Two decades of applied Kolmogorov complexity: in memoriam Andrei Nikolaevich Kolmogorov 1903-87

The authors provide an introduction to the main ideas of Kolmogorov complexity and survey the wealth of useful applications of this notion. It is based on a theory of information content of strings, intuitively, that the amount of information in a finite string is the size (i.e. number of bits) of the smallest program that, started with a blank memory, computes the string and then terminates. The following are treated: (1) application of the fact that some strings are compressible; this includes a strong version of Godel's incompleteness theorem; (2) lower-bound arguments that rest on application of the fact that certain strings cannot be compressed at all; applications range from Turing machines to electronic chips; (3) probability theory and a priori probability; applications range from foundational issues to the theory of learning and inductive inference in artificial intelligence. (4) Resource-bounded Kolmogorov complexity; applications range from NP-completeness and the P versus NP question to cryptography