We study the question of whether the class DisNP 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 ... more >>>

We study several properties of sets that are complete for NP. We prove that if $L$ is an NP-complete set and $S \not\supseteq L$ is a p-selective sparse set, then $L - S$ is many-one-hard for NP. We demonstrate existence of a sparse set $S \in \mathrm{DTIME}(2^{2^{n}})$ such that for ... more >>>

We investigate the class of disjoint NP-pairs under different reductions. The structure of this class is intimately linked to the simulation order of propositional proof systems, and we make use of the relationship between propositional proof systems and theories of bounded arithmetic as the main tool of our analysis. Specifically ... more >>>

We prove that every disjoint NP-pair is polynomial-time, many-one equivalent to the canonical disjoint NP-pair of some propositional proof system. Therefore, the degree structure of the class of disjoint NP-pairs and of all canonical pairs is identical. Secondly, we show that this degree structure is not superficial: Assuming there exist ... more >>>

For a proof system P we introduce the complexity class DNPP(P) of all disjoint NP-pairs for which the disjointness of the pair is efficiently provable in the proof system P. We exhibit structural properties of proof systems which make the previously defined canonical NP-pairs of these proof systems hard or ... more >>>

Disjoint NP-pairs are a well studied complexity theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint NP-pairs to disjoint k-tuples of NP-sets for k>1. We define subclasses of the class of all disjoint k-tuples of ... more >>>

In this paper we ask the question whether the extended Frege proof system EF satisfies a weak version of the deduction theorem. We prove that if this is the case, then complete disjoint NP-pairs exist. On the other hand, if EF is an optimal proof system, then the weak deduction ... more >>>

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) \le_m can(g) => f \le_s g] ... more >>>