Examining Fragments of the Quantified Propositional Calculus

Steven Perron
2008 Journal of Symbolic Logic (JSL)  
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
more » ... t unless the polynomial-time hierarchy collapses is the weakest theory in the S 2 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 S 2 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.
doi:10.2178/jsl/1230396765 fatcat:file2pnqujgvliksajephdbn34