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 Mat. Zametki, 2010, Volume 87, Issue 6, Pages 885–899 (Mi mz8734)

A Small Decrease in the Degree of a Polynomial with a Given Sign Function Can Exponentially Increase Its Weight and Length

V. V. Podolskiia, A. A. Sherstovb

a M. V. Lomonosov Moscow State University
b University of Texas in Austin

Abstract: A Boolean function $f\colon\{-1,+1\}^n\to\{-1,+1\}$ is called the sign function of an integer-valued polynomial $p(x)$ if $f(x)=\operatorname{sgn}(p(x))$ for all $x\in\{-1,+1\}^n$. In this case, the polynomial $p(x)$ is called a perceptron for the Boolean function $f$. The weight of a perceptron is the sum of absolute values of the coefficients of $p$. We prove that, for a given function, a small change in the degree of a perceptron can strongly affect the value of the required weight. More precisely, for each $d=1,2,…,n-1$, we explicitly construct a function $f\colon\{-1,+1\}^n\to\{-1,+1\}$ that requires a weight of the form $\exp\{\Theta(n)\}$ when it is represented by a degree $d$ perceptron, and that can be represented by a degree $d+1$ perceptron with weight equal to only $O(n^2)$. The lower bound $\exp\{\Theta(n)\}$ for the degree $d$ also holds for the size of the depth 2 Boolean circuit with a majority function at the top and arbitrary gates of input degree $d$ at the bottom. This gap in the weight values is exponentially larger than those that have been previously found. A similar result is proved for the perceptron length, i.e., for the number of monomials contained in it.

Keywords: Boolean function, integer-valued polynomial, sign function, perceptron, Boolean circuit, complexity theory, discrete Fourier transform, exponential gap

DOI: https://doi.org/10.4213/mzm8734

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English version:
Mathematical Notes, 2010, 87:6, 860–873

Bibliographic databases:

UDC: 519.712.3

Citation: V. V. Podolskii, A. A. Sherstov, “A Small Decrease in the Degree of a Polynomial with a Given Sign Function Can Exponentially Increase Its Weight and Length”, Mat. Zametki, 87:6 (2010), 885–899; Math. Notes, 87:6 (2010), 860–873

Citation in format AMSBIB
\Bibitem{PodShe10} \by V.~V.~Podolskii, A.~A.~Sherstov \paper A Small Decrease in the Degree of a Polynomial with a Given Sign Function Can Exponentially Increase Its Weight and Length \jour Mat. Zametki \yr 2010 \vol 87 \issue 6 \pages 885--899 \mathnet{http://mi.mathnet.ru/mz8734} \crossref{https://doi.org/10.4213/mzm8734} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2840383} \transl \jour Math. Notes \yr 2010 \vol 87 \issue 6 \pages 860--873 \crossref{https://doi.org/10.1134/S0001434610050263} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000279600700026} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77954407144} 

• http://mi.mathnet.ru/eng/mz8734
• https://doi.org/10.4213/mzm8734
• http://mi.mathnet.ru/eng/mz/v87/i6/p885

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This publication is cited in the following articles:
1. Podolskii V.V., “Lower Bound on Weights of Large Degree Threshold Functions”, Log. Meth. Comput. Sci., 9:2 (2013), 13
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