A Note on P - selective sets and on Adaptive

We study two properties of a complexity class Cwhether 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 C tt PF C O(log n)]). We show that if there exists a NP-hard p-selective set under truth-table reductions, then PF NP tt PF NP O(log n)]. As a consequence, it follows that if there exists a tt-hard p-selective set for NP, then for all k > 0; SAT 2 DTIME2 n= log k n ]. We show that if C ZPP 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 NP tt is eeectively included in PF NP kdlog ne ? 1].