On the power of deterministic reductions to C=P

Abstract

The counting class C=P, which captures the notion of “exact counting”, while extremely powerful under various nondeterministic reductions, is quite weak under polynomial-time deterministic reductions. We discuss the analogies between NP and co-C=P, which allow us to derive many interesting results for such deterministic reductions to co-C=P. We exploit these results to obtain some interesting oracle separations. Most importantly, we show that there exists an oracleA such that $$ \oplus P^A \nsubseteq P^{C_ = P^A } $$ and $$BPP^A \nsubseteq P^{C_ = P^A } $$ Therefore, techniques that would prove that C=P and PP are polynomial-time Turing equivalent, or that C=P is polynomial-time Turing hard for the polynomial-time hierarchy, would not relativize.