Problems complete for deterministic logarithmic space
Journal of Algorithms
Bounded arithmetic, propositional logic, and complexity theory
Bounded arithmetic, propositional logic, and complexity theory
Feasibly constructive proofs and the propositional calculus (Preliminary Version)
STOC '75 Proceedings of seventh annual ACM symposium on Theory of computing
Propositional logic for circuit classes
CSL'07/EACSL'07 Proceedings of the 21st international conference, and Proceedings of the 16th annuall conference on Computer Science Logic
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The proof system G$_{\rm 0}^{\rm *}$ of the quantified propositional calculus corresponds to NC1, 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.