Classes of bounded counting type and their inclusion relations

Abstract

Classes of bounded counting type are a generalization of complexity classes with finite acceptance types. The latter ones are defined via nondeterministic machines whose number of accepting paths up to a certain maximum is responsible for the question of acceptance of the input. For the classes of bounded counting type each computation path may have one of k possible results from the set {0,⋯, k-1} (k≥2), and we count the number of paths having result 1, as well as the number of paths having result 2, etc. Each result (except 0) is counted up to a certain maximum, and the vector formed by these numbers is responsible for the acceptance question.

In this paper we design and prove correctness of an algorithm deciding the question “Is there an oracle separating C ₁ from C ₂?” for arbitrary classes C ₁ and C ₂ of bounded counting type. For the special case of classes of finite acceptance types we can give a direct solution to the separability question, thus solving an open problem from [H94a].

Moreover, we note that a surprising consequence on relativizable closure properties of #P can be obtained from these investigations [H94c].

This research was supported by Deutsche Forschungsgemeinschaft, Grant number Wa 847/1-1, “k-wertige Schaltkreise”.