computational complexity

, Volume 4, Issue 4, pp 301–313 | Cite as

On the degree of boolean functions as real polynomials

  • Noam Nisan
  • Mario Szegedy
Article

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.

Key words

Approximation block sensitivity Boolean functions Fourier degree 

Subject classifications

68Q05 68Q99 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Aspens, R. Beigel, M. Furst, and S. Rudich, On the expressive power of voting polynomials.Proc. Twenty-third Ann. ACM Symp. Theor. Comput., 1991, 402–409.Google Scholar
  2. [2]
    R. Beigel, N. Reingold, and D. Spielman,The perceptron striles back. Technical report YALEU/DCS/TR-813, Yale University, 1990.Google Scholar
  3. [3]
    J. Brook and R. Smolenski, Polynomial threshold functions,AC° functions and spectral norms.Proc. 31st Ann. IEEE Symp. Found. Comput. Sci., 1990, 632–642.Google Scholar
  4. [4]
    E. W. Cheney,Introduction to approximation theory. McGraw-Hill Book Co., 1966.Google Scholar
  5. [5]
    M. Dietzfelbinger, M. Kutylowski, R. Reischuk, Exact Time Bounds for Computing Boolean Functions on PRAMs Without Simultaneous Writes.Symp. Parallel Algorithms and Architecture, 1990, 125–137.Google Scholar
  6. [6]
    H. Ehlich andK. Zeller, Schwankung von Polynomen zwischen Gitterpunkten.Mathematische Zeitschrift 86 (1964), 41–44.Google Scholar
  7. [7]
    J. von zur Gathen and J. Roche, Polynomials with two values.Preprint (1993).Google Scholar
  8. [8]
    J. Kahn, G. Kalai, and N. Linial, The influence of variables on Boolean functions.Proc. 29th Ann. IEEE Symp. Found. Comput. Sci., 1988, 68–80.Google Scholar
  9. [9]
    N. Linial, Y. Mansour, N. Nisan, Constant depth circuits, Fourier transform and Learnability.Proc. 30th Ann. IEEE Symp. Found. Comput. Sci., 1989, 574–579.Google Scholar
  10. [10]
    E. Kushilevitz and Y. Mansour, Learning decision trees using the Fourier transform.Proc. Twenty-third Ann. ACM Symp. Theor. Comput., 1991, 455–464.Google Scholar
  11. [11]
    M. Minsky andS. Papert,Perceptrons. MIT press, Cambridge, 1988 (Expanded edition). First edition appeared in 1968.Google Scholar
  12. [12]
    N. Nisan, CREW PRAM's and decision trees.Proc. Twenty-first Ann. ACM Symp. Theor. Comput., 1989, 327–335.Google Scholar
  13. [13]
    A.A. Razborov, Lower bounds for the size of circuits of bounded depth with basis (and,xor).Math. Notes of the Academy of Sciences of the USSR 41(4) (1987), 333–338.Google Scholar
  14. [14]
    T. J. Rivlin andE. W. Cheney, A comparison of Uniform Approximations on an interval and a finite subset thereof.SIAM J. Numer. Anal. 3(2) (1966), 311–320.Google Scholar
  15. [15]
    R. Smolenski, Algebraic methods in the theory of lower bounds for Boolean circuit complexity.Proc. Nineteenth Ann. ACM Symp. Theor. Comput., 1987, 77–82.Google Scholar
  16. [16]
    J. T. Schwartz, Fast probabilistic Algorithms for verification of Polynomial identities.J. Assoc. Comput. Mach. 27 (1980), 701–717.Google Scholar
  17. [17]
    I. Wegener,The complexity of Boolean functions. Wiley-Teubner Series in Comput. Sci., New York-Stuttgart, 1987.Google Scholar

Copyright information

© Birkhäuser Verlag 1994

Authors and Affiliations

  • Noam Nisan
    • 1
  • Mario Szegedy
    • 2
  1. 1.The Hebrew UniversityJerusalemIsrael
  2. 2.AT & T Bell LaboratoriesMurray HillUSA

Personalised recommendations