Two queries

Abstract

We consider the question whether two queries SAT are as powerful as one query. We show that if PNP[1] = PNP[2] then: Locally either NP = coNP or NP has polynomial-size circuits; PNP = PNP[1]; Σ2pqqΠ2p/1; Σ2p = UPNP[1]qqRPNP[1]; PH = BPPNP[1]. Moreover, we extend the work of Hemaspaandra, Hemaspaandra, and Hempel to show that if PΣ(2)(p)[1] = PΣ(2)(p)[2] then Σ2p = Π2p. We also give a relativized world, where PNP[1] = PNP[2], but NP≠coNP.