Quasi-algebraic decompositions of switching functions

Brayton (1982-90) and others have developed a rich theory of decomposition of switching functions based on algebraic manipulations of monomials. In this theory, a product g(X_g)·h(X_h ) is algebraic if X_g∩X_h=Ø. There are efficient methods for determining if a function has an algebraic product. If a function does not have an algebraic product, then there are good methods for obtaining a decomposition of the form f=g·h+r where g·h is an algebraic product. Algebraic decompositions have the desirable properties that they are canonical and preserve testability. In this paper we generalize the concept of an algebraic product to decompositions of the form f(X)=g(X_g)??h(X_h) where ?? is any binary Boolean operation and |X_g∩X_h|=k for some k⩾0. We call these decompositions quasi-algebraic decompositions. We begin by showing that we may restrict ourselves to the case where ?? is +(sum),·(product) or ⊕ (enclusive-or). We then give necessary and sufficient conditions for a function to have a quasi-algebraic decomposition for a given X_g and X_h. If a function has such a decomposition we show how to determine the functions g and h in a canonical manner. We also show that these decompositions are fully SSL testable. Finally, using standard benchmark circuits, we show that quasi algebraic decompositions occur often and are useful in reducing circuit size