Finding Irrefutable Certificates for S_2^p via Arthur and Merlin
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).
Symmetric alternation
promise-AM
Karp--Lipton theorem
learning circuits
157-168
Regular Paper
Venkatesan T.
Chakaravarthy
Venkatesan T. Chakaravarthy
Sambuddha
Roy
Sambuddha Roy
10.4230/LIPIcs.STACS.2008.1342
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