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On the Number of Solutions to a System of Boolean Equations

  • OPTIMIZATION, SYSTEM ANALYSIS, OPERATIONS RESEARCH
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Abstract

We consider questions concerning the solvability and the number of solutions of systems of Boolean equations. Many mathematical models arising in both operations research and cryptography are described in the language of such systems. This is due, in particular, to the fact that, in general, the problem of checking the compatibility of such systems of equations is NP-complete; therefore, the study of the qualitative properties of a system of Boolean equations provides additional information that permits one either to single out polynomially solvable particular cases or to increase the efficiency of enumeration schemes. The focus is on two aspects. The first one concerns the study of the existence and number of solutions of a Boolean equation and a system of equations when parametrizing the problem by the right-hand sides. Formulas and estimates are given for calculating this number both in general and in particular cases. Its maximum is also investigated depending on the specified parameter. The second aspect is devoted to a special case of the problem when the equation is given by the so-called continuous linear form. The properties of such forms and various criteria of continuity are studied.

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REFERENCES

  1. Papadimitriou, C.H. and Steiglitz, K., Combinatorial Optimization: Algorithms and Complexity, Englewood Cliffs, N.J.: Prentice-Hall, 1982. Translated under the title: Kombinatornaya optimizatsiya, Moscow: Mir, 1989.

    MATH  Google Scholar 

  2. Garey, M. and Johnson, D., Computers and Intractability, San Francisco : W. H. Freeman, 1979. Translated under the title: Vychislitel’nye mashiny i trudnoreshaemye zadachi, Moscow: Mir, 1982.

    MATH  Google Scholar 

  3. Schrijver, A., Theory of Linear and Integer Programming, Chichester: John Wiley & Sons, 1986. Translated under the title: Teoriya lineinogo i tselochislennogo programmirovaniya, Moscow: Mir, 1991.

    MATH  Google Scholar 

  4. Leontiev, V.K. and Gordeev, E.N., Generating functions in the knapsack problem, Dokl. Math., 2018, vol. 98, pp. 364–366. https://doi.org/10.1134/S1064562418050198

    Article  MATH  Google Scholar 

  5. Gordeev, E.N. and Leont’ev, V.K., On combinatorial properties of the knapsack problem, Comput. Math. Math. Phys., 2019, vol. 59, no. 8, pp. 1380–1388. https://doi.org/10.1134/S0965542519080074

    Article  MathSciNet  MATH  Google Scholar 

  6. Kellerer, H., Pferschy, U., and Pisinger, D., Knapsack Problems, Berlin: Springer, 2004.

    Book  Google Scholar 

  7. Leont’ev, V.K. and Tonoyan, G.P., Approximate solutions of systems of Boolean equations, Comput. Math. Math. Phys., 1993, vol. 33, no. 9, pp. 1221–1227.

    MathSciNet  MATH  Google Scholar 

  8. Leont’ev, V.K. and Tonoyan, G.P., On systems of Boolean equations, Comput. Math. Math. Phys., 2013, vol. 53, no. 5, pp. 632–639.

    Article  MathSciNet  Google Scholar 

  9. Kuzyurin, N.N. and Fomin, S.A., Effektivnye algoritmy i slozhnost’ vychislenii (Efficient Algorithms and Computational Complexity), Moscow: Mosk. Fiz.-Tekh. Inst., 2007.

    Google Scholar 

  10. Leontiev, V.K. and Gordeev, E.N., On the algebraic immunity of coding systems, Vopr. Kiberbezop., 2019, no. 1, pp. 59–89. https://doi.org/10.21681/2311-3456-2019-1-59-68

  11. Gordeev, E.N., Leontiev, V.K., and Medvedev, N.V., On the properties of Boolean polynomials relevant for cryptosystems, Vopr. Kiberbezop., 2017, no. 3, pp. 63–69. https://doi.org/10.21681/2311-3456-2017-3-63-69

  12. Mazurov, V.D. and Khachai, M.Yu., Committees of systems of linear inequalities, Autom. Remote Control, 2004, vol. 65, no. 2, pp. 193–203. https://doi.org/10.1023/B:AURC.0000014716.77510.61

    Article  MathSciNet  MATH  Google Scholar 

  13. Beresnev, V.L., An efficient algorithm for solving the problem of minimizing polynomials in Boolean variables with the connectedness property, Diskretn. Anal. Issled. Oper. Ser. 2 , 2005, vol. 12, no. 1, pp. 3–11.

    MathSciNet  MATH  Google Scholar 

  14. Leont’ev, V.K., On pseudo-Boolean polynomials, Comput. Math. Math. Phys., 2015, vol. 55, no. 11, pp. 1926–1932. https://doi.org/10.7868/S0044466915110113

    Article  MathSciNet  MATH  Google Scholar 

  15. Leontiev, V.K., Kombinatorika i informatsiya. Ch. 1. Kombinatornyi analiz (Combinatorics and Information. Part 1. Combinatorial Analysis), Moscow: Mosk. FIz.-Tekh. Inst., 2015.

    Google Scholar 

  16. Leontiev, V.K., Kombinatorika i informatsiya. Ch. 2. Informatsionnye modeli (Combinatorics and Information. Part 2. Information Models), Moscow: Mosk. Fiz.-Tekh. Inst., 2015.

    Google Scholar 

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Funding

This work was supported by the Russian Foundation for Basic Research, project no. 20-01-00645.

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Authors and Affiliations

  1. Dorodnicyn Computing Centre, Russian Academy of Sciences, Moscow, 119333, Russia

    V. K. Leontiev

  2. Bauman Moscow State Technical University, Moscow, 105005, Russia

    E. N. Gordeev

Authors
  1. V. K. Leontiev
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  2. E. N. Gordeev
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Correspondence to V. K. Leontiev or E. N. Gordeev.

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Translated by V. Potapchouck

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Leontiev, V.K., Gordeev, E.N. On the Number of Solutions to a System of Boolean Equations. Autom Remote Control 82, 1581–1596 (2021). https://doi.org/10.1134/S000511792109006X

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  • DOI: https://doi.org/10.1134/S000511792109006X

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