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On the set of continuously differentiable concave extensions of a Boolean function
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This paper is devoted to the study of the existence of extremal elements of the set of continuously differentiable concave extensions to the set 〖[0,1]〗^n of an arbitrary Boolean function f_B (x_1,…,x_n ), as well as finding the cardinality of the set of continuously differentiable concave extensions to 〖[0,1]〗^n of the Boolean function f_B (x_1,…,x_n ). As a result of the study, it is proved that, firstly, for any Boolean function f_B (x_1,…,x_n ) among its continuously differentiable concave extensions to 〖[0,1]〗^n there is no maximal element, secondly, if the Boolean function f_B (x_1,…,x_n ) has more than one essential variable, then among its continuously differentiable concave extensions to 〖[0,1]〗^n there is no minimal element, and if the Boolean function is constant or has only one essential variable, then among its continuously differentiable concave extensions to 〖[0,1]〗^n there is a unique minimal element, the explicit form of which is given in the paper. It was also established that the cardinality of the set of continuously differentiable concave extensions to 〖[0,1]〗^n of an arbitrary Boolean function f_B (x_1,…,x_n ) is equal to the continuum.
Tambov State University - G.R. Derzhavin
Title: On the set of continuously differentiable concave extensions of a Boolean function
Description:
This paper is devoted to the study of the existence of extremal elements of the set of continuously differentiable concave extensions to the set 〖[0,1]〗^n of an arbitrary Boolean function f_B (x_1,…,x_n ), as well as finding the cardinality of the set of continuously differentiable concave extensions to 〖[0,1]〗^n of the Boolean function f_B (x_1,…,x_n ).
As a result of the study, it is proved that, firstly, for any Boolean function f_B (x_1,…,x_n ) among its continuously differentiable concave extensions to 〖[0,1]〗^n there is no maximal element, secondly, if the Boolean function f_B (x_1,…,x_n ) has more than one essential variable, then among its continuously differentiable concave extensions to 〖[0,1]〗^n there is no minimal element, and if the Boolean function is constant or has only one essential variable, then among its continuously differentiable concave extensions to 〖[0,1]〗^n there is a unique minimal element, the explicit form of which is given in the paper.
It was also established that the cardinality of the set of continuously differentiable concave extensions to 〖[0,1]〗^n of an arbitrary Boolean function f_B (x_1,…,x_n ) is equal to the continuum.
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