Membership comparable sets

A set A is k(n) membership comparable if there is a polynomial time computable function that, given k(n) instances of A of length at most n, excludes one of the the 2(k(n)) possibilities for the memberships of the given strings in A. We shaw that if SAT is O(log n) membership comparable, then UniqueSAT is an element of P. This extends the work of Ogihara; Beigel, Kummer and Stephan; and Agrawal and Arvind [Ogi94, BKS94, AA94] and answers in the affirmative an open question suggested by Buhrman, Fortnow, and Torenvliet [BFT97]. Our proof also shows that if SAT is o(ll) membership comparable, then UniqueSAT can be solved in deterministic time 2(o(n)). Our main technical tool is an algorithm of Ar et al. [ALRS92] to reconstruct polynomials from noisy data through the use of bivariate polynomial factorization.