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International Symposium on Fundamentals of Computation Theory

FCT 2015: Fundamentals of Computation Theory pp 106–117Cite as

Constructive Relationships Between Algebraic Thickness and Normality

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,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 \(\varOmega \left( 2^{n^{1/6}}\right) \).

Contribution of the National Institute of Standards and Technology. The rights of this work are transferred to the extent transferable according to title 17 § 105 U.S.C.

Partially supported by the Danish Council for Independent Research, Natural Sciences.

M.G. Find—Most of this work was done while at the University of Southern Denmark.

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Notes

  1. 1.

    The constant 6.12 does not appear explicitly in these articles, however it can be derived using similar calculations as in the cited papers. This also follows from Theorem 6 later in this paper. We remark that 6.12 is not optimal.

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Author information

Authors and Affiliations

  1. Department of Mathematics of Computer Science, University of Southern Denmark, Odense, Denmark

    Joan Boyar

  2. Information Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, USA

    Magnus Gausdal Find

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  1. Joan Boyar
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  2. Magnus Gausdal Find
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Correspondence to Magnus Gausdal Find .

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Editors and Affiliations

  1. Université Paris Diderot, Paris, France

    Adrian Kosowski

  2. Université Bordeaux 1, Talence, France

    Igor Walukiewicz

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Boyar, J., Find, M.G. (2015). Constructive Relationships Between Algebraic Thickness and Normality. In: Kosowski, A., Walukiewicz, I. (eds) Fundamentals of Computation Theory. FCT 2015. Lecture Notes in Computer Science(), vol 9210. Springer, Cham. https://doi.org/10.1007/978-3-319-22177-9_9

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  • DOI: https://doi.org/10.1007/978-3-319-22177-9_9

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