Relating the bounded arithmetic and polynomial time hierarchies

Abstract

The bounded arithmetic theory S₂ is finitely axiomatized if and only if the polynomial hierarchy provably collapses. If T₂ⁱ equals S₂^{i + 1} then T₂ⁱ is equal to S₂ and proves that the polynomial time hierarchy collapses to ∑_{i + 3}^p, and, in fact, to the Boolean hierarchy over ∑_{i + 2}^p and to ∑_{i + 1}^p/poly.