prefixes x of ω . This criterion is also a consequence of the results mentioned above. [The original version of this work [2] contained a weaker (and cumbersome) version of Conidis’ result, and the proof used low basis theorem (in quite a strange way). The full version was formulated as a conjecture. This conjecture was later proved by Conidis. Bruno Bauwens (personal communication) noted that the proof can be obtained also by a simple modification of our original argument, and we reproduce Bauwens’ argument with his permission.] 1Laboratoire d’Informatique Fondamentale, CNRS & Université Aix-Marseille, France. Supported in part by ANR Sycomore and NAFIT ANR-08-EMER-008-01 grants. 2Andrej 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. 3Moscow State Lomonosov University, Russia. Supported in part by RFBR 05-01-02803-CNRS-a, 06-0100122-a. 4CNRS Poncelet Laboratory, Moscow 5IITP RAS, Moscow