Pseudorandomness for Approximate Counting and Sampling

Abstract

Abstract. We study computational procedures that use both randomness and nondeterminism. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to "boost" a given hardness assumption: We show that if there is a problem in EXP that cannot be computed by poly-size nondeterministic circuits then there is one which cannot be computed by poly-size circuits that make non-adaptive NP oracle queries. This in particular shows that the various assumptions used over the last few years by several authors to derandomize ArthurMerlin games (i.e., show AM = NP) are in fact all equivalent. We also define two new primitives that we regard as the natural pseudorandom objects associated with approximate counting and sampling of NP-witnesses. We use the "boosting" theorem and hashing techniques to construct these primitives using an assumption that is no stronger than that used to derandomize AM. We observe that Cai's proof that S P 2 ⊆ ZPP NP and the learning algorithm of Bshouty et al. can be seen as reductions to sampling that are not probabilistic. As a consequence they can be derandomized under an assumption which is weaker than the assumption that was previously known to suffice.

Chapter 8 Notes and References

“…Klivans and van Melkebeek [244] observed that the Impagliazzo-Wigderson pseudorandom generator construction is "black box" and used this to show that AM can be derandomized using functions that are worst-case hard for circuits with an NP oracle (Theorem 7.68). Subsequent work showed that one only needs worst-case hardness against a nonuniform analogue of NP ∩ co-NP [289,356,357].…”

Typically-correct derandomization

“…PRGs for non-deterministic circuits can be constructed if we require that the function h cannot be computed by non-deterministic circuits [KvM02,SU06]. Such PRGs are useful for derandomizing the class AM of "Arthur-Merlin protocols" [KvM02,MV05,SU05,SU06] and are also useful in other contexts (as we see later on).…”

SIGACT news complexity theory column 48

“…Another resource-bounded measure result regarding P/poly is a conditional one: Köbler and Lindner [40] showed that if µ p (NP) = 0, then P/poly has measure 0 in EXP NP . The full version of [35] uses derandomized approximate counting [70] to improve this to dim(P/poly | E NP ) = 0 under the same hypothesis. Is there a small span theorem for dimension?…”