Addition is exponentially harder than counting for shallow monotone circuits

TL;DR

This paper proves exponential lower bounds on the size of constant-depth monotone majority circuits computing a specific addition-based function, showing such circuits cannot efficiently simulate weighted threshold gates, and strengthens classical circuit complexity results.

Contribution

It establishes the first exponential size lower bounds for monotone majority circuits computing addition functions, answering longstanding open questions and matching upper bounds with new constructions.

Findings

Proved exponential lower bounds for depth-$d$ monotone majority circuits computing $U_{d,N}$.

Showed monotone majority circuits require exponential size to simulate weighted threshold gates.

Strengthened classical results by exhibiting functions in AC$^0$ requiring exponential size monotone circuits.

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 $x_1 + \cdots + x_n \geq t$. 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.