When Do Extra Majority Gates Help? Polylog(N) Majority Gates Are Equivalent To One
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@MISC{Beigel_whendo,
author = {Richard Beigel},
title = {When Do Extra Majority Gates Help? Polylog(N) Majority Gates Are Equivalent To One},
year = {}
}
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Abstract
. Suppose that a Boolean function f is computed by a constant depth circuit with 2 m AND-, OR-, and NOT-gates---and m majoritygates. We prove that f is computed by a constant depth circuit with 2 m O(1) AND-, OR-, and NOT-gates---and a single majority-gate, which is at the root. One consequence is that if a Boolean function f is computed by an AC 0 circuit plus polylog majority-gates, then f is computed by a probabilistic perceptron having polylog order. Another consequence is that if f agrees with the parity function on three-fourths of all inputs, then f cannot be computed by a constant depth circuit with 2 n o(1) AND-, OR-, and NOT-gates, and n o(1) majority-gates. Key words. Majority-gate; threshold-gate; symmetric gate; circuit; parity. Subject classifications. 68Q15. 1. Introduction One of the goals of complexity theory is to find ways to reduce the use of one resource. Typically this entails a modest increase in some other resources. Recently, quasipolynomial size...
