Computing Symmetric Boolean Functions by Circuits with Few Exact Threshold Gates

We consider constant depth circuits augmented with few exact threshold gates with arbitrary weights. We prove strong (up to exponential) size lower bounds for such circuits computing symmetric Boolean functions. Our lower bound is expressed in terms of a natural parameter, the balance, of symmetric functions. Furthermore, in the quasi-polynomial size setting our results provides an exact characterization of the class of symmetric functions in terms of their balance.

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