Lower bounds and the hardness of counting properties
Theoretical Computer Science
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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 UP_{O(1)}-hardness. The present paper raises the lower bound for nontrivial counting properties from UP_{O(1)}-hardness to FewP-hardness, i.e., from constant-ambiguity nondeterminism to polynomial-ambiguity nondeterminism. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard, and we prove that every P-constructibly semi-switching counting property of circuits is PP-hard.