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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  Full text: PDF file (610 kB) References: PDF file   HTML file

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
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• 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      •  Number of views: This page: 506 Full text: 64 References: 30 First page: 13 Contact us: math-net2020_01 [at] mi-ras ru Terms of Use Registration Logotypes © Steklov Mathematical Institute RAS, 2020