# Lectures 7+8

In the past two lectures we proved a beautiful theorem by Impagliazzo and Kabanets that states that a derandomization of ZEROP (aka PIT) will yield some kind of circuit lower bound (either NEXP not in P/poly or the Permanent not in VP). The proof uses many results, old and new, from complexity, such as Toda’s Theorem, the completness of the Permanent for Sharp-P, MA is contained in PH, and the non-deterministic time-hierarchy theorem.

Another central theorem used in the proof is a beautiful theorem by Impagliazzo, Kabanets and Wigderson: if NEXP is contained in P/poly then NEXP=MA. We already seen that the randomized version of NEXP, namely MA_EXP, is not contained in P/poly (essentially due to IP=PSPACE), so NEXP is a natural next goal for P/poly lower bounds. Moreover, note that if MA is replaced with its non-randomized version, NP, then this would have contradict the non-deterministic time-hierarchy theorem. This said, we do not even know how to prove that BPP is strictly contained in NEXP (which is an easier problem as BPP is contained in P/poly). The proof is based on the ingenious “easy witness” method introduced by Kabanets.

We also commented that the proof for the above theorem yields also a proof for the fact the Succinct-SAT has Succinct-witness under the assumption that NEXP is contained in P/poly. We will use this for proving Williams’ result, starting next time. You are invited to read Ryan’s informal description of his result, or even watch Ryan’s online talk.