<html><head><meta http-equiv="Content-Type" content="text/html charset=utf-8"></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space;" class=""><div class=""><u class="">*REMINDER*</u></div><div class=""><br class=""></div><div class="">Combinatorics & Theoretical Seminar</div> <br class="">Tuesday, April 18, 2017<br class="">Ryerson 251 @ 3pm<br class=""><b class=""><br class="">Andrej Bogdanov</b><br class="">Chinese University of Hong Kong<br class=""><a href="http://www.cs.washington.edu/homes/jrl/" class="">http://www.cs.washington.edu/homes/jrl/</a><br class=""><br class="">Title: “Small bias requires large formulas”<br class="">A small-biased function is a randomized function whose distribution of truth-tables is small-biased. We show that known explicit lower bounds on the size of (1) general Boolean formulas, (2) Boolean formulas of fan-in two, (3) de Morgan formulas, as well as (4) correlation lower bounds against small de Morgan formulas apply to small-biased functions. As a consequence, any strongly explicit small-biased generator is subject to the best known explicit formula lower bounds in all these models.<br class=""><br class="">On the other hand, we give a construction of a small-biased function that is tight with respect to lower bounds (1) and (2) for the relevant range of parameters. We interpret this construction as a natural-like barrier against substantially stronger lower bounds for general formulas.<br class=""><br class="">Host: Prof. Alexander Razborov<br class=""><br class="">Refreshments will be served before the talk in Ry 255.<br class=""><br class=""> <br class=""><br class=""> </body></html>