Complexity-Theoretic Analogs of Rice's Theorem

Rice's Theorem states that every nontrivial language property of the recursively enumerable sets is undecidable. Borchert and Stephan BS96] initiated a search for complexity-theoretic analogs of Rice's Theorem. In particular, they proved that every nontrivial counting property of circuits is UP-hard. We extend their result by proving that every nontrivial counting property of circuits is UP O(1)-hard; that is, we raise the lower bound from unambiguous nondeterminism to constant-ambiguity nondeterminism. We show that this conclusion cannot be strengthened to SPP-hardness unless unlikely complexity class containments hold. Nonetheless, we prove that every P-constructibly bi-innnite counting property of circuits is SPP-hard.