Kolmogorov Complexity: Recent Research in Moscow

The Kolmogorov complexity theory emerged in sixties; main definitions were independently found by Ray Solomonoff, A.N. Kolmogorov and G. Chaitin. The motivations for the definition were quite different. For Solomonoff the main goal was inductive inference theory. Kolmogorov's work was closely connected with foundations of probability theory and information theory. Chaitin's studied the length of programs for computing binary sequences. For the historical account as well as for the exact definitions of the main notions of Kolmogorov complexity theory we refer the reader to the book of Ming Li and Paul Vitanyi [6]. Let us say only that most basic questions connected with Kolmogorov complexity were solved in seventies; the main achievement, probably, was the definition of randomness for individual random sequence and the complexity characterization of random sequences (A.N. Kolmogorov, P. MartinLSf, L.A. Levin, C.P. Schnorr). At the same time it became clear that the philosophical notion of randomness should be studied in the framework of computational complexity theory. The notions of a pseudorandom number generator plays here a central role; since eighties a lot of philosophically interesting and technically deep results were obtMned. Nevertheless, there are some natural question about Kolmogorov complexity which are still unsolved. We believe that they deserve attention (for mathematical, if not philosophical reasons); the goal of this talk is to state some of the questions and the results obtained in Moscow during the last years.