Some fundamental properties of Boolean ring normal forms

Abstract

Boolean ring is an algebraic structure which uses exclusive ? or instead of the usual or. It yields a unique normal form for every Boolean function. In this paper we present several fundamental properties concerning Boolean rings. We present a simple method for deriving the Boolean ring normal form directly from a truth table. We also describe a notion of normal form of a Boolean function with a don't-care condition, and show an algorithm for generating such a normal form. We then discuss two Boolean ring based theorem proving methods for propositional logic. Finally we give some arguments on why the Boolean ring representation had not been used more extensively, and how it can be used in computing.