Tuples of Disjoint $\mathsf{NP}$ -Sets

Abstract

Disjoint \(\mathsf{NP}\) -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 \(\mathsf{NP}\) -pairs to disjoint k-tuples of \(\mathsf{NP}\) -sets for k≥2. We define subclasses of the class of all disjoint k-tuples of \(\mathsf{NP}\) -sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint \(\mathsf{NP}\) -pairs exist if and only if complete disjoint k-tuples of \(\mathsf{NP}\) -sets exist for all k≥2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems.