Geodesic
Application of election functions to estimate the number of monotone self-dual boolean functions
Modelirovanie i analiz informacionnyh sistem, Tome 29 (2022) no. 2, pp. 78-91.

Voir la notice de l'article provenant de la source Math-Net.Ru

One of the problems of modern discrete mathematics is R. Dedekind problem on the number of monotone boolean functions. For other precomplete classes, general formulas for the number of functions of the classes had been found, but it has not been found so far for the class of monotone boolean functions. Within the framework of this problem, there are problems of a lower level. One of them is the absence of a general formula for the number of boolean functions of intersection $MS$ of two classes — the class of monotone functions and the class of self-dual functions. In the paper, new lower bounds are proposed for estimating the cardinality of the intersection for both an even and an odd number of variables. It is shown that the election function of an odd number of variables is monotone and self-dual. The election function of an even number of variables is determined. Free election functions, which are functions with fictitious variables similar in properties to election functions, are introduced. Then the union of a set of election functions and a set of free election functions is considered, and the cardinality of this union is calculated. The resulting value of the cardinality is proposed as a lower bound for $|MS|$. For the class $MS$ of monotone self-dual functions of an even number of variables, the lower bound is improved over the bounds proposed earlier, and for functions of an odd number of variables, the lower bound for $|MS|$ is presented for the first time.
Keywords: functions of election, self-dual boolean functions, monotone boolean functions, the Dedekind problem, boolean functions with fictitious variables, functions of free election, equilibrium sets, disjunctive normal form. 
@article{MAIS_2022_29_2_a0,
     author = {L. Yu. Bystrov and E. V. Kuz'min},
     title = {Application of election functions to estimate the number of monotone self-dual boolean functions},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {78--91},
     publisher = {mathdoc},
     volume = {29},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2022_29_2_a0/}
}
TY  - JOUR
AU  - L. Yu. Bystrov
AU  - E. V. Kuz'min
TI  - Application of election functions to estimate the number of monotone self-dual boolean functions
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2022
SP  - 78
EP  - 91
VL  - 29
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2022_29_2_a0/
LA  - ru
ID  - MAIS_2022_29_2_a0
ER  - 
%0 Journal Article
%A L. Yu. Bystrov
%A E. V. Kuz'min
%T Application of election functions to estimate the number of monotone self-dual boolean functions
%J Modelirovanie i analiz informacionnyh sistem
%D 2022
%P 78-91
%V 29
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2022_29_2_a0/
%G ru
%F MAIS_2022_29_2_a0
L. Yu. Bystrov; E. V. Kuz'min. Application of election functions to estimate the number of monotone self-dual boolean functions. Modelirovanie i analiz informacionnyh sistem, Tome 29 (2022) no. 2, pp. 78-91. http://geodesic.mathdoc.fr/item/MAIS_2022_29_2_a0/

[1] D. J. Kleitman, “On Dedekind's Problem: The Number of Monotone Boolean Functions”, Proceedings of the American Mathematical Society, 21:3 (1969), 677–682 | MR | Zbl

[2] L. Y. Bystrov, V. S.Rublev, “Bulevy funkcii, ne prinadlezhashchie predpolnym klassam”, Zametki po informatike i matematike, 13 (2021), 22–26

[3] L. Haviarova, E. Toman, “The Number of Monotone and Self-Dual Boolean Functions”, JAMSI, 10:2 (2014), 93–111 | MR | Zbl

[4] S. V. Yablonskiy, Introduction into discrete mathematics, 5th ed., HSE, 2008 | MR

[5] G. P. Gavrilov, A. A. Sapozhenko, Zadachi i uprazhneniya po diskretnoj matematike, Fizmatlit, 2005 | MR

[6] A. A.Rubchinskiy, Diskretnye matematicheskie modeli. Nachal'nye ponyatiya i standartnye zadachi, uchebnoe posobie, Direct-Media, 2014

[7] D. N. Zhuk, “Ot dvuznachnoj k k-znachnoj logike”, Intellektual'nye sistemy. Teoriya i prilozheniya, 22:1, 131–149 | MR

[8] V. N. Semenchuk, Diskretnaya matematika, Kurs lekcij, Francysk Skaryna Homiel State University, 2007

[9] A. D. Korshunov, “Reshenie problemy dedekinda o chisle monotonnyh bulevyh funkcij”, Doklady Akademii Nauk SSSR, 233:4 (1977), 543–546 | MR