Lower bounds and the hardness of counting properties

Abstract

Rice's Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan [BSOO] started the search for complexity-theoretic analogs of Rice's Theorem, and proved that every nontrivial counting property of boolean circuits is UP-hard. Hemaspaandra and Rot he [HR00] improved the UP-hardness lower bound to UPO(i)-hardness. The present paper raises the lower bound for nontrivial counting properties from UPO(i)-hardness to FewP-hardness, i.e., from constant-ambiguity nondeterminism to polynomial-ambiguity nondeterminism. Furthermore, we prove that this lower bound is rather tight with respect to relativizable techniques, i.e., no rel-ativizable technique can raise this lower bound to FewP-≤1-ttp-hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard.