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On the planar monotone computation of threshold functions

Preliminary version
  • W. F. McColl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 182)

Abstract

Let T k (n) , 1≤kn, 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 kth 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 n2−3 and of depth at least 2n−3+ [log2(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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1984

Authors and Affiliations

  • W. F. McColl
    • 1
  1. 1.Department of Computer ScienceUniversity of WarwickCoventryEngland

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