Counting classes: thresholds, parity, mods, and fewness

Abstract

Counting classes consist of languages defined in terms of the number of accepting computations of nondeterministic polynomial-time Turing machines. Well-known examples of counting classes are NP, co-NP, ⊕ P, and PP. Every counting class is a subset of P #P[1], the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine. Using closure properties of #P, we systematically develop a complexity theory for counting classes defined in terms of thresholds and moduli. An unexpected result is that MOD k i P = MOD k P for prime k. Finally, we improve a result of Cai and Hemachandra (1990) by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in Valiant and Vazirani (1986)).