Amplifying ZPP^SAT[1] and the Two Queries Problem

This paper shows a complete upward collapse in the Polynomial Hierarchy (PH) if for ZPP, two queries to a SAT oracle is equivalent to one query. That is, ZPP^SAT[1] = ZPP^SAT||[2] rArr ZPP^SAT[1] = PH. These ZPP machines are required to succeed with probability at least 1/2 + 1/p(n) on inputs of length n for some polynomial p(n). This result builds upon recent work by Tripathi who showed a collapse of PH to S₂ ^P. The use of the probability bound of 1/2 + 1/p(n) is justified in part by showing that this bound can be amplified to 1 - 2^-nk for ZPP^SAT[1] computations. This paper also shows that in the deterministic case, P^SAT[1] = P^SAT||[2] rArr PH sube ZPP^SAT[1] where the ZPP^SAT[1] machine achieves a probability of success of 1/2 - 2^-nk.