### Separating \${AC}^0\$ from Depth-2 Majority Circuits

Alexander A. Sherstov
2009 SIAM journal on computing (Print)
We construct a function in AC 0 that cannot be computed by a depth-2 majority circuit of size less than exp(Θ(n 1/5 )). This solves an open problem due to Krause and Pudlák (1997) and matches Allender's classic result (1989) 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
more » ... anslates 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(−Ω(n 1/5 )), thereby establishing the separations Σ cc 2 ⊆ PP cc and Π cc 2 ⊆ PP cc in communication complexity. Introduction. A natural and important computational model is that of a polynomial-size circuit of majority gates. This model has been extensively studied for the past two decades [14, 15, 26, 27, 38, [43] [44] [45] . Research has shown that majority circuits of depth 2 and 3 already possess surprising computational power. Indeed, it is a longstanding open problem [21] to exhibit a Boolean function that cannot be computed by a depth-3 majority circuit of polynomial size. To illustrate, the arithmetic operations of powering, multiplication, and division on n-bit integer arguments can all be computed by depth-3 majority circuits of polynomial size [45] . An even more striking example is the addition of n n-bit integers, which is computable by a depth-2 majority circuit of polynomial size [45]. Depth-2 majority circuits of polynomial size can also compute every symmetric function (such as parity) and every DNF and CNF formula of polynomial size. This chief goal of this paper is to relate the computational power of majority circuits to that of AC 0 , another extensively studied class, which consists of polynomialsize constant-depth circuits of and, or, not gates. A well-known result due to Allender [1] states that every function in AC 0 can be computed by a depth-3 majority circuit of quasipolynomial size. For over ten years, it has been an open problem to determine whether Allender's simulation is optimal. Specifically, Krause and Pudlák [21, §6] ask whether every function in AC 0 can be computed by a depth-2 majority circuit of quasipolynomial size. We solve this open problem completely, even in the more general setting of majority-of-threshold circuits (i.e., depth-2 circuits in which a majority gate receives inputs from arbitrary linear threshold gates): There is a function f : {−1, 1} n → {−1, 1}, explicitly given and computable by an AC 0 circuit of depth 3, whose computation requires a majority vote of exp(Ω(n 1/5 )) linear threshold gates. In other words, Allender's simulation is optimal in a strong sense. The lower bound in Theorem 1.1 is an exponential improvement over previous work. The best * This paper appeared in preliminary form in