On the Existence and Properties of Convex Extensions of Boolean Functions

Pub Date : 2024-07-05 DOI:10.1134/s0001434624030210

D. N. Barotov

引用次数: 0

Abstract

We study the problem of the existence of a convex extension of any Boolean function \(f(x_1,x_2,\dots,x_n)\) to the set \([0,1]^n\). A convex extension \(f_C(x_1,x_2,\dots,x_n)\) of an arbitrary Boolean function \(f(x_1,x_2,\dots,x_n)\) to the set \([0,1]^n\) is constructed. On the basis of the constructed convex extension \(f_C(x_1,x_2,\dots,x_n)\), it is proved that any Boolean function \(f(x_1,x_2,\dots,x_n)\) has infinitely many convex extensions to \([0,1]^n\). Moreover, it is proved constructively that, for any Boolean function \(f(x_1,x_2,\dots,x_n)\), there exists a unique function \(f_{DM}(x_1,x_2,\dots,x_n)\) being its maximal convex extensions to \([0,1]^n\).

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论布尔函数凸扩展的存在和性质

Abstract 我们研究了任意布尔函数 \(f(x_1,x_2,\dots,x_n))向集合 \([0,1]^n\)的凸扩展的存在性问题。构造了任意布尔函数 \(f(x_1,x_2,\dots,x_n)\) 到集合 \([0,1]^n\) 的凸扩展 \(f_C(x_1,x_2,\dots,x_n)\)。在所构造的凸扩展（f_C(x_1,x_2,\dots,x_n)）的基础上，证明了任何布尔函数（f(x_1,x_2,\dots,x_n)）都有无穷多个凸扩展到（[0,1]^n\ ）。此外，构造证明了对于任何布尔函数 \(f(x_1,x_2,\dots,x_n)\)，都存在一个唯一的函数 \(f_{DM}(x_1,x_2,\dots,x_n)\)，它是\([0,1]^n\)的最大凸扩展。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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