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

We obtain several results that distinguish self-reducibility of a language L with the question of whether search reduces to decision for L. We prove that if NE intersects co-NE ≠ E, then there exists a set L 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. We obtain results that distinguish adaptively randomly self-reducible sets from nonadaptively randomly self-reducible sets, results concerning tradeoffs in multiprover interactive proof systems, and results that distinguish checkable language from those that are nonadaptively checkable. Many of our results depend on new techniques for constructing p-selective sets.