Examining fragments of the quantified propositional calculus

Abstract

When restricted to proving Σ_i^q formulas, the quantified propositional proof system G_i^* is closely related to the Σ_i^b theorems of Buss’s theory S₂ⁱ. Namely, G_i^* has polynomial-size proofs of the translations of theorems of S₂ⁱ, and S₂ⁱ proves that G_i^* is sound. However, little is known about G_i^* when proving more complex formulas. In this paper, we prove a witnessing theorem for G_i^* similar in style to the KPT witnessing theorem for T₂ⁱ. This witnessing theorem is then used to show that S₂ⁱ proves G_i^* is sound with respect to Σ_i+1^q 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 G_i p-simulates G_i+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 G_i^* and S₂ⁱ does not extend to more complex formulas.