Counting classes: Thresholds, parity, mods, and fewness

Abstract

Counting classes are classes of languages defined in terms of the number of accepting computations of non-deterministic polynomial-time Turing machines. Well known examples of counting classes are NP, co-NP, ⊕P, and PP. Every counting class consists of languages in 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.We perform an in-depth investigation of counting classes defined in terms of thresholds and moduli. We show that the computational power of a threshold machine is a monotone function of the threshold. Then we show that the class MODZkP is at least as large as FewP. Finally, we improve a result of Cai and Hemachandra 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 [21]).KeywordsTuring MachineClosure PropertyCounting ClassSatisfiable FormulaNondeterministic MachineThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.