Prominent classes of the most general subsumptive solutions of Boolean equations

Abstract

In a recent seminal paper, Rudeanu derived necessary and sufficient conditions for the most general form of the subsumptive general solution of a Boolean equation. The present paper investigates four prominent classes of the aforementioned most general form. These include the conventional (eliminants) class, the don’t-care class, the Rudeanu class, and the complete-sum (CS) class. We show that each of these classes satisfies the Rudeanu conditions, and hence fits into the frame of the most general form of the subsumptive solution. The four classes are of variable but comparable computational complexities. The conventional (eliminants) class is the least likely to produce a compact solution, while the don’t-care class is the most likely to produce such a solution. Three examples are presented to demonstrate solutions produced by the four classes and compare them form the point of view of compactness of solutions, which determines the ease of generating the tree of particular solutions. A remaining open question that has yet to be settled is whether these four classes are exhaustive, or in other words, whether there are general solution classes other than them. We demonstrate that the don’t-care class is never inferior (and occasionally superior) to the other classes. In fact, this class seeks global minimality and has a better control on the pertinent variables and algebra generators. Both the CS class and the Rudeanu class have an advantage over the eliminants method. The CS class utilizes a canonical representation that explicitly shows all information in the simplest form. The Rudeanu class enhances the eliminants class by incorporating the consensus (Boolean resolution) concept in a way similar to that of the CS approach.