Proving SAT does not have small circuits with an application to the two queries problem

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/sup NP[1]/=P/sup NP[2]/, then the polynomial-time hierarchy collapses to S/sub 2//sup P//spl sube//spl Sigma//sub 2//sup p//spl cap//spl Pi//sub 2//sup p/. Even showing that the hierarchy collapsed to /spl Sigma//sub 2//sup p/ remained open.