On the minimal fourier degree of symmetric Boolean functions

Abstract

In this paper we give a new upper bound on the minimal degree of a nonzero Fourier coefficient in any non-linear symmetric Boolean function. Specifically, we prove that for every non-linear and symmetric f: {0, 1}k → {0, 1} there exists a set; \(\not 0 \ne S \subset [k]\) such that ¦S¦ = O(Γ(k)+√k, and \(\hat f(S) \ne 0\) where Γ(m)≤m 0.525 is the largest gap between consecutive prime numbers in {1,..., m}. As an application we obtain a new analysis of the PAC learning algorithm for symmetric juntas, under the uniform distribution, of Mossel et al. [10].

Our bound on the degree is a significant improvement over the previous result of Kolountzakis et al. [8] who proved that ¦S¦=O(k=log k).

We also show a connection between lower-bounding the degree of non-constant functions that take values in {0,1,2} and the question that we study here.

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Correspondence to Amir Shpilka.

Additional information

A preliminary version appeared in CCC 2011 [13].

This research was partially supported by the Israel Science Foundation (grant number 339/10).

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Shpilka, A., Tal, A. On the minimal fourier degree of symmetric Boolean functions. Combinatorica 34, 359–377 (2014). https://doi.org/10.1007/s00493-014-2875-z

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Keywords

  • Boolean Function
  • Prime Number
  • Symmetric Function
  • Minimal Degree
  • Fourier Spectrum