Game interpretation of Kolmogorov complexity

Details Game interpretation of Kolmogorov complexity Game interpretation of Kolmogorov complexity

2010-03-24

arxiv.org/abs/1003.4712v1

Mathematics

Logic

11 pages. Presented in 2009 at the conference on randomness in Madison.

Scientific paper

The Kolmogorov complexity function K can be relativized using any oracle A, and most properties of K remain true for relativized versions. In section 1 we provide an explanation for this observation by giving a game-theoretic interpretation and showing that all "natural" properties are either true for all sufficiently powerful oracles or false for all sufficiently powerful oracles. This result is a simple consequence of Martin's determinacy theorem, but its proof is instructive: it shows how one can prove statements about Kolmogorov complexity by constructing a special game and a winning strategy in this game. This technique is illustrated by several examples (total conditional complexity, bijection complexity, randomness extraction, contrasting plain and prefix complexities).

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