Abstract

The well-known lower bound for the maximum number of prime implicants of a Boolean function (the length of the reduced DNF) differs by \(\Theta (\sqrt {n}) \) times from the upper bound and is asymptotically attained at a symmetric belt function with belt width \(n/3 \). To study the properties of connected Boolean functions with many prime implicants, we introduce the notion of a locally extremal function in a certain neighborhood in terms of the number of prime implicants. Some estimates are obtained for the change in the number of prime implicants as the values of the belt function range over a \(d \)-neighborhood. We prove that the belt function for which the belt width and the number of the lower layer of unit vertices are asymptotically equal to \(n/3 \) is locally extremal in some neighborhood for \(d \le \Theta (n) \) and not locally extremal if \(d \ge 2^{\Theta (n)} \). A similar statement is true for the functions that have prime implicants of different ranks. The local extremality property is preserved after applying some transformation to the Boolean function that preserves the distance between the vertices of the unit cube.

中文翻译：

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具有局部极值素数的连通布尔函数

摘要

布尔函数的最大素数的最大已知下限（缩减的DNF的长度）与上限的差为（\ Theta（\ sqrt {n}）\）倍，并且渐近地达到带宽度为\（n / 3 \）的对称带函数。为了研究具有许多素数蕴涵项的布尔函数的性质，我们以素数蕴涵数的形式介绍了某个邻域中局部极值函数的概念。由于带函数的值范围在\（d \）邻域内，因此得出了一些素数蕴含量变化的估计值 。我们证明了带函数，其中带宽度和单位顶点的下层数渐近等于 \（n / 3 \）在\（d \ le \ Theta（n）\）的某个邻域中是局部极值 ，而在\（d \ ge 2 ^ {\ Theta（n）} \）的情况下不是局部极值。对于具有不同等级的主要蕴涵的函数，类似的说法也适用。在对布尔函数进行一些转换后，保留了局部极值属性，该函数保留了单位立方体顶点之间的距离。