P-Selective Sets and Reducing Search to Decision vs Self-Reducibility

AbstractWe distinguish self-reducibility of a languageLwith the question of whether search reduces to decision forL. Results include: (i) If NE≠E, then there exists a setLin NP−P such that search reduces to decision forL, search doesnotnonadaptively reduce to decision forLandLis not self-reducible. (ii) If UE≠E, then there exists a languageL∈UP−P such that search nonadaptively reduces to decision for L, but L is not self-reducible. (iii) If UE∩co-UE≠E, then there is a disjunctive self-reducible languageL∈UP−P for which search doesnotnonadaptively reduce to decision. We prove that if NE⊈BPE, then there is a languageL∈NP−BPP such thatLis randomly self-reducible,notnonadaptively randomly self-reducible, andnotself-reducible. We obtain results concerning trade-offs in multiprover interactive proof systems and results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results are proven by constructing p-selective sets. We obtain a p-selective set that isnot⩽Ptt-equivalent to any tally language, and we show that if P=PP, then every p-selective set is ⩽PT-equivalent to a tally language. Similarly, if P=NP, then every cheatable set is ⩽Pm-equivalent to a tally language. We construct a recursive p-selective tally set that isnotcheatable