P-selective sets and reducing search to decision vs self-reducibility

We distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. Results include: (i) If NE ≠ E, then there exists a set t in NP - P such that search reduces to decision for L, search does not nonadaptively reduce to decision for L and L is not self-reducible, (ii) If UE ≠ E, then there exists a language L ∈ 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 language L ∈ UP - P for which search does not nonadaptively reduce to decision. We prove that if NE ⊈ BPE, then there is a language L ∈ NP - BPP such that L is randomly self-reducible, not nonadaptively randomly self-reducible, and not self-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 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.