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    Addition is exponentially harder than counting for shallow monotone circuits

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    Let Uk,NU_{k,N} denote the Boolean function which takes as input kk strings of NN bits each, representing kk numbers a(1),…,a(k)a^{(1)},\dots,a^{(k)} in {0,1,…,2Nβˆ’1}\{0,1,\dots,2^{N}-1\}, and outputs 1 if and only if a(1)+β‹―+a(k)β‰₯2N.a^{(1)} + \cdots + a^{(k)} \geq 2^N. Let THRt,n_{t,n} denote a monotone unweighted threshold gate, i.e., the Boolean function which takes as input a single string x∈{0,1}nx \in \{0,1\}^n and outputs 11 if and only if x1+β‹―+xnβ‰₯tx_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 Uk,NU_{k,N}. More precisely, we show that for any constant dβ‰₯2d \geq 2, any depth-dd monotone majority circuit computing Ud,NU_{d,N} must have size 2Ξ©(N1/d)\smash{2^{\Omega(N^{1/d})}}. Since Uk,NU_{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-dd, size-2O(N1/d)2^{O(N^{1/d})} monotone majority circuit for Ud,NU_{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 AC0^0 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-33 AC0^0 but requires exponential size monotone circuits of any constant depth, even if the circuits are composed of THR gates
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