Solution of Boolean equations via atomic decomposition into independent switching equations

Abstract

ABSTRACTThis paper considers the problem of solving a system of Boolean equations over a finite (atomic) Boolean algebra other than the two-valued one, referred to herein as a ‘big’ Boolean algebra. The paper suggests the replacement of this system of equations by a single Boolean equation, and then proposes a novel method for solving this equation by using its atomic decomposition into several independent switching equations. This method has many advantages including the efficient derivation of a complete compact listing of all particular solutions in a form similar to the recently developed permutative additive form, but obtained in a more direct fashion without using parameters. Many detailed examples are used to illustrate the proposed new method. The examples demonstrate how the consistency condition might force a collapse of the underlying Boolean algebra into a subalgebra, and also how to list a huge number of particular solutions in a very small space.