On the complexity of some two-person perfect-information games

Abstract

We present a number of two-person games, based on simple combinatorial ideas, for which the problem of deciding whether the first player can win is complete in polynomial space. This provides strong evidence, although not absolute proof, that efficient general algorithms for deciding the winner of these games do not exist. The existence of a polynomial-time algorithm for deciding any one of these games would imply the unexpected result that polynomial-time algorithms exist for (a) all the rest of these games, (b) all NP-complete problems and (c) in general, any problem decidable by a polynomial tape bounded Turing machine.