A transition calculus for Boolean functions

Abstract

A transition calculus is developed for describing and analyzing the dynamic behavior of logic circuits. Boolean partial derivatives are introduced that are more powerful and applicable to a wider class of problems than the Boolean difference. The partial derivatives are used to define a Boolean differential which provides a concise method for describing the effect on a switching function of changes in its variables. It is shown that a nonconstant function is uniquely determined by its differential, and integration techniques are developed for finding a function when its differential is known. The useful concepts of exact integrals, compatible integrals, and integration by parts are introduced and the conditions for their existence are established. Algorithms for both differentiation and integration are simply implemented using Karnaugh maps.