On the Number of Solutions to a System of Boolean Equations

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 pro...

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Bibliographic Details
Published inAutomation and remote control Vol. 82; no. 9; pp. 1581 - 1596
Main Authors Leontiev, V. K., Gordeev, E. N.
Format Journal Article
LanguageEnglish
Published Moscow Pleiades Publishing 01.09.2021
Springer Nature B.V
Subjects
Boolean
Boolean algebra
CAE) and Design
Calculus of Variations and Optimal Control; Optimization
Computer-Aided Engineering (CAD
Control
Cryptography
Enumeration
Mathematics
Mathematics and Statistics
Mechanical Engineering
Mechatronics
Operations Research
Optimization
Robotics
System Analysis
Systems Theory
Boolean programming problem
continuous linear form
linear transformation
Boolean equations
NP-completeness
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Summary: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.
ISSN:0005-1179
1608-3032
DOI:10.1134/S000511792109006X