Kolmogorov complexity in perspective

We survey the diverse approaches to the notion of information content: from Shannon entropy to Kolmogorov complexity. The main applications of Kolmogorov complexity are presented: namely, the mathematical notion of randomness (which goes back to the 60’s with the work of MartinLöf, Schnorr, Chaitin, Levin), and classification, which is a recent idea with provocative implementation by Vitanyi and Cilibrasi. . Note. Following Robert Soare’s recommendations in [35], which have now gained large agreement, we shall write computable and computably enumerable in place of the old fashioned recursive and recursively enumerable. Notation. By log(x) we mean the logarithm of x in base 2. By ⌊x⌋ we mean the “floor” of x, i.e. the largest integer ≤ x. Similarly, ⌈x⌉ denotes the “ceil” of x, i.e. the smallest integer ≥ x. Recall that the length of the binary representation of a non negative integer n is 1 + ⌊logn⌋. 1 Three approaches to the quantitative definition of information A title borrowed from Kolmogorov’s seminal paper, 1965 [22]. 1.1 Which information ? 1.1.1 About anything... About anything can be seen as conveying information. As usual in mathematical modelization, we retain only a few features of some real entity or process, and associate to them some finite or infinite mathematical objects. For instance,