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).