Abstract
In this paper, we consider the formula complexity of the majority function over a basis consisting only of small (bounded-size) majority gates. Using Valiant's amplification technique, we show that there is a formula of sizeO(n 4.29) when only the gateM 3 (the majority gate on three inputs) is used. Then, based on a result of Boppana, we show that not only is our result optimal with respect to the amplification technique, there is no smaller formula over the basis of all monotone 3-input functions (again with respect to amplification). Finally, we show that no better bounds are possible even with respect to more general input distributions. In particular, we show that it is not possible to use amplification to “bootstrap”, that is, use smaller majority functions in the initial distribution to find an optimal formula for a larger majority function.
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Gupta, A., Mahajan, S. Using amplification to compute majority with small majority gates. Comput Complexity 6, 46–63 (1996). https://doi.org/10.1007/BF01202041
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DOI: https://doi.org/10.1007/BF01202041