A Propositional Proof System for Log Space

Abstract

The proof system G $_{\rm 0}^{\rm *}$ of the quantified propositional calculus corresponds to NC ¹, and G $_{\rm 1}^{\rm *}$ 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 $_{\rm 1}^{\rm *}$ 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$_{\rm 0}^{B}$-rec into a family of tautologies that have polynomial size GL ^* proofs. Σ$_{\rm 0}^{B}$-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$_{\rm 0}^{B}$-rec, and put Σ$_{\rm 0}^{B}$-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$_{\rm 0}^{B}$-rec. This is done by giving a log space algorithm that witnesses GL ^* proofs.