Separating Ac from Depth-2 Majority Circuits∗

We construct a function in AC that cannot be computed by a depth-2 majority circuit of size less than exp(Θ(n1/5)). This solves an open problem due to Krause and Pudlák (1997) and matches Allender’s classic result (1989) that AC can be efficiently simulated by depth-3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This technique, the Degree/Discrepancy Theorem, is of independent interest. It translates lower bounds on the threshold degree of any Boolean function into upper bounds on the discrepancy of a related function. Upper bounds on the discrepancy, in turn, immediately imply lower bounds on communication and circuit size. In particular, we exhibit the first known function in AC with exponentially small discrepancy, exp(−Ω(n1/5)), thereby establishing the separations Σcc 2 6⊆ PP cc and Πcc 2 6⊆ PP cc in communication complexity.