Abstract
Let $U_{k,N}$ denote the Boolean function which takes as input $k$ strings of $N$ bits each, representing $k$ numbers $a^{(1)},\dots,a^{(k)}$ in $\{0,1,\dots,2^{N}-1\}$, and outputs 1 if and only if $a^{(1)} + \cdots + a^{(k)} \geq 2^N.$ Let THR$_{t,n}$ denote a monotone unweighted threshold gate, i.e., the Boolean function which takes as input a single string $x \in \{0,1\}^n$ and outputs $1$ if and only if $x_1 + \cdots + x_n \geq t$. The function $U_{k,N}$ may be viewed as a monotone function that performs addition, and THR$_{t,n}$ may be viewed as a monotone function that performs counting. We refer to circuits that are composed of THR gates as monotone majority circuits.
The main result of this paper is an exponential lower bound on the size of bounded-depth monotone majority circuits that compute $U_{k,N}$. More precisely, we show that for any constant $d \geq 2$, any depth-$d$ monotone majority circuit computing $U_{d,N}$ must have size $\smash{2^{\Omega(N^{1/d})}}$. Since $U_{k,N}$ can be computed by a single monotone weighted threshold gate (that uses exponentially large weights), our lower bound implies that constant-depth monotone majority circuits require exponential size to simulate monotone weighted threshold gates. This answers a question posed by Goldmann and Karpinski (STOC'93) and recently restated by Hastad (2010, 2014). We also show that our lower bound is essentially best possible, by constructing a depth-$d$, size-$2^{O(N^{1/d})}$ monotone majority circuit for $U_{d,N}$.
As a corollary of our lower bound, we significantly strengthen a classical theorem in circuit complexity due to Ajtai and Gurevich (JACM'87). They exhibited a monotone function that is in AC$^0$ but requires super-polynomial size for any constant-depth monotone circuit composed of unbounded fan-in AND and OR gates. We describe a monotone function that is in depth-$3$ AC$^0$ but requires exponential size monotone circuits of any constant depth, even if the circuits are composed of THR gates.
Joint work with Xi Chen (Columbia) and Rocco Servedio (Columbia).