О некоторых особенностях задачи разрешимости систем булевых уравнений

Abstract

The purpose of the article is to present new results on combinatorial characteristics of systems of Boolean equations, on which such properties of systems as compatibility, solvability, number of solutions and a number of others depend. The research method is the reduction of applied problems to combinatorial models with the subsequent application of classical methods of combinatorics: the method of generating functions, the method of coefficients, methods for obtaining asymptotics, etc. Obtained result. In this paper, we obtain results concerning the solvability of systems of Boolean equations. The complexity of the problem of “ transformation” of an incompatible system into a joint one is analyzed. An approach to solving the problem of separating the minimum number of joint subsystems from an incompatible system is described and justified. The problem is reduced to the problem of finding the minimum covering set. The system compatibility criterion is obtained. Using the method of coefficients, formulas for finding and estimating the number of solutions for parameterizing the problem on the right-hand sides of equations are derived. The maximum of this number is also investigated depending on the parameter. Formulas for the number of solutions for two special cases are obtained: with a restriction on the number of equations and on the size of the problem parameters