The Valiant-Vazirani theorem says that if there is a polynomial time algorithm (deterministic or randomized) for distinguishing between a SAT formula that has exactly one satisfying assignment, and an unsatisfiable formula - then NP=RP. This theorem is proved by showing that UNIQUE-SAT is NP-hard under randomized reductions.
Subject to plausible derandomization conjectures, the Theorem can be strengthened to "an efficient solution to UNIQUE-SAT implies NP = P".
My first instinct was to think that implied there exists a deterministic reduction from 3SAT to UNIQUE-SAT, but it's not clear to me how this particular reduction can be derandomized.
My question is: what is believed or known about "derandomizing reductions"? Is it/should it be possible? What about in the case of V-V?
Since UNIQUE-SAT is complete for PromiseNP under randomized reductions, can we use a derandomization tool to show that "a deterministic polynomial time solution to UNIQUE-SAT implies that PromiseNP = PromiseP?