Lower Bound on Weights of Large Degree Threshold Functions

  • Vladimir V. Podolskii
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7318)

Abstract

An integer polynomial p of n variables is called a threshold gate for the Boolean function f of n variables if for all x ∈ {0, 1} n f(x) = 1 if and only if p(x) ≥ 0. The weight of a threshold gate is the sum of its absolute values.

In this paper we study how large weight might be needed if we fix some function and some threshold degree. We prove \(2^{\Omega(2^{2n/5})}\) lower bound on this value. The best previous bound was \(2^{\Omega(2^{n/8})}\) [12].

In addition we present substantially simpler proof of the weaker \(2^{\Omega(2^{n/4})}\) lower bound. This proof is conceptually similar to other proofs of the bounds on weights of nonlinear threshold gates, but avoids a lot of technical details arising in other proofs. We hope that this proof will help to show the ideas behind the construction used to prove these lower bounds.

Keywords

Boolean Function Minimal Weight Complexity Measure Ordinal Number Circuit Complexity 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Babai, L., Hansen, K.A., Podolskii, V.V., Sun, X.: Weights of exact threshold functions. In: Hliněný, P., Kučera, A. (eds.) MFCS 2010. LNCS, vol. 6281, pp. 66–77. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Beigel, R.: Perceptrons, PP, and the polynomial hierarchy. Computational Complexity 4, 339–349 (1994)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Buhrman, H., Vereshchagin, N.K., de Wolf, R.: On computation and communication with small bias. In: Proc. of the 22nd Conf. on Computational Complexity (CCC), pp. 24–32 (2007)Google Scholar
  4. 4.
    Håstad, J.: On the size of weights for threshold gates. SIAM J. Discret. Math. 7(3), 484–492 (1994)MATHCrossRefGoogle Scholar
  5. 5.
    Klivans, A.R., Servedio, R.A.: Learning DNF in time \(2^{\tilde O(n^{1/3})}\). J. Comput. Syst. Sci. 68(2), 303–318 (2004)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Krause, M., Pudlák, P.: Computing Boolean functions by polynomials and threshold circuits. Comput. Complex. 7(4), 346–370 (1998)MATHCrossRefGoogle Scholar
  7. 7.
    Minsky, M.L., Papert, S.A.: Perceptrons: Expanded edition. MIT Press, Cambridge (1988)MATHGoogle Scholar
  8. 8.
    Muroga, S.: Threshold logic and its applications. Wiley Interscience, Chichester (1971)MATHGoogle Scholar
  9. 9.
    Muroga, S., Toda, I., Takasu, S.: Theory of majority decision elements. Journal of the Franklin Institute 271(5), 376–418 (1961)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Myhill, J., Kautz, W.H.: On the size of weights required for linear-input switching functions. IRE Trans. on Electronic Computers 10(2), 288–290 (1961)CrossRefGoogle Scholar
  11. 11.
    Parberry, I.: Circuit complexity and neural networks. MIT Press, Cambridge (1994)MATHGoogle Scholar
  12. 12.
    Podolskii, V.V.: Perceptrons of large weight. Probl. Inf. Transm. 45, 46–53 (2009)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Podolskii, V.V., Sherstov, A.A.: A small decrease in the degree of a polynomial with a given sign function can exponentially increase its weight and length. Mathematical Notes 87, 860–873 (2010)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Razborov, A.A.: On small depth threshold circuits. In: Proceedings of the Third Scandinavian Workshop on Algorithm Theory, pp. 42–52. Springer, London (1992)Google Scholar
  15. 15.
    Saks, M.E.: Slicing the hypercube. Surveys in Combinatorics, pp. 211–255 (1993)Google Scholar
  16. 16.
    Sherstov, A.A.: Communication lower bounds using dual polynomials. Bulletin of the EATCS 95, 59–93 (2008)MathSciNetMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Vladimir V. Podolskii
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

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