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Derandomizing Arthur-Merlin games using hitting sets. (English) Zbl 1085.68058
Summary: We prove that AM (and hence Graph Nonisomorphism) is in NP if for some $$\epsilon > 0$$, some language in $$\text{NE}\cap\text{coNE}$$ requires nondeterministic circuits of size $$2^{\epsilon n}$$ 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 $$\epsilon > 0$$, if there is a language in E which requires nondeterministic circuits of size $$2^{\epsilon n}$$, then $$\text{P}=\text{BPP}$$. This differs from Impagliazzo and Wigderson’s theorem “only” by replacing deterministic circuits with nondeterministic ones.

##### MSC:
 68Q15 Complexity classes (hierarchies, relations among complexity classes, etc.) 68Q17 Computational difficulty of problems (lower bounds, completeness, difficulty of approximation, etc.)
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