Nondeterministic functions and the existence of optimal proof systems

We provide new characterizations of two previously studied questions on nondeterministic function classes: Q1: Do nondeterministic functions admit efficient deterministic refinements?Q2: Do nondeterministic function classes contain complete functions? We show that Q1 for the class is equivalent to the question whether the standard proof system for is p-optimal, and to the assumption that every optimal proof system is p-optimal. Assuming only the existence of a p-optimal proof system for , we show that every set with an optimal proof system has a p-optimal proof system. Under the latter assumption, we also obtain a positive answer to Q2 for the class .An alternative view on nondeterministic functions is provided by disjoint sets and tuples. We pursue this approach for disjoint -pairs and its generalizations to tuples of sets from and with disjointness conditions of varying strength. In this way, we obtain new characterizations of Q2 for the class . Question Q1 for is equivalent to the question of whether every disjoint -pair is easy to separate. In addition, we characterize this problem by the question of whether every propositional proof system has the effective interpolation property. Again, these interpolation properties are intimately connected to disjoint -pairs, and we show how different interpolation properties can be modeled by -pairs associated with the underlying proof system.