On the formalisation of Kolmogorov complexity

Abstract

Kolmogorov complexity is an essential tool in the study of algorithmic information theory, and is used in the fields of Artificial Intelligence, cryptography, and coding theory. The formalisation of the theorems of Kolmogorov complexity is also key to proving results in the theory of Intelligent Agents, specifically the results in Universal Artificial Intelligence. In this paper, we present a mechanisation of some of these fundamental results. In particular, building on HOL4's existing model of computability, we provide a formal definition of the complexity of a binary string, and then prove (i) that Kolmogorov complexity is uncomputable; (ii) the Kolmogorov Complexity invariance theorem; (iii) the Kraft and Kolmogorov-Kraft inequalities; and (iv) key Kolmogorov Complexity inequalities.