Advertisement

Proving lower bounds on the monotone complexity of Boolean functions

  • Ingo Wegener
Section VI: Complexity Of Boolean Functions
Part of the Lecture Notes in Computer Science book series (LNCS, volume 171)

Abstract

The hardware of computers may be well modelled by Boolean networks computing Boolean functions f:{0, 1}n → {0, 1}m. Though by counting arguments it is easy to show that nearly all Boolean functions may be computed only by networks of exponential size until now one is not able to prove nonlinear lower bounds for explicitly defined functions. Therefore one has restricted oneself to monotone networks where only -and ∨-gates are available. We repeat shortly the known methods for the proof of nonlinear lower bounds for the monotone network complexity of functions of n inputs and outputs. Afterwards we describe the method of using value functions, which was introduced by the author [13] to prove a bound of size n2/logn, until now the largest bound of this kind. For the Boolean convolution we present a new Ω(n3/2)-lower bound proved recently by one of my students (Weiß [14]). There one combines the elimination method and some information flow arguments. Finally we discuss how one may combine the above mentioned methods in order to get perhaps better bounds for the Boolean convolution.

Keywords

Boolean Function Boolean Network Elimination Method World Record Boolean Matrix 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

6. References

  1. [01]
    N. Blum: An Ω(n4/3) lower bound on the monotone network complexity of the n-th degree convolution, 22-th Symp. on Foundations of Comp. Science (1981) 101–108Google Scholar
  2. [02]
    E.A. Lamagna/J.E. Savage: Combinational complexity of some monotone functions, Proc. 15th SWAT Conference, New Orleans (1974) 140–144Google Scholar
  3. [03]
    K. Mehlhorn: Some remarks on Boolean sums, Acta Informatica 12 (1979), 371–375Google Scholar
  4. [04]
    K. Mehlhorn/Z. Galil: Monotone switching circuits and Boolean matrix product, Computing 16 (1976) 99–111Google Scholar
  5. [05]
    E.I. Nechiporuk: On a Boolean matrix, Systems Res.Theory 21 (1971) 236–239Google Scholar
  6. [06]
    M.S. Paterson: Complexity of monotone networks for Boolean matrix product, Theoret. Comput. Sci. 1 (1975) 13–20Google Scholar
  7. [07]
    N. Pippenger: On another Boolean matrix, IBM Research Report 6914 (1977)Google Scholar
  8. [08]
    N. Pippenger/L.G. Valiant: Shifting graphs and their applications, J.ACM 23 (1976) 423–432Google Scholar
  9. [09]
    V.R. Pratt: The power of negative thinking in multiplying Boolean matrices, SIAM J.Comput. 4 (1975) 326–330Google Scholar
  10. [10]
    R.E. Tarjan: Complexity of monotone networks for computing conjunctions, Ann. Discrete Math. 2 (1978) 121–133Google Scholar
  11. [11]
    I. Wegener: Switching functions whose monotone complexity is nearly quadratic, Theoret. Comput. Sci. 9 (1979) 83–97Google Scholar
  12. [12]
    I. Wegener: A new lower bound on the monotone network complexity of Boolean sums, Acta Informat. 13 (1980) 109–114Google Scholar
  13. [13]
    I. Wegener: Boolean functions whose monotone complexity is of size n2/logn, Theoret. Comput. Sci. 21 (1982) 213–224Google Scholar
  14. [14]
    J. Weiß: Eine n3/2-untere Schranke für die Boolesche Konvolution Diplomarbeit, Univ. Bielefeld (1982)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • Ingo Wegener
    • 1
  1. 1.FB-20 Informatik, Johann Wolfgang Goethe-UniversitätFrankfurt a.M.Fed. Rep. of Germany

Personalised recommendations