On extremal elements and the cardinality of the set of continuously differentiable convex extensions of a Boolean function
In this paper we study the existence of the maximal and minimal elements of the set of continuously differentiable convex extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ and the cardinality of the set of continuously differentiable convex extensions to $[0,1]^n$...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Yaroslavl State University
2025-06-01
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| Series: | Моделирование и анализ информационных систем |
| Subjects: | |
| Online Access: | https://www.mais-journal.ru/jour/article/view/1935 |
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| Summary: | In this paper we study the existence of the maximal and minimal elements of the set of continuously differentiable convex extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ and the cardinality of the set of continuously differentiable convex extensions to $[0,1]^n$ of the Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$. As a result of the study, it was established that the cardinality of the set of continuously differentiable convex extensions to $[0,1]^n$ of an arbitrary Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ is equal to the continuum. It is argued that for any Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$, there is no minimal element among its continuously differentiable convex extensions to $[0,1]^n$. It is proved that for any Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$, the set of its continuously differentiable convex extensions to $[0,1]^n$ has a maximal element only if the number of essential variables of the given Boolean function $f_{B}(x_1,x_2,\ldots,x_n)$ is less than 2. |
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| ISSN: | 1818-1015 2313-5417 |