Arthur and Merlin as Oracles

Abstract

We study some problems solvable in deterministic polynomial time given oracle access to the (promise version of) Arthur–Merlin class. Our main results are the following:

$$\circ\quad{\rm BPP}^{{\rm NP}}_{||} \subseteq {{\rm P}^{{{\rm pr}{\rm AM}}}_{||}}.$$

$$\circ\quad{{\rm S}_2^p} \subseteq {{\rm P}^{{{\rm pr}{\rm AM}}}}.$$

In addition to providing new upper bounds for the classes ${{{\rm S}_2^p}}$ and ${{\rm BPP}^{{\rm NP}}_{||}}$ , these results are interesting from a derandomization perspective. In conjunction with the hitting set generator construction of Miltersen and Vinodchandran (Computational Complexity 14(3), 2005), we get that ${{{\rm S}_2^p} = {{\rm P}^{\rm NP}}}$ and ${{\rm BPP}^{{\rm NP}}_{||} = {\rm P}^{{\rm NP}}_{||}}$ , under the hardness hypothesis associated with derandomizing the class AM. This gives an alternative proof of the same result obtained by Shaltiel and Umans (Computational Complexity 15(4), 2007).

We also show that if NP has polynomial size circuits, then the polynomial time hierarchy (PH) collapses as PH =  P^prMA. Under the same hypothesis, we also derive a FP^prMA algorithm for learning circuits for SAT; this improves the ZPP^NP algorithm for the same problem by Bshouty et al. (JCSS 52(3), 1996).

Finally, we design a FP^prAM algorithm for the problem of finding near-optimal strategies for succinctly presented zero-sum games. For the same problem, Fortnow et al. (Computational Complexity 17(3), 2008a) described a ZPP^NP algorithm. One advantage of our FP^prAM algorithm is that it can be derandomized using the construction of Miltersen and Vinodchandran yielding a FP^NP algorithm, under a hardness hypothesis used for derandomizing AM.