# Lecture 9

Today we finally proved Williams’ result that unconditionally separates NEXP from ACC0. As emphasized, the idea of the proof is more general and gives a way to transform fast algorithms for CKT-SAT against a class C to a lower bound against C.

The proof today was modulu an important part we will do next time, namely the structural result (initiated by Yao and improved by Beigel-Tarui) for ACC^0 circuits that in turn yields fast CKT-SAT algorithms for ACC^0 circuits.

Next time we will show this result and by that, in fact, prove Williams’ result up to (1) constructions of PRG under hardness assumptions (2) Toda’s Theorem (3) The completeness of the Permanent for Sharp P, and (4) Completeness of Succinct-SAT for NTIME(2^n/n^c). It is very likely that the first three results will appear in the end of the semester talk, so we can safely say we proved Williams’ result assuming no background in complexity, other than the very very basics (and (4), but come on..).