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Examining Fragments of the Quantified Propositional Calculus

Published online by Cambridge University Press:  12 March 2014

Steven Perron
Steven Perron*
Affiliation:
University of Toronto, Department of Computer Science, M5S 3G4, Toronto, Ontario, Canada, E-mail: [email protected]
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Abstract

When restricted to proving formulas, the quantified propositional proof system is closely related to the theorems of Buss's theory . Namely, has polynomial-size proofs of the translations of theorems of , and proves that is sound. However, little is known about when proving more complex formulas. In this paper, we prove a witnessing theorem for similar in style to the KPT witnessing theorem for . This witnessing theorem is then used to show that proves is sound with respect to formulas. Note that unless the polynomial-time hierarchy collapses is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi, p-simulates with respect to all quantified propositional formulas. We finish by proving that S2 can be axiomatized by plus axioms stating that the cut-free version of is sound. All together this shows that the connection between and does not extend to more complex formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 73 , Issue 3 , September 2008 , pp. 1051 - 1080
Copyright
Copyright © Association for Symbolic Logic 2008

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