P-Selective Sets, and Reducing Search to Decision vs. Self-Reducability

We obtain several results that distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. These include: (i) If NE 6 = E, then there exists a set L in NP ? P such that search reduces to decision for L, search does not nonadaptively reduces to decision for L, and L is not self-reducible. 1 (ii) If UE 6 = E, then there exists a language L 2 UP ? P such that search nonadaptively reduces to decision for L, but L is not self-reducible. (iii) If UE \ co-UE 6 = E, then there is a disjunctive self-reducible language L 2 UP ? P for which search does not nonadaptively reduce to decision. (iv) There is an oracle relative to which satissable assignments cannot be computed nonadaptively relative to SAT. We prove that if NE 6 6 BPE, then there is a language L 2 NP?BPP such that L is randomly self-reducible, not nonadaptively randomly self-reducible, and not self-reducible. We obtain results concerning tradeoos in multiprover interactive proof systems, and obtain results that distinguish checkable languages from those that are nonadaptively checkable. Many of our results depend on new techniques for constructing p-selective sets. We obtain a p-selective set that is not P tt-equivalent to any tally language, and we show that if P = PP, then every p-selective set is P T-equivalent to a tally language. Similarly, if P = NP, then every cheatable set is P m-equivalent to a tally language. We construct a recursive p-selective tally set that is not cheatable.