Proving SAT does not have Small Circuits with an Application to the Two

We show that if SAT does not have small circuits, then there must exist a small number of satisfiable formulas such that every small circuit fails to compute satisfiability correctly on at least one of these formulas. We use this result to show that if PNP[1] = PNP[2], then the polynomial-time hierarchy collapses to Sp 2 ⊆ Σp 2 ∩Πp 2. Even showing that the hierarchy collapsed to Σp 2 remained open prior to this paper.

• Type: Conference
• Year: 2003
• Where: COCO
• Authors: Lance Fortnow, Aduri Pavan, Samik Sengupta
• Keywords: COCO 2003, Polynomial-time Hierarchy Collapses, Satisfiable Formulas, Small Circuit