Lower Bounds and the Hardness of Counting Properties

Rice’s Theorem states that all nontrivial language properties of recursively enumerable sets are undecidable. Borchert and Stephan [BS00] 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 Rothe [HR00] improved the UP-hardness lower bound to UPO(1)-hardness. The present paper raises the lower bound for nontrivial counting properties from UPO(1)-hardness to FewPhardness, i.e., from constant-ambiguity nondeterminism to polynomialambiguity nondeterminism. Furthermore, we prove that this lower bound is rather tight with respect to relativizable techniques, i.e., no relativizable technique can raise this lower bound to FewP-≤p1-tt-hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard.