Arthur and Merlin as Oracles

Abstract

We study some problems solvable in deterministic polynomial time given oracle access to the promise version of the Arthur-Merlin class AM. The main result is that ${{\rm BPP}^{\rm NP}_{||}} \subseteq {{\rm P}^{{{\rm pr}{\rm AM}}}_{||}}$. An important property of the class ${{\rm P}^{{{\rm pr}{\rm AM}}}_{||}}$ is that it can be derandomized as ${{\rm P}^{{{\rm pr}{\rm AM}}}_{||}}={{\rm P}^{\rm NP}_{||}}$, under a natural hardness hypothesis used for derandomizing the class AM; this directly follows from a result due to Miltersen and Vinodchandran [10]. As a consequence, we get that ${{\rm BPP}^{{\rm NP}}_{||}} = {{\rm P}^{\rm NP}_{||}}$, under the above hypothesis. This gives an alternative (and perhaps, a simpler) proof of the same result obtained by Shaltiel and Umans [16], using different techniques. Next, we present an FPprAMalgorithm for finding near-optimal strategies of a succinctly presented zero-sum game. For the same problem, Fortnow et al. [7] described a ZPPNPalgorithm. As a by product of our algorithm, we also get an alternative proof of the result by Fortnow et. al. One advantage with an FPprAMalgorithm is that it can be directly derandomized using the Miltersen-Vinodchandran construction [10]. As a consequence, we get an FPNPalgorithm for the above problem, under the hardness hypothesis used for derandomizing AM.