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Lecture 10

January 17, 2013

Today we proved the structural result on ACC0 circuits by Beigel and Tarui (this line of work was initated by Yao, and has enojyed further results). See also, the web addendum of Arora-Barak. We actually made a slight change in the proof and used small-bias sets instead of pairwise indepedent family of hash functions, so to reduce the fan-in of the Or gates (recall our construction of small-bias sets from Lecture 5). This gives a slight advantage in the parameters, as well as being somewhat simpler and more natural. One additionally change we made was to use the construction of Gopalan, Guruswami and Lipton for the modulus amplifying polynomials.

We also showed how this structural result implies the better-than-naive algorithm for ACC0 CKT-SAT, which was the missing piece in our proof of Williams’ result. So, finally, we finished this part of the course!

The course will have two more lectures, both devoted to Matrix Rigidity. In the next lecture we will learn about Matrix Rigidity and its relation to linear-circuit lower bounds. We will also see some classic results. In the last lecture we will cover a more recent and beautiful result by Zeev Dvir (who was a PhD student at Weizmann).

Those of you who don’t have a lecture to scribe, please mail me your preferred end-of-the-semester talk that you want to scribe.

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