Game arguments in computability theory and algorithmic information theory

Abstract

We provide some examples showing how game-theoretic arguments (the approach that goes back to Lachlan and was developed by An. Muchnik) can be used in computability theory and algorithmic information theory. To illustrate this technique, we start with a proof of a classical result, the unique numbering theorem of Friedberg, translated to the game language. Then we provide game-theoretic proofs for three other results: (1) the gap between conditional complexity and total conditional complexity; (2) Epstein-Levin theorem relating a priori and prefix complexity for a stochastic set (for which we provide a new game-theoretic proof) and ( 3) some result about information distances in algorithmic information theory (obtained by two of the authors [A.M. and M.V.] several years ago but not yet published).