Some notes on threshold circuits, and multiplication in depth 4

  • Thomas Hofmeister
  • Walter Hohberg
  • Susanne Köhling
Part of the Lecture Notes in Computer Science book series (LNCS, volume 529)


It is known that two n-bit numbers can be added by polynomial-size ∧- ∨-circuits of depth 3 [W2] and multiplied by threshold-circuits of depth 10. Two tricks for the reduction of depth in threshold circuits are formalized. Further, threshold circuits for the addition of m numbers of length n, mn, in depth 3 (with O(n2) gates and O(nm3 + n3m) wires) and for the multiplication of two n-bit numbers in depth 4 (O(n2) gates and O(n4) wires) are presented.


circuits computational complexity design of algorithms multiplication parallel algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

5. References

  1. [A]
    E. Allender, A Note on the Power of Threshold Circuits, Tech. Rept., Dep. of Computer Science, Rutgers University, New Brunswick (1989).Google Scholar
  2. [BS]
    J. Bruck and R. Smolensky, Polynomial Threshold Functions, AC0Functions and Spectral Norms, Tech. Rept. RJ 7140 (67387) 11/15/89, IBM San Jose, CA (1989).Google Scholar
  3. [CSV]
    A. K. Chandra, L. Stockmeyer, and U. Vishkin, Constant Depth Reducibility, SIAM J. Comput. 13 (1984) 423–439.Google Scholar
  4. [FSS]
    M. Furst, J. B. Saxe, and M. Sipser, Parity, Circuits, and the Polynomial-Time Hierarchy, Mathematical Systems Theory 17 (1984) 13–27.Google Scholar
  5. [HMPST]
    A. Hajnal, W. Maass, P. Pudlak, M. Szegedy, and G. Turan, Threshold Circuits of Bounded Depth, Proc. 28. IEEE Symp. on Foundations of Computer Science (1987) 99–110.Google Scholar
  6. [H]
    S. T. Hu, Threshold Logic, University of California Press, Berkeley, 1965.Google Scholar
  7. [M]
    S. Muroga, Threshold Logic and Its Applications, John Wiley, New York, 1971.Google Scholar
  8. [PS]
    I. Parberry and G. Schnitger, Parallel Computation with Threshold Functions, J. Comput. System Sci. 36 (1988) 278–302.Google Scholar
  9. [R]
    A. A. Razborov, Lower Bounds on the Size of Bounded-Depth Networks over the Basis {∧,⊕}, Tech. Rept., Moscow State Univ. (1986).Google Scholar
  10. [SB]
    K.-Y. Siu and J. Bruck, On the Dynamic Range of Linear Threshold Functions, Tech. Rept. RJ 7237 (68004) 1/4/90, IBM, San Jose, CA (1990).Google Scholar
  11. [W1]
    I. Wegener, The Complexity of the Parity Function in Unbounded Fan-in, Unbounded Depth Circuits, to appear in Theoretical Computer Science, 1990.Google Scholar
  12. [W2]
    I. Wegener, Unbounded Fan-in Circuits, to appear in Advances in the Theory of Computation and Computational Mathematics, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1991

Authors and Affiliations

  • Thomas Hofmeister
    • 1
  • Walter Hohberg
    • 1
  • Susanne Köhling
    • 1
  1. 1.Lehrstuhl Informatik IIUniversität DortmundDortmund 50

Personalised recommendations