Chapter 11 - vector boolean algebra and vector boolean calculus; theory and applications

Abstract

This chapter discusses binary-vector Boolean algebra, which is a generalization of ordinary Boolean algebra. In this algebra, every element is represented by a binary vector and in addition to ordinary AND, OR, and NOT operations, a new operation called the rotation operation, which rotates, the components of a binary vector is introduced. The NOT or COMPLEMENTATION operation is extended to a general operation called the generalized complement, which includes the total complement, the null complement, and partial complements. Because of this generalization, all axioms and theorems of ordinary Boolean algebra are generalized. It is shown that DeMorgan's theorem, Shannon's theorem, and expansion theorem are generalized into general forms, which include their corresponding ordinary version as a special case. It is shown that any multivalued logic truth table can be represented by a vector Boolean function. Compact canonical forms of this function are presented. These forms may be used in describing digital systems with multi-based inputs and outputs.