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**1 - 2**of**2**### Quantified Propositional Logspace Reasoning

, 2008

"... 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 G ∗ 1 with cuts restricted to ΣCNF(2) formu ..."

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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 G ∗ 1 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 the final formula. To show that GL ∗ is strong enough to capture log space reasoning, we translate theorems of V L into a family of tautologies that have polynomial-size GL ∗ proofs. V L is a theory of bounded arithmetic that is known to correspond to logarithmic-space reasoning. To do the translation, we find an appropriate axiomatization of V L, and put V L 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 V L. This is done by giving a logarithmic-space algorithm that witnesses GL ∗ proofs. 1

### Propositional Logic for Circuit Classes

"... Abstract. By introducing a parallel extension rule that is aware of inde-pendence of the introduced extension variables, a calculus for quantified propositional logic is obtained where heights of derivations correspond to heights of appropriate circuits. Adding an uninterpreted predicate on bit-stri ..."

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Abstract. By introducing a parallel extension rule that is aware of inde-pendence of the introduced extension variables, a calculus for quantified propositional logic is obtained where heights of derivations correspond to heights of appropriate circuits. Adding an uninterpreted predicate on bit-strings (analog to an oracle in relativised complexity classes) this statement can be made precise in the sense that the height of the most shallow proof that a circuit can be evaluated is, up to an additive con-stant, the height of that circuit. The main tool for showing lower bounds on proof heights is a variant of an iteration principle studied by Takeuti. This reformulation might be of independent interest, as it allows for polynomial size formulae in the relativised language that require proofs of exponential height. 1 Introduction and Related Work In systems like “extended Frege ” there is a rule that allows one to introduce a new variable by a defining clause p ↔ A. If several variables are to be introduced,