**Authors:** Richard Beigel and John Gill

**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, PARITYP, 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_{ki}P = MOD_{k}P
for prime *k*.

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 Valiant & Vazirani '86).