Logical operations and Kolmogorov complexity. II

Abstract

For Part I, see Theoretical Computer Science (to be published). Investigates the Kolmogorov complexity of the problem (a/spl rarr/c)/spl and/(b/spl rarr/d), defined as the minimum length of a program that, given a, outputs c and, given b, outputs d. We prove that, unlike all known problems of this kind, its complexity is not expressible in terms of the Kolmogorov complexity of a, b, c and d, their pairs, triples, etc. This solves the problem posed in Part I. We then consider the following theorem: there are two strings, whose mutual information is large but which have no common information in a strong sense. This theorem was proven by A. Muchnik et al. (1999) via a non-constructive argument. We present a constructive proof, thus solving a problem posed by Muchnik et al. We give also an interpretation of both results in terms of Shannon entropy.