Computing Solutions Uniquely collapses the Polynomial Hierarchy

Is there an NP function that, when given a satissable formula as input, outputs one satisfying assignment uniquely? That is, can a nondeterministic function cull just one satisfying assignment from a possibly exponentially large collection of assignments? We show that if there is such a nondeterministic function, then the polynomial hierarchy collapses to ZPP NP (and thus, in particular, to NP NP). As the existence of such a function is known to be equivalent to the statement \every NP function has an NP reenement with unique outputs," our result provides the strongest evidence yet that NP functions cannot be reened. We prove our result via a result of independent interest. We say that a set A is NPSV-selective (NPMV-selective) if there is a 2-ary partial NP function with unique values (a 2-ary partial NP function) that decides which of its inputs (if any) is \more likely" to belong to A; this is a nondeterministic analog of the recursion-theoretic notion of the semi-recursive sets and the extant complexity-theoretic notion of P-selectivity. Our hierarchy collapse result follows by combining the easy observation that every set in NP is NPMV-selective with the following result: If A 2 NP is NPSV-selective, then A 2 (NP \ coNP)=poly. Relatedly, we prove that if A 2 NP is NPSV-selective, then A is Low2. We prove that the polynomial hierarchy collapses even further, namely to NP, if all coNP sets are NPMV-selective. This follows from a more general result we prove: Every self-reducible NPMV-selective set is in NP.