An Investigation of the Laws of Thought
An Investigation of the Laws of Thought
Boolean Differential Calculus and its Application to Switching Theory
IEEE Transactions on Computers
IEEE Transactions on Computers
Analyzing Errors with the Boolean Difference
IEEE Transactions on Computers
Boolean Difference for Fault Detection in Asynchronous Sequential Machines
IEEE Transactions on Computers
Fault-Tolerant Computing: An Introduction and an Overview
IEEE Transactions on Computers
An Efficient Algorithm for Generating Complete Test Sets for Combinational Logic Circuits
IEEE Transactions on Computers
Uncertainty, Energy, and Multiple-Valued Logics
IEEE Transactions on Computers
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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.