Disjoint NP-Pairs

We study the question of whether the class DisjNP of disjoint pairs (A, B) of NP-sets contains a complete pair. The question relates to the question of whether optimal proof systems exist, and we relate it to the previously studied question of whether there exists a disjoint pair of NP-sets that is NP-hard. We show under reasonable hypotheses that nonsymmetric disjoint NP-pairs exist, which provides additional evidence for the existence of P-inseparable disjoint NP-pairs. We construct an oracle relative to which the class of disjoint NP-pairs does not have a complete pair; an oracle relative to which optimal proof systems exist, and hence complete pairs exist, but no pair is NP-hard; and an oracle relative to which complete pairs exist, but optimal proof systems do not exist.