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The Development of Suitable Inequalities and Their Application to Systems of Logical Equations

Author

Listed:
  • Dostonjon Numonjonovich Barotov

    (Department of Data Analysis and Machine Learning, Financial University under the Government of the Russian Federation, 109456 Moscow, Russia)

  • Ruziboy Numonjonovich Barotov

    (Department of Mathematical Analysis, Khujand State University, 1 Mavlonbekova, Khujand 735700, Tajikistan)

  • Vladimir Soloviev

    (Department of Data Analysis and Machine Learning, Financial University under the Government of the Russian Federation, 109456 Moscow, Russia)

  • Vadim Feklin

    (Department of Data Analysis and Machine Learning, Financial University under the Government of the Russian Federation, 109456 Moscow, Russia)

  • Dilshod Muzafarov

    (Department of Mathematical Analysis, Khujand State University, 1 Mavlonbekova, Khujand 735700, Tajikistan)

  • Trusunboy Ergashboev

    (Department of Mathematical Analysis, Khujand State University, 1 Mavlonbekova, Khujand 735700, Tajikistan)

  • Khudoyberdi Egamov

    (Department of Higher and Applied Mathematics, Khujand State University, 1 Mavlonbekova, Khujand 735700, Tajikistan)

Abstract

In this paper, two not-difficult inequalities are invented and proved in detail, which adequately describe the behavior of discrete logical functions x o r ( x 1 , x 2 , … , x n ) and a n d ( x 1 , x 2 , … , x n ) . Based on these proven inequalities, infinitely differentiable extensions of the logical functions x o r ( x 1 , x 2 , … , x n ) and a n d ( x 1 , x 2 , … , x n ) were defined for the entire ℝ n . These suitable extensions were applied to systems of logical equations. Specifically, the system of m logical equations in a constructive way without adding any equations (not field equations and no others) is transformed in ℝ n first into an equivalent system of m smooth rational equations ( S m S R E ) so that the solution of S m S R E can be reduced to the problem minimization of the objective function, and any numerical optimization methods can be applied since the objective function will be infinitely differentiable. Again, we transformed S m S R E into an equivalent system of m polynomial equations ( S m P E ) . This means that any symbolic methods for solving polynomial systems can be used to solve and analyze an equivalent S m P E . The equivalence of these systems has been proved in detail. Based on these proofs and results, in the next paper, we plan to study the practical applicability of numerical optimization methods for S m S R E and symbolic methods for S m P E .

Suggested Citation

  • Dostonjon Numonjonovich Barotov & Ruziboy Numonjonovich Barotov & Vladimir Soloviev & Vadim Feklin & Dilshod Muzafarov & Trusunboy Ergashboev & Khudoyberdi Egamov, 2022. "The Development of Suitable Inequalities and Their Application to Systems of Logical Equations," Mathematics, MDPI, vol. 10(11), pages 1-9, May.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:11:p:1851-:d:826255
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    References listed on IDEAS

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    1. Dostonjon Numonjonovich Barotov & Ruziboy Numonjonovich Barotov, 2022. "Polylinear Transformation Method for Solving Systems of Logical Equations," Mathematics, MDPI, vol. 10(6), pages 1-10, March.
    2. Dostonjon Barotov & Aleksey Osipov & Sergey Korchagin & Ekaterina Pleshakova & Dilshod Muzafarov & Ruziboy Barotov & Denis Serdechnyy, 2021. "Transformation Method for Solving System of Boolean Algebraic Equations," Mathematics, MDPI, vol. 9(24), pages 1-12, December.
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