On Polynomial Approximations to AC^0

In this talk, we will discuss some questions related to polynomial approximations of AC0. A classic result due to Tarui (1991) and Beigel, Reingold, and Spielman (1991), states that any AC0 circuit of size s and depth d has an -error probabilistic polynomial over the reals of degree at most (log(s/))ˆ{O(d)}. We will have a re-look at this construction and show how to improve the bound to (log s)ˆ{O(d)}log(1/), which is much better for small values of . As an application of this result, we show that (log s)ˆ{O(d)}log(1/)-wise independence fools AC0, improving on Tal’s strengthening of Braverman’s theorem that (log(s/))ˆ{O(d)}-wise independence fools AC0. Time permitting, we will also discuss some lower bounds on the best polynomial approximations to AC0. Joint work with Srikanth Srinivasan Organizer(s): Pranjal Awasthi and Swastik Kopparty