Optimal proof systems for propositional logic and complete sets

Abstract

A polynomial time computable function h : Σ* → Σ* whose range is the set of tautologies in Propositional Logic (TAUT), is called a proof system. Cook and Reckhow defined this concept in [5] and in order to compare the relative strength of different proof systems, they considered the notion of p-simulation. Intuitively a proof system h p-simulates a second one h′ if there is a polynomial time computable function γ translating proofs in h′ into proofs in h. A proof system is called optimal if it p-simulates every other proof system. The question of whether p-optimal proof systems exist is an important one in the field. Krajíček and Pudlák [13,12] proved a sufficient condition for the existence of such optimal systems, showing that if the deterministic and nondeterministic exponential time classes coincide, then p-optimal proof systems exist. They also gave a condition implying the existence of optimal proof systems (a related concept to the one of p-optimal systems). In this paper we improve this result obtaining a weaker sufficient condition for this fact. We show that if a particular class of sets with low information content in nondeterministic double exponential time is included in the corresponding deterministic class, then p-optimal proof systems exist. We also show some complexity theoretical consequences that follow from the assumption of the existence of p-optimal systems. We prove that if p-optimal systems exist then the class UP (and some other related complexity classes) have many-one complete languages, and that many-one complete sets for NP ∩ SPARSE follow from the existence of optimal proof systems.