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 PFNP[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.