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 rel-ativizable technique can raise this lower bound to FewP- ≤ p 1-tt-hardness. We also prove a Rice-style theorem for NP, namely that every nontrivial language property of NP sets is NP-hard.

We consider the problems of deciding whether the joint distribution sampled by a given cir-cuit satisfies certain statistical properties such as being i.i.d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordi-nates, and many other variants. We give a proof that simultaneously shows all these problems are C P-complete, by showing that the following promise problem (which is a restriction of all the above problems) is C P-complete: Given a circuit, distinguish the case where the output distribution is uniform and the case where every pair of coordinates is neither uncorrelated nor identically distributed. This completeness result holds even for samplers that are depth-3 circuits. We also consider circuits that are d-local, in the sense that each output bit depends on at most d input bits. We give linear-time algorithms for deciding whether a 2-local sampler’s joint distribution is fully independent, and whether it is exchangeable. We also show that for general circuits, certain approximation versions of the problems of deciding full independence and exchangeability are SZK-complete. We also introduce a bounded-error version of C P, which we call BC P, and we investigate its structural properties. 1