Adaptive versus nonadaptive queries to NP and p-selective sets

Abstract.

We prove that P = NP follows if for some $k > 0$, the class ${\rm PF}^{\rm NP}_{tt}$ of functions that are computable in polynomial time by nonadaptively accessing an oracle in NP is effectively included in PF^NP[k⌈log n⌉— 1], the class of functions that are computable in polynomial 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 $\cal C$: whether there exists a truthtable hard p-selective language for $\cal C$, and whether polynomially-many nonadaptive queries to $\cal C$ can be answered by making O(log n)-many adaptive queries to $\cal C$ (in symbols, whether ${\rm PF}^{\cal C}_{tt} \subseteq {\rm PF}^{\cal C}[O({\rm log}\,n)]$). Among these, we show that if there exists an NP-hard p-selective set under truth-table reductions, then ${\rm PF}^{\rm NP}_{tt} \subseteq {\rm PF}^{\rm NP}[O({\log}\,n)]$. However, if ${\cal C} \supseteq {\rm ZPP}^{\rm NP}$, then these two properties are equivalent.