Examining fragments of the quantified propositional calculus

When restricted to proving Σqi formulas, the quantified propositional proof system Gi is closely related to the Σ b i theorems of Buss’s theory S i 2. Namely, G ∗ i has polynomial-size proofs of the translations of theorems of S 2, and S i 2 proves that G ∗ i 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 T i 2 . This witnessing theorem is then used to show that S i 2 proves G ∗ i is sound with respect to Σqi+1 formulas. Note that unless the polynomial-time hierarchy collapses S 2 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 G ∗ i+1. We finish by proving that S2 can be axiomatized by S 1 2 plus axioms stating that the cut-free version of G0 is sound. All together this shows that the connection between G ∗ i and S i 2 does not extend to more complex formulas. §