# Lecture 4

Most of the lecture today was devoted to prove Razborov-Smolensky Theorem ’87, following chapter 14 in Arora-Barak.

We then switched to talk about randomization in computation. The goal is to introduce the first topic of this course, which is, to my opinion, the second greatest result in complexity theory (right after the PCP Theorem – someone should convience someone to give a course on this at Weizmann..), namely, Hardness-Randomness tradeoff (or as I like to call it – Hardomness).

We gave an example of a randomized algorithm for the Polynomial Identity Testing problem, on the way introduced to algebraic circuits. The analysis is based on the famous Schwartz-Zippel Lemma. We omitted a proof for the lemma. A standard proof can be found in chapter 7 of Arora-Barak. See an alternative proof by Dana Moshkovitz (who was a Ph.D. student at Weizmann). There is a great, recent survey talk by Ran Raz about arithmetic circuits (on my computer the talk sometimes gets stuck).

We also mentioned that the Permanent is complete for the class VNP, which is an algebraic analoug (though non uniform) to the class NP, introduced by Valiant. This will be a project for the end of the semester talk. For now, you can view a great talk available online by Amir Yehudayoff on this (Amir was also a Ph.D. student at Weizmann).