A note on p-selective sets and on adaptive versus nonadaptive queries to NP

We study 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 (in symbols, is PF_tt^C⊆PF^C[O(log n)]). We show that if there exists an NP-hard p-selective set under truth-table reductions, then PF_tt^NP⊆PF^NP[O(log n)]. We observe that if C⊇P^NP, then these two properties are equivalent. Also, we show that if there exists a truth-table complete standard-left cut in NP, then the polynomial hierarchy collapses to P ^NP. We prove that P=NP follows if for some k>0, the class PF_tt^NP is “effectively included” in PF ^NP[k[log n]-1]