Around Kolmogorov complexity: basic notions and results

TL;DR

This paper provides a concise survey of fundamental concepts and results in algorithmic information theory, including Kolmogorov complexity and randomness, with proofs and applications in computational complexity and incompleteness.

Contribution

It offers a clear overview of basic notions and main results in algorithmic information theory, filling a gap in accessible literature.

Findings

Proves symmetry of information and links between a priori probability and prefix complexity.

Establishes criteria for randomness based on complexity measures.

Demonstrates applications in computational complexity and incompleteness theorems.

Abstract

Algorithmic information theory studies description complexity and randomness and is now a well known field of theoretical computer science and mathematical logic. There are several textbooks and monographs devoted to this theory where one can find the detailed exposition of many difficult results as well as historical references. However, it seems that a short survey of its basic notions and main results relating these notions to each other, is missing. This report attempts to fill this gap and covers the basic notions of algorithmic information theory: Kolmogorov complexity (plain, conditional, prefix), Solomonoff universal a priori probability, notions of randomness (Martin-L\"of randomness, Mises--Church randomness), effective Hausdorff dimension. We prove their basic properties (symmetry of information, connection between a priori probability and prefix complexity, criterion of randomness in terms of complexity, complexity characterization for effective dimension) and show some applications (incompressibility method in computational complexity theory, incompleteness theorems). It is based on the lecture notes of a course at Uppsala University given by the author.