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Convex Continuation of a Boolean Function and Its Applications

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Abstract

A convex continuation of an arbitrary Boolean function to the set \( [0,1]^n \) is constructed. Moreover, it is proved that for any Boolean function \( f(x_1,x_2,\dots ,x_n) \) that has no neighboring points on the set \( \mathrm{supp} f \), the constructed function \( f_C(x_1,x_2, \dots ,x_n) \) is the only totally maximally convex continuation to \( [0,1]^n \). Based on this, in particular, it is constructively stated that the problem of solving an arbitrary system of Boolean equations can be reduced to the problem of minimizing a function any local minimum of which in the desired region is a global minimum, and thus for this problem the problem of local minima is completely resolved.

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Funding

This work was supported by ongoing institutional funding. No additional grants to carry out or direct this particular research were obtained.

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Authors and Affiliations

  1. Financial University under the Government of the Russian Federation, Moscow, 109456, Russia

    D. N. Barotov

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  1. D. N. Barotov
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Correspondence to D. N. Barotov.

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Translated by V. Potapchouck

CONFLICT OF INTEREST. The author of this work declares that he has no conflicts of interest.

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Barotov, D.N. Convex Continuation of a Boolean Function and Its Applications. J. Appl. Ind. Math. 18, 1–9 (2024). https://doi.org/10.1134/S1990478924010010

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  • DOI: https://doi.org/10.1134/S1990478924010010

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