Abstract

Abstract. The proof system G ∗ 0 of the quantified propositional calculus corresponds to NC 1, and G ∗ 1 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 G ∗ 1 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 B 0-rec into a family of tautologies that have polynomial size GL ∗ proofs. Σ B 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 B 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 B 0-rec. This is done by giving a log space algorithm that witnesses GL ∗ proofs. 1

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