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A Propositional Proof System for Log Space

  • Steven Perron
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3634)

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.

Keywords

Normal Form Free Variable Propositional Calculus Proof System Propositional Formula 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Steven Perron
    • 1
  1. 1.Department of Computer ScienceUniversity of Toronto 

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