COMPUTING SOLUTIONS UNIQUELY COLLAPSES THE POLYNOMIAL HIERARCHY

Is there an hrp function that, when given a satisfiable formula as inp ut, outputs one satisfying assignment uniquely? That is, can a nondete rministic 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 col lapses to ZPP(NP) (and thus, in particular, to NPNP). Because the exis tence of such a function is known to be equivalent to the statement '' every NP function has an NP refinement with unique outputs,'' our resu lt provides the strongest evidence yet that NP functions cannot be ref ined. 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 p artial NP function with unique values (a 2-ary partial NP function) th at decides which of its inputs (if any) is ''more likely'' to belong t o A; this is a nondeterministic analog of the recursion-theoretic noti on of the semirecursive sets and the extant complexity-theoretic notio n 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 epsilon NP is NPSV-selective, then A epsilon (N P boolean AND coNP)/poly. Relatedly, we prove that if A epsilon NP is NPSV-selective, then A is Low(2). We prove that the polynomial hierarc hy collapses even further, namely to NP, if all coNP sets are NPMV-sel ective. This follows from a more general result we prove: Every self-r educible NPMV-selective set is in NP.