A Propositional Proof System for Log Space
نویسنده
چکیده
The proof system G0 of the quantified propositional calculus corresponds to NC, and G1 corresponds to P , but no formula-based proof system that corresponds log space reasoning has ever been developed. This paper does this by developing GL∗. We begin by defining a class ΣCNF (2) of quantified formulas that can be evaluated in log space. Then GL∗ is defined as G1 with cuts restricted to ΣCNF (2) formulas and no cut formula that is not quantifier free contains a non-parameter free variable. To show that GL∗ is strong enough to capture log space reasoning, we translate theorems of Σ 0 -rec into a family of tautologies that have polynomial size GL∗ proofs. Σ 0 -rec is a theory of bounded arithmetic that is known to correspond to log space. To do the translation, we find an appropriate axiomatization of Σ 0 -rec, and put Σ B 0 -rec proofs into a new normal form. To show that GL∗ is not too strong, we prove the soundness of GL∗ in such a way that it can be formalized in Σ 0 -rec. This is done by giving a log space algorithm that witnesses GL∗ proofs.
منابع مشابه
On the polynomial-space completeness of intuitionistic propositional logic
We present an alternative, purely semantical and relatively simple, proof of the Statman’s result that both intuitionistic propositional logic and its implicational fragment are PSPACE -complete.
متن کاملAtomic polymorphism
It has been known for six years that the restriction of Girard’s polymorphic system F to atomic universal instantiations interprets the full fragment of the intuitionistic propositional calculus. We firstly observe that Tait’s method of “convertibility” applies quite naturally to the proof of strong normalization of the restricted Girard system. We then show that each β-reduction step of the fu...
متن کاملThe Deduction Rule and Linear and Near-Linear Proof Simulations
We introduce new proof systems for propositional logic, simple deduction Frege systems, general deduction Frege systems and nested deduction Frege systems, which augment Frege systems with variants of the deduction rule. We give upper bounds on the lengths of proofs in Frege proof systems compared to lengths in these new systems. As applications we give near-linear simulations of the propositio...
متن کاملQuantified Propositional Logspace Reasoning
In this paper, we develop a quantified propositional proof systems that corresponds to logarithmic-space reasoning. We begin by defining a class ΣCNF (2) of quantified formulas that can be evaluated in log space. Then our new proof system GL is defined as G1 with cuts restricted to ΣCNF (2) formulas and no cut formula that is not quantifier free contains a free variable that does not appear in ...
متن کاملOn the correspondence between arithmetic theories and propositional proof systems - a survey
Bounded arithmetic is closely related to propositional proof systems, and this relation has found many fruitful applications. The aim of this paper is to explain and develop the general correspondence between propositional proof systems and arithmetic theories, as introduced by Kraj́ıček and Pudlák [41]. Instead of focusing on the relation between particular proof systems and theories, we favour...
متن کامل