I have a basic question about circuit complexity. Apparently no example of a language that requires super-polynomial circuits to decide is known. (This is despite the fact that an easy counting argument shows that there must be many such languages.)
On the hand it is known that P/poly does not contain all of EXPSPACE. I was wondering why did this not imply that any EXPSPACE-complete problem must have super-polynomial circuit complexity. I figured it was because I might not be able to reduce any problem to a complete problem using polynomial-sized circuits. Is that correct?
On the other hand, I should be able to reduce any problem in NP to SAT using polynomial-sized circuits. So am I correct if I claim that the existence of polynomial sized circuits for SAT implies that P/poly contains NP?