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

We 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.