Polynomial Threshold Functions and Boolean Threshold Circuits

  • Kristoffer Arnsfelt Hansen
  • Vladimir V. Podolskii
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8087)


We initiate a comprehensive study of the complexity of computing Boolean functions by polynomial threshold functions (PTFs) on general Boolean domains. A typical example of a general Boolean domain is {1,2} n . We are mainly interested in the length (the number of monomials) of PTFs, with their degree and weight being of secondary interest.

First we motivate the study of PTFs over the {1,2} n domain by showing their close relation to depth two threshold circuits. In particular we show that PTFs of polynomial length and polynomial degree compute exactly the functions computed by polynomial size THR ∘ MAJ circuits. We note that known lower bounds for THR ∘ MAJ circuits extends to the likely strictly stronger model of PTFs. We also show that a “max-plus” version of PTFs are related to AC 0 ∘ THR circuits.

We exploit this connection to gain a better understanding of threshold circuits. In particular, we show that (super-logarithmic) lower bounds for 3-player randomized communication protocols with unbounded error would yield (super-polynomial) size lower bounds for THR ∘ THR circuits.

Finally, having thus motivated the model, we initiate structural studies of PTFs. These include relationships between weight and degree of PTFs, and a degree lower bound for PTFs of constant length.


Boolean Function Communication Complexity Sign Rank Threshold Function Constant Length 
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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  1. 1.
    Basu, S., Bhatnagar, N., Gopalan, P., Lipton, R.J.: Polynomials that sign represent parity and Descartes’ rule of signs. Computat. Complex. 17(3), 377–406 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beigel, R.: The polynomial method in circuit complexity. In: CCC 1993, pp. 82–95. IEEE Computer Society Press (1993)Google Scholar
  3. 3.
    Beigel, R.: Perceptrons, PP, and the polynomial hierarchy. Comput. Complex. 4(4), 339–349 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bruck, J., Smolensky, R.: Polynomial threshold functions, AC0 functions, and spectral norms. SIAM J. Comput. 21(1), 33–42 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Buhrman, H., Vereshchagin, N.K., de Wolf, R.: On computation and communication with small bias. In: CCC 2007, pp. 24–32 (2007)Google Scholar
  6. 6.
    Butkovič, P.: Max-linear Systems: Theory and Algorithms. Springer (2010)Google Scholar
  7. 7.
    Forster, J.: A linear lower bound on the unbounded error probabilistic communication complexity. J. Comput. Syst. Sci. 65(4), 612–625 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Forster, J., Krause, M., Lokam, S.V., Mubarakzjanov, R., Schmitt, N., Simon, H.U.: Relations between communication complexity, linear arrangements, and computational complexity. In: Hariharan, R., Mukund, M., Vinay, V. (eds.) FSTTCS 2001. LNCS, vol. 2245, pp. 171–182. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  9. 9.
    Goldmann, M.: On the power of a threshold gate at the top. Inform. Process. Lett. 63(6), 287–293 (1997)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Goldmann, M., Håstad, J., Razborov, A.A.: Majority gates vs. general weighted threshold gates. Comput. Complex. 2(4), 277–300 (1992)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hajnal, A., Maass, W., Pudlák, P., Szegedy, M., Turán, G.: Threshold circuits of bounded depth. J. Comput. Syst. Sci. 46(2), 129–154 (1993)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hansen, K.A., Miltersen, P.B.: Some meet-in-the-middle circuit lower bounds. In: Fiala, J., Koubek, V., Kratochvíl, J. (eds.) MFCS 2004. LNCS, vol. 3153, pp. 334–345. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Hansen, K.A., Podolskii, V.V.: Exact threshold circuits. In: CCC 2010, pp. 270–279. IEEE Computer Society (2010)Google Scholar
  14. 14.
    Hansen, K.A., Podolskii, V.V.: Polynomial threshold functions and boolean threshold circuits. ECCC TR13-021 (2013)Google Scholar
  15. 15.
    Håstad, J., Goldmann, M.: On the power of small-depth threshold circuits. Comput. Complex. 1, 113–129 (1991)CrossRefzbMATHGoogle Scholar
  16. 16.
    Klivans, A.R., O’Donnell, R., Servedio, R.A.: Learning intersections and thresholds of halfspaces. J. Comput. Syst. Sci. 68(4), 808–840 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klivans, A.R., Servedio, R.A.: Learning DNF in time \(2^{{\widetilde{O}}(n^{1/3})}\). J. Comput. Syst. Sci. 68(2), 303–318 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Krause, M., Pudlák, P.: On the computational power of depth-2 circuits with threshold and modulo gates. Theor. Comput. Sci. 174(1–2), 137–156 (1997)CrossRefzbMATHGoogle Scholar
  19. 19.
    Krause, M., Pudlák, P.: Computing boolean functions by polynomials and threshold circuits. Comput. Complex. 7(4), 346–370 (1998)CrossRefzbMATHGoogle Scholar
  20. 20.
    Minsky, M., Papert, S.: Perceptrons: An Introduction to Computational Geometry. MIT Press (1969)Google Scholar
  21. 21.
    Muroga, S.: Threshold Logic and its Applications. John Wiley & Sons, Inc. (1971)Google Scholar
  22. 22.
    Muroga, S., Toda, I., Takasu, S.: Theory of majority decision elements. Journal of the Franklin Institute 271, 376–418 (1961)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Nisan, N.: The communication complexity of threshold gates. In: Miklós, D., Szönyi, T., Sós, V.T. (ed.) Combinatorics, Paul Erdős is Eighty, Mathematical Studies, vol. 1, pp. 301–315. Bolyai Society (1993)Google Scholar
  24. 24.
    Paturi, R., Simon, J.: Probabilistic communication complexity. J. Comput. Syst. Sci. 33(1), 106–123 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Razborov, A., Wigderson, A.: n Ω(logn) lower bounds on the size of depth-3 threshold circuits with AND gates at the bottom. Inf. Proc. Lett. 45(6), 303–307 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Razborov, A.A., Sherstov, A.A.: The sign-rank of AC0. SIAM J. Comput. 39(5), 1833–1855 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Saks, M.E.: Slicing the hypercube. In: Walker, K. (ed.) Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 187. Cambridge University Press (1993)Google Scholar
  28. 28.
    Sherstov, A.A.: Communication lower bounds using dual polynomials. Bulletin of the EATCS 95, 59–93 (2008)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sherstov, A.A.: Separating AC0 from depth-2 majority circuits. SIAM J. Comput. 38(6), 2113–2129 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Speyer, D., Sturmfels, B.: Tropical Mathematics. ArXiv math/0408099 (2004),

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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Kristoffer Arnsfelt Hansen
    • 1
  • Vladimir V. Podolskii
    • 2
  1. 1.Aarhus UniversityDenmark
  2. 2.Steklov Mathematical InstituteRussia

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