The utility of a Kolmogorov complexity method in combinatorial theory is demonstrated by several examples.

The Kolmogorov complexity function K can be relativized using any oracle A, and most properties of K remain true for relativized versions KA. In section 1 we provide an explanation for this observation by giving a game-theoretic interpretation and showing that all “natural” properties are either true for all KA or false for all KA if we restrict ourselves to sufficiently powerful oracles A. Thi...

We compare the elementary theories of Shannon information and Kolmogorov complexity, the extent to which they have a common purpose, and where they are fundamentally different. We discuss and relate the basic notions of both theories: Shannon entropy versus Kolmogorov complexity, the relation of both to universal coding, Shannon mutual information versus Kolmogorov (‘algorithmic’) mutual inform...

The goal of statistics is to provide explanations (models) of observed data. We are given some data and have to infer a plausible probabilistic hypothesis explaining it. Consider, for example, the following scenario. We are given a “black box”. We have turned the box on (only once) and it has produced a sequence x of million bits. Given x, we have to infer a hypothesis about the black box. Clas...

In this paper, we survey a few recent applications of Kolmogorov complexity to lower bounds in several models of computation. We consider KI complexity of Boolean functions, which gives the complexity of finding a bit where inputs differ, for pairs of inputs that map to different function values. This measure and variants thereof were shown to imply lower bounds for quantum and randomized decis...

Born about three decades ago, Kolmogorov Complexity Theory (KC) led to important discoveries that, in particular, give a new understanding of the fundamental problem: interrelations between classical continuum mathematics and reality (physics, biology, engineering sciences,. . .). Speciically, in addition to the equations, physicists use the following additional diicult-to-formalize property: t...

Kolmogorov complexity and Shannon entropy are conceptually different measures. However, for any recursive probability distribution, the expected value of Kolmogorov complexity equals its Shannon entropy, up to a constant. We study if a similar relationship holds for Rényi and Tsallis entropies of order α, showing that it only holds for α = 1. Regarding a time-bounded analogue relationship, we s...

The idea to find the “maximal number that can be named” can be traced back to Archimedes (see his Psammit [1]). From the viewpoint of computation theory the natural question is “which number can be described by at most n bits”? This question led to the definition of the so-called “busy beaver” numbers (introduced by T. Rado). In this note we consider different versions of the busy beaver-like n...

Solovay [19] proved that there exists a computable upper bound f of the prefix-free Kolmogorov complexity function K such that f(x) = K(x) for infinitely many x. In this paper, we consider the class of computable functions f such that K(x) ≤ f(x)+O(1) for all x and f(x) ≤ K(x) + O(1) for infinitely many x, which we call Solovay functions. We show that Solovay functions present interesting conne...

We survey diverse approaches to the notion of information: from Shannon entropy to Kolmogorov complexity. Two of the main applications of Kolmogorov complexity are presented: randomness and classification. The survey is divided in two parts published in a same volume. Part II is dedicated to the relation between logic and information system, within the scope of Kolmogorov algorithmic informatio...