The composition complexity of majority
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We study the complexity of computing majority as a composition of local functions: Maj_n = h(g_1,…,g_m), where each g_j :{0,1}^n→{0,1} is an arbitrary function that queries only k ≪ n variables and h : {0,1}^m →{0,1} is an arbitrary combining function. We prove an optimal lower bound of m ≥Ω( n/klog k ) on the number of functions needed, which is a factor Ω(log k) larger than the ideal m = n/k. We call this factor the composition overhead; previously, no superconstant lower bounds on it were known for majority. Our lower bound recovers, as a corollary and via an entirely different proof, the best known lower bound for bounded-width branching programs for majority (Alon and Maass '86, Babai et al. '90). It is also the first step in a plan that we propose for breaking a longstanding barrier in lower bounds for small-depth boolean circuits. Novel aspects of our proof include sharp bounds on the information lost as computation flows through the inner functions g_j, and the bootstrapping of lower bounds for a multi-output function (Hamming weight) into lower bounds for a single-output one (majority).
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