On the existence of complete disjoint NP-pairs

Disjoint NP-pairs are an interesting model of computation with important applications in cryptography and proof complexity. The question whether there exists a complete disjoint NP-pair was posed by Razborov in 1994 and is one of the most important problems in the field. In this paper we prove that there exists a many-one hard disjoint NP-pair which is computed with access to a very weak oracle (a tally NP-oracle). In addition, we exhibit candidates for complete NP-pairs and apply our results to a recent line of research on the construction of hard tautologies from pseudorandom generators

Inseparability and Strong Hypotheses for Disjoint NP Pairs

This paper investigates the existence of inseparable disjoint pairs of NP languages and related strong hypotheses in computational complexity. Our main theorem says that, if NP does not have measure 0 in EXP, then there exist disjoint pairs of NP languages that are P-inseparable, in fact TIME(2^(n^k))-inseparable. We also relate these conditions to strong hypotheses concerning randomness and genericity of disjoint pairs

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 SAT 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 SAT, 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

The Informational Content of Canonical Disjoint NP-Pairs

We investigate the connection between propositional proof systems and their canonical pairs. It is known that simulations between proof systems translate to reductions between their canonical pairs. We focus on the opposite direction and study the following questions. Q1: Where does the implication [can(f) ≤ pp m can(g) ⇒ f ≤s g] hold, and where does it fail? Q2: Where can we find proof systems of different strengths, but equivalent canonical pairs? Q3: What do (non-)equivalent canonical pairs tell about the corresponding proof systems? Q4: Is every NP-pair (A, B), where A is NP-complete, strongly many-one equivalent to the canonical pair of some proof system? In short, we show that Q1 and Q2 can be answered with ‘everywhere’, which generalizes previous results by Pudlák and Beyersdorff. Regarding Q3, inequivalent canonical pairs tell that the proof systems are not “very similar”, while equivalent, P-inseparable canonical pairs tell that they are not “very different”. We can relate Q4 to the open problem in structural complexity that asks whether unions of disjoint NP-complete sets are NP-complete. This demonstrates a new connection between proof systems, disjoint NP-pairs, and unions of disjoint NP-complete sets.