Examining The Fragments of G

Abstract

When restricted to proving Sigma_i ^q formulas, the quantified prepositional proof system G_i* is closely related to the Sigma_i ^b theorems of Buss's theory S₂ ⁱ. Namely, G_i* has polynomial- size proofs of the translations of theorems of S₂ ⁱ, and S₂ ⁱ proves that G_i* is sound. However, little is known about G* 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 T₂ ⁱ. This witnessing theorem is then used to show that S₂ ⁱ proves G* is sound with respect to prenex Sigma_i+1 ^q formulas. Note that unless the polynomial hierarchy collapses S₂ ⁱ is the weakest theory in the S₂ ⁱ 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. This is followed by a proof that Gi p-simulates G*_i+1. We finish by proving that S₂ can be axiomatized by S₂ ¹ plus axioms stating that the cut-free version of G* is sound. All together this shows that the connection between G_i* and S₂ ⁱ does not extend to more complex formulas.