We show that if SAT does not have small circuits, then there must exist a small number of 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 P^NP[1]=P^NP[2], then the polynomial-time hierarchy collapses to S²_P &sube &Sigma₂^p &cap &Pi₂^p$. Even showing that the hierarchy collapsed to &Sigma₂^p$ remained open prior to this paper.

• Journal of Computer and System Sciences. Special issue for selected papers from the 18th IEEE Conference on Computational Complexity. To appear.
  
• Proceedings of the 18th IEEE Conference on Computational Complexity, 2003. 347-350.