On pseudorandomness and resource-bounded measure

In this paper we extend a key result of Nisan and Wigderson NW94] to the nondeterministic setting: for all > 0 we show that if there is a language in E = DTIME(2 O(n)) that is hard to approximate by nondeterministic circuits of size 2 n , then there is a pseudorandom generator that can be used to derandomize BP NP (in symbols, BP NP = NP). By applying this extension we are able to answer some open questions in Lut97] regarding the derandomization of the classes BP P k and BP P k under plausible measure theoretic assumptions. As a consequence, if P 2 does not have p-measure 0, then AM \ coAM is low for P 2. Thus, in this case, the graph isomorphism problem is low for P 2. By using the Nisan-Wigderson design of a pseudorandom generator we unconditionally show the inclusion MA ZPP NP and that MA \ coMA is low for ZPP NP .