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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 complexitytheoretic analogs of Rice’s Theorem, and proved that every nontrivial counting property of boolean circuits is UPhard. Hemasp ..."
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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 complexitytheoretic analogs of Rice’s Theorem, and proved that every nontrivial counting property of boolean circuits is UPhard. Hemaspaandra and Rothe [HR00] improved the UPhardness 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 constantambiguity 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 ≤ p 1tthardness. We also prove a Ricestyle theorem for NP, namely that every nontrivial language property of NP sets is NPhard.
The Complexity of Deciding Statistical Properties of Samplable Distributions
, 2013
"... We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies certain statistical properties such as being i.i.d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordinates, and ma ..."
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We consider the problems of deciding whether the joint distribution sampled by a given circuit satisfies certain statistical properties such as being i.i.d., being exchangeable, being pairwise independent, having two coordinates with identical marginals, having two uncorrelated coordinates, and many other variants. We give a proof that simultaneously shows all these problems are C Pcomplete, by showing that the following promise problem (which is a restriction of all the above problems) is C Pcomplete: 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 depth3 circuits. We also consider circuits that are dlocal, in the sense that each output bit depends on at most d input bits. We give lineartime algorithms for deciding whether a 2local 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 SZKcomplete. We also introduce a boundederror version of C P, which we call BC P, and we investigate its structural properties. 1