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