Constructive Relationships Between Algebraic Thickness and Normality

Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9210)

Abstract

We study the relationship between two measures of Boolean functions; algebraic thickness and normality. For a function f, the algebraic thickness is a variant of the sparsity, the number of nonzero coefficients in the unique \(\mathbb {F}_2\) polynomial representing f, and the normality is the largest dimension of an affine subspace on which f is constant. We show that for \(0 < \epsilon <2\), any function with algebraic thickness \(n^{3-\epsilon }\) is constant on some affine subspace of dimension \(\varOmega \left( n^{\frac{\epsilon }{2}}\right) \). Furthermore, we give an algorithm for finding such a subspace. We show that this is at most a factor of \(\varTheta (\sqrt{n})\) from the best guaranteed, and when restricted to the technique used, is at most a factor of \(\varTheta (\sqrt{\log n})\) from the best guaranteed. We also show that a concrete function, majority, has algebraic thickness \(\varOmega \left( 2^{n^{1/6}}\right) \).

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Copyright information

© Springer International Publishing Switzerland (outside the US)  2015

Authors and Affiliations

  1. 1.Department of Mathematics of Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.Information Technology LaboratoryNational Institute of Standards and TechnologyGaithersburgUSA

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