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