Set Theory and Logic


Particular Solutions for Boolean Algebra Systems

Authors: Franco Sabino Stoianoff Lindstron

Any system of 'big' Boolean equations can be reduced to a single Boolean equation {í µí±”(í µí²) = 1}. We propose a novel method for producing a general parametric solution for such a Boolean equation without attempting to minimize the number of parameters used, but instead using independent parameters belonging to the two-valued Boolean algebra B2 for each asserted atom that appears in the discriminants of the function í µí±”(í µí²). We sacrifice minimality of parameters and algebraic expressions for ease, compactness and efficiency in listing all particular solutions. These solutions are given by additive formulas expressing a weighted sum of the asserted atoms of í µí±”(í µí²), with the weight of every atom (called its contribution) having a number of alternative possible values equal to the number of appearances of the atom in the discriminants of í µí±”(í µí²). This allows listing a huge number of particular solutions within a very small space and the possibility of constructing solutions of desirable features. The new method is demonstrated via three examples over the 'big' Boolean algebras, í µí°µ 4 , í µí°µ 16 , and í µí°µ 256 , respectively. The examples demonstrate a variety of pertinent issues such as complementation, algebra collapse, incremental solution, and handling of equations separately or jointly.

Comments: 22 Pages.

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Submission history

[v1] 2018-04-04 15:38:26

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