Separating ${AC}^0$ from Depth-2 Majority Circuits
Alexander A. Sherstov · SIAM Journal on Computing · 2009
We construct a function in ${AC}^0$ that cannot be computed by a depth-2 majority circuit of size less than $\exp(\Theta(n^{1/5}))$. This solves an open problem due to Krause and Pudlák [Theoret. Comput. Sci., 174 (1997), pp. 137–156] and matches Allender's classic result [A note on the power of threshold circuits, in Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science (FOCS), Research Triangle Park, NC, 1989, pp. 580–584] that ${AC}^0$ 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 ${AC}^0$ with exponentially small discrepancy, $\exp(-\Omega(n^{1/5}))$, thereby establishing the separations $\Sigma_2^{cc} ot\subseteq{PP}^{cc}$ and $\Pi_2^{cc} ot\subseteq{PP}^{cc}$ in communication complexity.