We prove that
AM (and hence Graph Nonisomorphism) is in
NP if for some ε > 0, some language in
NE ∩
coNE requires nondeterministic circuits of size 2
ε n This improves results of Arvind and Köbler and of Klivans and van Melkebeek who proved the same conclusion, but under stronger hardness assumptions.
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 hitting set approach to derandomization. As a spin-off, we show that this approach is strong enough to give an easy proof of the following implication: for some ε > 0, if there is a language in E which requires nondeterministic circuits of size 2ε n , then P = BPP. This differs from Impagliazzo and Wigderson’s theorem “only” by replacing deterministic circuits with nondeterministic ones.
