On the minimal fourier degree of symmetric Boolean functions


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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  1. [1]

    N. Alon, O. Goldreich, J. Håstad and R. Peralta: Simple construction of almost k-wise independent random variables, Random Structures and Algorithms 3 (1992), 289–304.

    Article  MATH  MathSciNet  Google Scholar 

  2. [2]

    R. C. Baker, G. Harman and J. Pintz: The difference between consecutive primes, II, Proceedings of the London Mathematical Society 83 (2001), 532–562.

    Article  MATH  MathSciNet  Google Scholar 

  3. [3]

    A. Blum and P. Langley: Selection of relevant features and examples in machine learning, Artif. Intell. 97 (1997), 245–271.

    Article  MATH  MathSciNet  Google Scholar 

  4. [4]

    A. L. Blum: Relevant examples and relevant features: Thoughts from computational learning theory, in: In AAAI Fall Symposium on ‘Relevance’, 1994.

    Google Scholar 

  5. [5]

    H. Cramér: On the order of magnitude of the difference between consecutive prime numbers, Acta Arithmetica 2 (1936), 23–46.

    Google Scholar 

  6. [6]

    G. Cohen, A. Shpilka and A. Tal: On the degree of univariate polynomials over the integers, in: Innovations in Theoretical Computer Science (ITCS), 409–427, 2012.

    Google Scholar 

  7. [7]

    J. von zur Gathen and J. R. Roche: Polynomials with two values, Combinatorica 17 (1997), 345–362.

    Article  MATH  MathSciNet  Google Scholar 

  8. [8]

    M. N. Kolountzakis, R. J. Lipton, E. Markakis, A. Mehta and N. K. Vishnoi: On the Fourier spectrum of symmetric Boolean functions, Combinatorica 29 (2009), 363–387.

    Article  MATH  MathSciNet  Google Scholar 

  9. [9]

    D. E. Knuth: The Art of Computer Programming, Volume III: Sorting and Searching, Addison-Wesley, 1973.

    Google Scholar 

  10. [10]

    E. Mossel, R. O’Donnell and R. A. Servedio: Learning functions of k relevant variables, J. Comput. Syst. Sci. 69 (2004), 421–434.

    Article  MATH  MathSciNet  Google Scholar 

  11. [11]

    N. Nisan and M. Szegedy: On the degree of Boolean functions as real polynomials. Computational Complexity 4 (1994), 301–313.

    Article  MATH  MathSciNet  Google Scholar 

  12. [12]

    T. Siegenthaler: Correlation-immunity of nonlinear combining functions for cryptographic applications, IEEE TIT 30 (1984), 776–780.

    MATH  MathSciNet  Google Scholar 

  13. [13]

    A. Shpilka and A. Tal: On the minimal fourier degree of symmetric boolean functions, in: Proceedings of the 26th Annual IEEE Conference on Computational Complexit (CCC), 200–209, 2011.

    Google Scholar 

  14. [14]

    G. Valiant: Finding correlations in subquadratic time, with applications to learning parities and juntas, in: FOCS, 11–20, 2012.

    Google Scholar 

  15. [15]

    G. Z. Xiao and J. L. Massey: A spectral characterization of correlation-immune combining functions, IEEE TIT 34 (1988), 569–571.

    MATH  MathSciNet  Google Scholar 

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Corresponding author

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