# 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 MODZ_{k}P 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]).

## Keywords

Turing Machine Closure Property Counting Class Satisfiable Formula Nondeterministic Machine## Preview

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