Finding Irrefutable Certificates for S_2^p via Arthur and Merlin

Abstract

We show that $S_2^psubseteq P^{prAM}$, where $S_2^p$ is the
   symmetric alternation class and $prAM$ refers to the promise
   version of the Arthur-Merlin class $AM$.  This is derived as a
   consequence of our main result that presents an $FP^{prAM}$
   algorithm for finding a small set of ``collectively irrefutable
   certificates'' of a given $S_2$-type matrix.  The main result also
   yields some new consequences of the hypothesis that $NP$ has
   polynomial size circuits.  It is known that the above hypothesis
   implies a collapse of the polynomial time hierarchy ($PH$) to
   $S_2^psubseteq ZPP^{NP}$ (Cai 2007, K"obler and Watanabe 1998).
   Under the same hypothesis, we show that $PH$ collapses to
   $P^{prMA}$.  We also describe an $FP^{prMA}$ algorithm for learning
   polynomial size circuits for $SAT$, assuming such circuits exist.
   For the same problem, the previously best known result was a
   $ZPP^{NP}$ algorithm (Bshouty et al.  1996).