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 (Math. Logic Quart. 46 (4) (2000) 489-504) 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 (Theoret. Comput. Sci. 244 (1-2) (2000) 205-217) 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 FewP-hardness, i.e., from constant-ambiguity nondeterminism to polynomial-ambiguity nondeterminism. Furthermore, we prove that no relativizable technique can raise this lower bound toFewP-less than or equal to(1-tt)(P)-hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard, and for a broad class of leaf-language classes we prove a sufficient condition for the natural analog of Rice's Theorem to hold. (C) 2004 Elsevier B.V. All rights reserved.