OPTIMAL METHOD FOR SOLVING SPECIAL CLASSES OF SYSTEMS OF NONLINEAR EQUATIONS OF THE SECOND DEGREE
DOI:
https://doi.org/10.26577/JMMCS.2023.v118.i2.02Keywords:
Zhegalkin polynomial, linear Boolean functions, homogeneous-identity matrices, polynomial length, disjunctive normal formsAbstract
In order to simplify the notation and reduce the time for solving systems of Boolean equations, a method is proposed that is optimal for solving a separate class of systems of nonlinear Boolean equations of the second degree. In the class of systems of non-linear Boolean equations under study, logical formulas are divided completely or partially into some linear factors. As a result, logical formulas are reduced to a product of linear polynomials, on the basis of which a system of linear Boolean equations is obtained, which is solved an order of magnitude easier than a system of second-order Boolean equations. It is considered some problems of minimization of special disjunctive normal forms obtained from the Zhegalkin polynomial of the second degree of a special class.
References
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[4] A. Kabulov, I. Saymanov, I. Yarashov and F. Muxammadiev, "Algorithmic method of security of the Internet of Things based on steganographic coding,"2021 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS), Toronto, ON, Canada, 2021, pp. 1-5, doi: 10.1109/IEMTRONICS52119.2021.9422588.
[5] A. Kabulov, I. Normatov, E. Urunbaev and F. Muhammadiev, "Invariant Continuation of Discrete Multi-Valued Functions and Their Implementation,"2021 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS), Toronto, ON, Canada, 2021, pp. 1-6, doi: 10.1109/IEMTRONICS52119.2021.9422486.
[6] A.Kabulov, I. Normatov, A.Seytov and A.Kudaybergenov, "Optimal Management of Water Resources in Large Main Canals with Cascade Pumping Stations,"2020 IEEE International IOT, Electronics and Mechatronics Conference (IEMTRONICS), Vancouver, BC, Canada, 2020, pp. 1-4, doi: 10.1109/IEMTRONICS51293.2020.9216402.
[7] Kabulov, A.V., Normatov, I.H. (2019). About problems of decoding and searching for the maximum upper zero of discrete monotone functions. Journal of Physics: Conference Series, 1260(10), 102006. doi:10.1088/1742-6596/1260/10/102006
[8] Kabulov, A.V., Normatov, I.H. Ashurov A.O. (2019). Computational methods of minimization of multiple functions. Journal of Physics: Conference Series, 1260(10), 10200. doi:10.1088/1742-6596/1260/10/102007
[9] Yablonskii S.V. Vvedenie v diskretnuyumatematiku: Ucheb. posobiedlyavuzov. -2e izd., pererab. idop. -M.:Nauka. Glavnayaredaksiyafiziko-matematicheskoy literature, -384 s.
[10] Djukova, E.V., Zhuravlev, Y.I. Monotone Dualization Problem and Its Generalizations: Asymptotic Estimates of the Number of Solutions. Comput. Math. and Math. Phys. 58, 2064–2077 (2018). https://doi.org/10.1134/S0965542518120102
[11] Leont’ev, V.K. Symmetric boolean polynomials. Comput. Math. and Math. Phys. 50, 1447–1458 (2010). https://doi.org/10.1134/S0965542510080142
[12] Nisan, N. and Szegedy, M. (1991). On the Degree of Boolean Functions as Real Polynomials, in preparation.
[13] RamamohanPaturi. 1992. On the degree of polynomials that approximate symmetric Boolean functions (preliminary version). In Proceedings of the twenty-fourth annual ACM symposium on Theory of Computing (STOC ’92). Association for Computing Machinery, New York, NY, USA, 468–474. https://doi.org/10.1145/129712.129758.
[14] Gu J., Purdom P., Franco J., Wah B.W. Algorithms for the satisfiability (SAT) problem:A Survey // DIMACS Series in Discrete Mathematics and Theoretical Computer Science. 1997.Vol. 35. P. 19–152.
[15] Goldberg E., Novikov Y. BerkMin: A Fast and Robust SAT Solver // Automation andTest in Europe (DATE). 2002. P. 142–149.
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Published
2023-06-30
How to Cite
Bayzhumanov, A. A. (2023). OPTIMAL METHOD FOR SOLVING SPECIAL CLASSES OF SYSTEMS OF NONLINEAR EQUATIONS OF THE SECOND DEGREE. Journal of Mathematics, Mechanics and Computer Science, 118(2), 11–20. https://doi.org/10.26577/JMMCS.2023.v118.i2.02
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Mathematics
