Vector Boolean Algebra and Calculus

Abstract

In this paper, binary-vector Boolean algebra, a generalization of ordinary Boolean algebra [6], is introduced. 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 (rightward or leftward) the components of a binary vector is introduced. Moreover, the NOT or COMPLEMENTATION operation is extended to a more general operation called the generalized complement which includes the total complement (ordinary complement), the null complement (no complement), and newly introduced partial complements. Because of this generalization, all axioms and theorems of ordinary Boolean algebra are generalized. In particular, it is shown that DeMorgan's theorem, Shannon's theorem, and the expansion theorem are generalized into more general forms which include their corresponding ordinary version as a special case. It is also shown that any multivalued logic truth table can be represented by a vector Boolean function. Three compact canonical (sum-of-products and product-of-sums) forms of this function are presented.