A Novel Method for Compact Listing of All Particular Solutions of a System of Boolean Equations
Ali Muhammad Ali Rushdi, Waleed Ahmad · British Journal of Mathematics & Computer Science · 2017
Any system of 'big' Boolean equations can be reduced to a single Boolean equation { ( ) = 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 ( ) .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 ( ), 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 ( ).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, , , and , respectively.The examples demonstrate a variety of pertinent issues such as complementation, algebra collapse, incremental solution, and handling of equations separately or jointly.