Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems

D. M. Stull

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arXiv · 1701.04428

Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems

D. M. Stull

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Abstract

We prove a downward separation for $\mathsf{\Sigma}_2$-time classes. Specifically, we prove that if $\Sigma_2$E does not have polynomial size non-deterministic circuits, then $\Sigma_2$SubEXP does not have \textit{fixed} polynomial size non-deterministic circuits. To achieve this result, we use Santhanam's technique on augmented Arthur-Merlin protocols defined by Aydinlio\u{g}lu and van Melkebeek. We show that augmented Arthur-Merlin protocols with one bit of advice do not have fixed polynomial size non-deterministic circuits. We also prove a weak unconditional derandomization of a certain type of promise Arthur-Merlin protocols. Using Williams' easy hitting set technique, we show that $\Sigma_2$-promise AM problems can be decided in $\Sigma_2$SubEXP with $n^c$ advice, for some fixed constant $c$.

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D. M. Stull. 2017-01-16. Some Results on Circuit Lower Bounds and Derandomization of Arthur-Merlin Problems. https://arxiv.org/abs/1701.04428

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