On small depth threshold circuits

  • Alexander A. Razborov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 621)


In this talk we will consider various classes defined by small depth polynomial size circuits which contain threshold gates and parity gates. We will describe various inclusions between many classes defined in this way and also classes whose definitions rely upon spectral properties of Boolean functions.


Boolean Function Polynomial Size Majority Gate Threshold Gate Parity Gate 
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. [AB91]
    N. Alon and J. Bruck. Explicit constructions of depth-2 majority circuits for comparison and addition. Technical Report RJ 8300 (75661), IBM Research Division, August 1991.Google Scholar
  2. [Bar86]
    D. A. Barrington. A note on a theorem of Razborov. Technical report, University of Massachusetts, 1986.Google Scholar
  3. [BCH86]
    P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994–1003, 1986.Google Scholar
  4. [BOH90]
    Y. Brandman, A. Orlitsky, and J. Hennesy. A spectral lower bound technique for the size of decision trees and two-level AND/OR circuits. IEEE Transactions on Computers, 39(2):282–287, February 1990.Google Scholar
  5. [Bru90]
    J. Bruck. Harmonic analysis of polynomial threshold functions. SIAM Journal on Discrete Mathematics, 3(2):168–177, May 1990.Google Scholar
  6. [BS92]
    J. Bruck and R. Smolensky. Polynomial threshold functions, AC 0 functions and spectral norms. SIAM Journal on Computing, 21(1):33–42, February 1992.Google Scholar
  7. [CSV84]
    A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13:423–439, 1984.Google Scholar
  8. [GHR92]
    M. Goldmann, J. Håstad, and A. Razborov. Majority gates vs. general weighted threshold gates. In Proceedings of the 7 th Structure in Complexity Theory Annual Conference, 1992.Google Scholar
  9. [HG90]
    J. Håstad and M. Goldmann. On the power of small-depth threshold circuits. In Proceedings of the 31st IEEE FOCS, pages 610–618, 1990.Google Scholar
  10. [HHK91]
    T. Hofmeister, W. Honberg, and S. Köling. Some notes on threshold circuits and multiplication in depth 4. Information Processing Letters, 39:219–225, 1991.Google Scholar
  11. [HKP91]
    J. Hertz, R. Krogh, and A. Palmer. An Introduction to the Theory of Neural Computation. Addison-Wesley, 1991.Google Scholar
  12. [HMP+87]
    A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and G. Turán. Threshold circuits of bounded depth. In Proceedings of 28th IEEE FOCS, pages 99–110, 1987.Google Scholar
  13. [KKN88]
    J. Kahn, G. Kalai, and Linial N. The influence of variables on Boolean functions. In Proceedings of the 29 th IEEE Symposium on Foundations of Computer Science, pages 68–80, 1988.Google Scholar
  14. [KM91]
    E. Kushilevitz and Y. Mansour. Learning decision trees using the Fourier spectrum. In Proceedings of the 23rd ACM STOC, pages 455–464, 1991.Google Scholar
  15. [Kra91]
    M. Krause. Geometric arguments yield better bounds for threshold circuits and distributed computing. In 6-th Structure in Complexity Theory Conference, pages 314–322, 1991.Google Scholar
  16. [KW91]
    M. Krause and S. Waack. Variation ranks of communication matrices and lower bounds for depth two circuits having symmetric gates with unbounded fan-in. In Proceedings of the 32th IEEE Symposium on Foundations of Computer Science, pages 777–782, 1991.Google Scholar
  17. [LMN89]
    N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transforms and learnability. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pages 574–579, 1989.Google Scholar
  18. [MK61]
    J. Myhill and W.H. Kautz. On the size of weights required for linear-input switching functions. IRE Trans. on Electronic Computers, EC10(2):288–290, June 1961.Google Scholar
  19. [MSS91]
    W. Maass, G. Schnitger, and E. Sontag. On the computational power of sigmoid versus boolean threshold circuits. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science, pages 767–776, 1991.Google Scholar
  20. [Pip87]
    N. Pippenger. The complexity of computations by networks. IBM J. Res. Develop., 31:235–243, 1987.Google Scholar
  21. [Raz87]
    A. Razborov. Lower bounds on the size of bounded-depth networks over a complete basis with logical addition. Mathematical Notes of the Academy of Sciences of the USSR, 41(4):598–607, 1987. English translation in 41:4, pages 333–338.Google Scholar
  22. [SB91]
    K.-I. Siu and J. Bruck. On the power of threshold circuits with small weights. SIAM Journal on Discrete Mathematics, 4(3):423–435, 1991.Google Scholar
  23. [SBKH91]
    K.-I. Siu, J. Bruck, T. Kailath, and T. Hofmeister. Depth-efficient neural networks for division and related problems. Technical Report RJ 7946, IBM Research, January 1991. To appear in IEEE Trans. Information Theory.Google Scholar
  24. [Smo87]
    R. Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In Proceedings of the 19th ACM Symposium on Theory of Computing, pages 77–82, 1987.Google Scholar
  25. [SR92]
    K.-Y. Siu and V. Roychowdhury. On optimal depth threshold circuits for multiplication and related problems. Manuscript, 1992.Google Scholar
  26. [Yao79]
    A. Yao. Some complexity questions related to distributive computing. In Proceedings of the 11th ACM STOC, pages 209–213, 1979.Google Scholar
  27. [Yao90]
    A. Yao. On ACC and threshold circuits. In Proceedings of the 31th IEEE FOCS, pages 619–627, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Alexander A. Razborov
    • 1
  1. 1.Steklov Mathematical InstituteMoscowRussia

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