In this paper, we revisit a central concept in Kolmogorov complexity in which one would equate program-size complexity with information content. Despite the fact that Kolmogorov complexity has been widely accepted as an objective measure of the information content of a string, it has been the subject of many criticisms including the fundamental one directed by logicians and philosophers towards...

Most assertions involving Shannon entropy have their Kolmogorov complexity counterparts. A general theorem of Romashchenko [4] states that every information inequality that is valid in Shannon’s theory is also valid in Kolmogorov’s theory, and vice verse. In this paper we prove that this is no longer true for ∀∃-assertions, exhibiting the first example where the formal analogy between Shannon e...

This paper aims to provide a minimal introduction to algorithmic randomness. In particular, we cover the equivalent 1-randomness and MartinLöf randomness. After a brief review of relevant concepts in computability, we develop the basic theory of Kolmogorov complexity, including the KC theorem and more general notion of information content measures. We then build two natural definitions of rando...

1. Introduction. What makes one object more complex than another? Kolmogorov complexity, or program-size complexity, provides one of many possible answers to this fundamental question. In this theory, whose foundations have been developed independently by R. the complexity of an object is defined as the length of its shortest effective description, which is the minimum number of symbols that mu...

We argue that Martin-LL of tests provide useful techniques in proofs involving Kolmogorov complexity. Our examples cover more areas: (1) logical deenability; we prove an analogue of the 0-1 Law for various logics in terms of structures with high Kolmogorov complexity, (2) probabilistic computation: we display a relation between the Kolmogorov complexity of the random strings on which a probabil...

2 What we have learned? A personal pick 4 2.1 From randomness to complexity . . . . . . . . . . . . . . . . . . . . . 4 2.2 Formalization of randomness: infinite strings . . . . . . . . . . . . . 5 2.3 Random versus lawless . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.4 Randomness and finite strings: incompressibility . . . . . . . . . . . 7 2.5 Representation and Kolmogorov complexi...