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, ZPPSAT[1] = ZPPSAT‖[2] = ⇒ ZPPSAT[1] = PH. These ZPP machines are required to succeed with prob-ability 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 [16] who showed a collapse of PH to SP2. The use of the probability bound of 1/2 + 1/p(n) is justified in part by showing that this bound can be ampli-fied to 1−2−nk for ZPPSAT[1] computations. This paper also shows that in the deterministic case, PSAT[1] = PSAT‖[2] = ⇒ PH ⊆ ZPPSAT[1] where the ZPPSAT[1] machine achieves a probability of success of 1/2−2−nk.