Adaptive versus nonadaptive queries to NP and P-selective sets

We prove that P = NP follows if for some k > 0, the class PFttNP of functions that are computable in polynomial time by nonadaptively accessing an oracle in NP is effectively included in PFNP[k[log n]-1], the class of functions that are computable in polynomial time by making at most k[log n] - 1 queries to an oracle in NP. We draw several observations and relationships between the following two properties of a complexity class C: whether there exists a truth-table hard p-selective language for C, and whether polynomially-many nonadaptive queries to C can be answered by making O(log n)-many adaptive queries to C tin symbols, whether PFttC subset of or equal to PFC[O(log n)]). Among these, we show that if there exists an NP-hard p-selective set under truth-table reductions, then PFttNP subset of or equal to PFNP[O(log n)]. However, if C superset of or equal to ZPP(NP), then these two properties are equivalent.