Reducibility Classes of P-Selective Sets

A set is P-selective (Selman, 1979) if there is a polynomial-time semidecision algorithm for the set-an algorithm that given any two strings decides which is " more likely " to be in the set. This paper establishes a strict hierarchy among the various reductions and equivalences to P-selective sets. Given the large number of important problems that do not appear to have easy solutions, researchers have explored more flexible approaches to efficient set recognition (or near-recognition): almost polynomial time [ 141, average polynomial time (see [6]), implicit membership testability [7], near-testability [5], P-closeness [ 16,201, P-selectivity [17], and others. One such notion, that of the P-selective sets, has proven usetil in many contexts, such as characterizing P [4] and understanding whether SAT may have unique solutions [lo]. Intuitively, a set is P-selective if there is a 2-ary polynomial-time function that chooses which of its inputs is " more likely " (or, actually , " not less likely ") to belong to the set.