On the Existence and Properties of Convex Extensions of Boolean Functions

IF 0.6 4区数学 Q3 MATHEMATICS

Mathematical Notes 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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来源期刊

Mathematical Notes 数学-数学

CiteScore

0.90

自引率

16.70%

发文量

179

审稿时长

24 months

期刊介绍： Mathematical Notes is a journal that publishes research papers and review articles in modern algebra, geometry and number theory, functional analysis, logic, set and measure theory, topology, probability and stochastics, differential and noncommutative geometry, operator and group theory, asymptotic and approximation methods, mathematical finance, linear and nonlinear equations, ergodic and spectral theory, operator algebras, and other related theoretical fields. It also presents rigorous results in mathematical physics.