Abstract
Let T (n)k , 1≤k≤n, be the monotone symmetric Boolean function of n arguments defined by
T (n)k 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 (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 .
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© 1984 Springer-Verlag Berlin Heidelberg
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McColl, W.F. (1984). On the planar monotone computation of threshold functions. In: Mehlhorn, K. (eds) STACS 85. STACS 1985. Lecture Notes in Computer Science, vol 182. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0024011
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DOI: https://doi.org/10.1007/BFb0024011
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