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)


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) .


Output Node Input Node Threshold Function Planar Circuit Prime Implicant 
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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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