On the planar monotone computation of threshold functions

Abstract

Let T ⁽ⁿ⁾_k , 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 ⁽ⁿ⁾_k 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 ⁽ⁿ⁾₁ , T ⁽ⁿ⁾₂ and their duals T ⁽ⁿ⁾_n , T ⁽ⁿ⁾_n−1 respectively, can be realized by such highly restricted circuits. The complexity of planar monotone circuits for T ⁽ⁿ⁾₂ 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⁾_n−1 .