Abstract

One of the problems of modern discrete mathematics is Dedekind’s problem on the number of monotone Boolean functions. For other precomplete classes, general formulas for the number of functions of the classes had been found, but it has not been found so far for the class of monotone Boolean functions. Within the framework of this problem, there are problems of a lower level. One of them is the absence of a general formula for the number of Boolean functions of intersection \(MS\) of two classes—the class of monotone functions and the class of self-dual functions. In the paper, new lower bounds are proposed for estimating the cardinality of the intersection for both an even and an odd number of variables. It is shown that the majority voting function of an odd number of variables is monotone and self-dual. The majority voting function of an even number of variables is determined. Free voting functions, which are functions with fictitious variables similar in properties to majority voting functions, are introduced. Then the union of a set of majority voting functions and a set of free voting functions is considered, and the cardinality of this union is calculated. The resulting value of the cardinality is proposed as a lower bound for \(\left| {MS} \right|\). For the class \(MS\) of monotone self-dual functions of an even number of variables, the lower bound is improved over the bounds proposed earlier, and for functions of an odd number of variables, the lower bound for \(\left| {MS} \right|\) is presented for the first time.

中文翻译：

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应用多数表决函数估计单调自对偶布尔函数的个数

摘要

现代离散数学的问题之一是戴德金关于单调布尔函数数的问题。对于其他预完备类，已经找到了该类函数个数的通用公式，但对于单调布尔函数类，迄今为止尚未找到。在这个问题的框架内，还有一些较低层次的问题。其中之一是缺乏两类——单调函数类和自对偶函数类——交集\(MS\)的布尔函数个数的通式。在本文中，提出了新的下界来估计偶数和奇数变量的交集基数。结果表明，奇数个变量的多数投票函数是单调且自对偶的。确定偶数个变量的多数投票函数。引入了自由投票函数，它是具有与多数投票函数类似的属性的虚拟变量的函数。然后考虑一组多数投票函数和一组自由投票函数的并集，并计算该并集的基数。建议将基数的结果值作为\(\left| {MS} \right|\) 的下界。对于偶数个变量的单调自对偶函数类\(MS\)，其下界比之前提出的界限有所改进，而对于奇数个变量的函数，\(\left | {MS} \right|\)首次呈现。