Derandomizing Arthur-Merlin Games Using Hitting Sets

Peter Bro Miltersen, N. V. Vinodchandran

Abstract: We prove that AM (and hence Graph Nonisomorphism) is in NP if for some /spl epsiv/>0, some language in NE/spl cap/ coNE requires nondeterministic circuits of size 2/sup en/. This improves results of Arvind and Kobler (1997) and of Klivans and Van Melkebeek (1999) who have proven the same conclusion, but under stronger hardness assumptions, namely, either the existence of a language in NE/spl cap/ coNE which cannot be approximated by nondeterministic circuits of size less than 2/sup en/ or the existence of a language in NE/spl cap/ coNE which requires oracle circuits of size 2/sup en/ with oracle gates for SAT (satisfiability). The previous results on derandomizing AM were based on pseudorandom generators. In contrast, our approach is based on a strengthening of Andreev, Clementi and Rolim's (1996) hitting set approach to derandomization. As a spin-off we show that this approach is strong enough to give an easy (if the existence of explicit dispersers can be assumed known) proof of the following implication: for some /spl epsiv/>0, if there is a language in E which requires nondeterministic circuits of size 2/sup en/, then P=BPP. This differs from Impagliazzo and Wigderson's (1995) theorem "only" by replacing deterministic circuits with nondeterministic ones.