On the degree of boolean functions as real polynomials

Abstract

Every Boolean function may be represented as a real polynomial. In this paper, we characterize the degree of this polynomial in terms of certain combinatorial properties of the Boolean function.

Our first result is a tight lower bound of Ω(logn) on the degree needed to represent any Boolean function that depends onn variables.

Our second result states that for every Boolean functionf, the following measures are all polynomially related:

  • o The decision tree complexity off.

  • o The degree of the polynomial representingf.

  • o The smallest degree of a polynomialapproximating f in theL max norm.

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Nisan, N., Szegedy, M. On the degree of boolean functions as real polynomials. Comput Complexity 4, 301–313 (1994). https://doi.org/10.1007/BF01263419

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Key words

  • Approximation
  • block sensitivity
  • Boolean functions
  • Fourier degree

Subject classifications

  • 68Q05
  • 68Q99