The complexity of depth-3 circuits computing symmetric Boolean functions

We give tight lower bounds for the size of depth-3 circuits with limited bottom fanin computing symmetric Boolean functions. We show that any depth-3 circuit with bottom fanin k which computes the Boolean function Exactn/(k+1)n, has at least (1 + 1/k)n/(n + 1) gates. We show that for k = o(√n) this lower bound is essentially tight, by generalizing a known upper bound on the size of depth-3 circuits with bottom fanin 2, computing symmetric Boolean functions.

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