Low level complexity for combinatorial games

Abstract

There have been numerous attempts to discuss the time complexity of problems and classify them into hierarchical classes such as P, NP, PSPACE, EXP, etc. A great number of familiar problems have been reported which are complete in NP (nondeterministic polynomial time). Even and Tarjan considered generalized Hex and showed that the problem to determine who wins the game if each player plays perfectly is complete in polynomial space. Shaefer derived some two-person game from NP complete problems which are complete in polynomial space. A rough discussion such as to determine whether or not a given problem belongs to NP is independent of the machine model and the way of defining the size of problems, since any of the commonly used machine models can be simulated by any other with a polynomial loss in running time and by no matter what criteria the size is defined, they differ from each other by polynomial order. However, in precise discussion, for example, in the discussion whether the computation of a problem requires O(n k) time or O(nk+l) time, the complexity heavily depends on machine models and the definition of size of problems. From these points, we introduce somewhat stronger notion of the reducibility.