The expressive power of voting polynomials

Abstract

We consider the problem of approximating a Boolean functionf∶{0,1}n→{0,1} by the sign of an integer polynomialp of degreek. For us, a polynomialp(x) predicts the value off(x) if, wheneverp(x)≥0,f(x)=1, and wheneverp(x)<0,f(x)=0. A low-degree polynomialp is a good approximator forf if it predictsf at almost all points. Given a positive integerk, and a Boolean functionf, we ask, “how good is the best degreek approximation tof?” We introduce a new lower bound technique which applies to any Boolean function. We show that the lower bound technique yields tight bounds in the casef is parity. Minsky and Papert [10] proved that a perceptron cannot compute parity; our bounds indicate exactly how well a perceptron canapproximate it. As a consequence, we are able to give the first correct proof that, for a random oracleA, PPA is properly contained in PSPACEA. We are also able to prove the old AC0 exponential-size lower bounds in a new way. This allows us to prove the new result that an AC0 circuit with one majority gate cannot approximate parity. Our proof depends only on basic properties of integer polynomials.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    R. Beigel: Relativized counting classes: Relations among thresholds, parity, and mods.Journal of Computer and System Sciences 42(1) February 1991, 76–96.

    Google Scholar 

  2. [2]

    R. Beigel, N. Reingold, andD. Spielman: The perceptron strikes back.Proceedings of the 6th Annual Conference on Structure in Complexity Theory (1991), 286–291.

  3. [3]

    C. H. Bennett, andJ. Gill: Relative to a random oracleA, PA≠NPA≠co-NPA with probability 1.SIAM Journal on Computing 10(1) February 1981, 96–113.

    Google Scholar 

  4. [4]

    J. Bruck: Harmonic analysis of polynomial threshold functions.SIAM Journal on Discrete Mathematics 3(2) May 1990, 168–177.

    Google Scholar 

  5. [5]

    J. Bruck, andR. Smolensky: Polynomial threshold functions, AC0 functions and spectral norms. InProceedings of the 31st Annual Symposium on Foundations of Computer Science (1990), 632–641.

  6. [6]

    V. Chvátal:Linear Programming. W. H. Freeman and Company, (1983).

  7. [7]

    M. Furst, J. Saxe, andM. Sipser: Parity, circuits and the polynomial time hierarchy.Mathematical Systems Theory 17 (1984), 13–27.

    Google Scholar 

  8. [8]

    C. Gotsman: On boolean functions, polynomials, and algebraic threshold functions. Unpublished manuscript.

  9. [9]

    N. Linial, Y. Mansour andN. Nisan: Constant depth circuits, fourier transform, and learnability. InProceedings of the 30th Annual Symposium on Foundations of Computer Science (1989), 574–579.

  10. [10]

    M. L. Minsky, andS. Papert:Perceptrons. MIT Press, Cambridge, MA, (1988), Expanded Edition. The first edition appeared in 1968.

    Google Scholar 

  11. [11]

    R. Paturi, andM. E. Saks: On threshold circuits for parity. InProceedings of the 31st Annual Symposium on Foundations of Computer Science (1990), 397–404.

  12. [12]

    A. A. Razborov: Lower bounds for the size of circuits of bounded depth with basis {⋏,⊕}.Math. Notes of the Academy of Sciences of the USSR 41(4) September 1987, 333–338.

    Google Scholar 

  13. [13]

    A. Schrijver:Theory of Linear and Integer Programming. John Wiley and Sons., (1986).

  14. [14]

    R. Smolensky: Algebraic methods in the theory of lower bounds for boolean circuit complexity. InProceedings of the 19th Annual ACM Symposium on Theory of Computing (1987), 77–82.

  15. [15]

    E. Stiemke: Über positive Lösungen homogener linearer Gleichungen.Mathematische Annalen 76 (1915), 340–342.

    Google Scholar 

  16. [16]

    J. Tarui: Randomized polynomials, threshold circuits, and the polynomial hierarchy. Manuscript, August 1990.

  17. [17]

    L. G. Valiant, andV. V. Vazirani: NP is as easy as detecting unique solutions. InProceedings of the 17th Annual ACM Symposium on Theory of Computing (1985).

  18. [18]

    A. Yao: Separating the polynomial-time hierarchy by oracles. InProceedings of the 26th Annual IEEE Symposium on Foundations of Computer Science (1985), 1–10.

Download references

Author information

Affiliations

Authors

Additional information

Supported in part by NSF grants CCR-8808949 and CCR-8958528.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Aspnes, J., Beigel, R., Furst, M. et al. The expressive power of voting polynomials. Combinatorica 14, 135–148 (1994). https://doi.org/10.1007/BF01215346

Download citation

AMS subject classification code (1991)

  • 68 Q 15
  • 05 E 35
  • 42 C 10
  • 94 C 10