The Kolmogorov complexity of infinite words

We present a brief survey of results on relations between the Kolmogorov complexity of infinite strings and several measures of information content (dimensions) known from dimension theory, information theory or fractal geometry. Special emphasis is laid on bounds on the complexity of strings in constructively given subsets of the Cantor space. Finally, we compare the Kolmogorov complexity to the subword complexity of infinite strings. The aim of this paper is to briefly survey several results on the Kolmogorov complexity of infinite strings. We focus on those results which can be derived by elementary methods from the Kolmogorov complexity of finite strings (words) and counting arguments for sets of finite strings (languages) as e.g. structure functions and the concept of entropy of languages. The concept of Kolmogorov or program size complexity was introduced in the papers by Solomonoff [27], Kolmogorov [14] and Chaitin [6] in the sixties (for the complete history see the textbooks by Calude [4] or Li and Vitányi [15]). It measures the information content of a (finite) string as the size of the smallest program that computes the string, that is, the complexity of a string is the amount of information necessary to print the string. The original intention of Kolmogorov was to give an alternative approach to information theory not depending on probability theory. A first fact proving evidence of this intention was P. Martin-Löf’s [19] characterisation of infinite random strings. Roughly speaking, if an infinite string is random then most of its initial words have maximum Kolmogorov complexity, that is, have a ? This paper was presented at the 7th Workshop ”Descriptional Complexity of Formal Systems‘‘, June 30 July 2, 2005, Como, Italy Email address: email: [email protected] (Ludwig Staiger). Preprint submitted to Elsevier Science 26 April 2006 Electronic Colloquium on Computational Complexity, Report No. 70 (2006)