# On the planar monotone computation of threshold functions

Preliminary version

Conference paper

First Online:

## Abstract

Let

*T*_{k}^{(n)}, 1≤*k*≤*n*, be the monotone symmetric Boolean function of*n*arguments defined by$$T_k^{\left( n \right)} \left( {x_1 , x_2 ,....,x_n } \right) = 1iff\sum\limits_{i = 1}^n {x_1 \geqslant k} .$$

*T* _{k} ^{(n)} is called the **k**^{th} threshold function. In this paper we consider the problem of realizing threshold functions by planar monotone circuits. It is shown that for *n*≥5, only *T* _{1} ^{(n)} , *T* _{2} ^{(n)} and their duals *T* _{n} ^{(n)} , *T* _{n−1} ^{(n)} respectively, can be realized by such highly restricted circuits. The complexity of planar monotone circuits for *T* _{2} ^{(n)} is also investigated. It is shown that any such circuit must be of size at least *n*^{2}−3 and of depth at least 2*n*−3+ [*log*_{2}(*n*−1)], and that both of these bounds can be simultaneously achieved. By duality, these results also hold for *T* _{n−1} ^{(n)} .

## Keywords

Output Node Input Node Threshold Function Planar Circuit Prime Implicant
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.

## References

- [1]L.M.Adleman. The circuit complexity of the threshold 2 function. Unpublished manuscript, Dept. of Mathematics, M.I.T. (1979).Google Scholar
- [2]P.A.Bloniarz. The complexity of monotone Boolean functions and an algorithm for finding shortest paths in a graph. (Ph.D. thesis.)
*Technical Report No.*238, Laboratory for Computer Science, M.I.T. (1979).Google Scholar - [3]P.E.Dunne. A 2.5
*n*lower bound on the monotone network complexity of*T*_{3}^{n}.*Theory of Computation Report No.*62, University of Warwick (1984).Google Scholar - [4]P.E. Dunne. Lower bounds on the monotone network complexity of threshold functions. Unpublished manuscript, Dept. of Computer Science, University of Warwick (1984).Google Scholar
- [5]M.J. Fischer, A.R. Meyer and M.S. Paterson. Ω(
*n*log*n*) lower bounds on the length of Boolean formulas.*SIAM J. Comput.*11 (1982), 416–427.Google Scholar - [6]J.Friedman. Constructing
*O*(*n*log*n*) size monotone formulae for the k-th elementary symmetric polynomial of*n*Boolean variables. To appear in*Proc.*25*th IEEE Symp. on Foundations of Computer Science*(1984).Google Scholar - [7]L.M. Goldschlager. A space efficient algorithm for the monotone planar circuit value problem.
*Inf. Proc. Lett.*10 (1980), 25–27.Google Scholar - [8]L.S. Khasin. Complexity bounds for the realization of monotonic symmetrical functions by means of formulas in the basis V,&,-.
*Dokl. Akad. Nauk. SSSR*189 (1969), 752–755;*Soviet Phys. Dokl.*14 (1970), 1149–1151.Google Scholar - [9]V.M. Khrapchenko. Method of determining lower bounds for the complexity of P-schemes.
*Mat. Zametki*10 (1971), 83–92Google Scholar - [9]a
*Mathematical Notes of the Academy of Sciences of the USSR*10 (1971), 474–479.Google Scholar - [10]M. Kleiman and N. Pippenger. An explicit construction of short monotone formulae for the monotone symmetric functions.
*Theoret. Comput. Sci.*7 (1978), 325–332.Google Scholar - [11]R.E. Krichevskii. Complexity of contact circuits realizing a function of logical algebra.
*Dokl. Akad. Nauk SSSR*151 (1963), 803–806;*Soviet Phys. Dokl.*8 (1964), 770–772.Google Scholar - [12]E.A. Lamagna. The complexity of monotone networks for certain bilinear forms, routing problems, sorting, and merging.
*IEEE Trans. Computers C*-28 (1979), 773–782.Google Scholar - [13]R.J. Lipton and R.E. Tarjan. Applications of a planar separator theorem.
*SIAM J. Comput.*9 (1980), 615–627.Google Scholar - [14]W.F. McColl. Planar crossovers.
*IEEE Trans. Computers C*-30 (1981), 223–225.Google Scholar - [15]D.E. Muller and F.P. Preparata. Bounds to complexities of networks for sorting and for switching.
*J. Assoc. Comput. Mach.*22 (1975), 195–201.Google Scholar - [16]M.S.P. Paterson. New bounds on formula size.
*Proc. 3rd GI Conf. Theoret. Comput. Sci.*, Lecture Notes in Computer Science Vol.48, Springer-Verlag (1977), 17–26.Google Scholar - [17]L.J. Stockmeyer. On the combinational complexity of certain symmetric Boolean functions.
*Math. Syst. Theory*10 (1977), 323–336.Google Scholar - [18]L. G. Valiant. Short monotone formulae for the majority function. To appear in
*J. of Algorithms*.Google Scholar - [19]A.C. Yao. Bounds on selection networks.
*SIAM J. Comput.*9 (1980), 566–582.Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 1984