Nondeterministic functions and the existence of optimal proof systems
Theoretical Computer Science
The deduction theorem for strong propositional proof systems
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
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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.