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September 2008 Examining fragments of the quantified propositional calculus
Steven Perron
J. Symbolic Logic 73(3): 1051-1080 (September 2008). DOI: 10.2178/jsl/1230396765

Abstract

When restricted to proving Σiq formulas, the quantified propositional proof system Gi* is closely related to the Σib theorems of Buss’s theory S2i. Namely, Gi* has polynomial-size proofs of the translations of theorems of S2i, and S2i proves that Gi* is sound. However, little is known about Gi* when proving more complex formulas. In this paper, we prove a witnessing theorem for Gi* similar in style to the KPT witnessing theorem for T2i. This witnessing theorem is then used to show that S2i proves Gi* is sound with respect to Σi+1q formulas. Note that unless the polynomial-time hierarchy collapses S2i is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that G1* is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi p-simulates Gi+1* with respect to all quantified propositional formulas. We finish by proving that S2 can be axiomatized by S21 plus axioms stating that the cut-free version of G0* is sound. All together this shows that the connection between Gi* and S2i does not extend to more complex formulas.

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Steven Perron. "Examining fragments of the quantified propositional calculus." J. Symbolic Logic 73 (3) 1051 - 1080, September 2008. https://doi.org/10.2178/jsl/1230396765

Information

Published: September 2008
First available in Project Euclid: 27 December 2008

zbMATH: 1165.03047
MathSciNet: MR2444286
Digital Object Identifier: 10.2178/jsl/1230396765

Rights: Copyright © 2008 Association for Symbolic Logic

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Vol.73 • No. 3 • September 2008
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