Finding Irrefutable Certificates for S2p via Arthur and Merlin

We show that Sp2 ⊆ P , where Sp2 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 algorithm for finding a small set of “collectively irrefutable certificates” of a given S2-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 Sp2 ⊆ ZPP NP [5, 14]. Under the same hypothesis, we show that PH collapses to P. We also describe an FP algorithm for learning polynomial size circuits for SAT, assuming such circuits exist. For the same problem, the previously best known result was a ZPP algorithm [4].