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.