  A dichotomy theorem for counting problems due to Creignou and Hermann states
that or any nite set S of logical relations, the counting problem #SAT(S) is
either in FP, or #P-complete. In the present paper we show a dichotomy theorem
for polynomial evaluation. That is, we show that for a given set S, either
there exists a VNP-complete family of polynomials associated to S, or the
associated families of polynomials are all in VP. We give a concise
characterization of the sets S that give rise to "easy" and "hard" polynomials.
We also prove that several problems which were known to be #P-complete under
Turing reductions only are in fact #P-complete under many-one reductions.

Computational Complexity