Multiple-valued logic, in which the number of discrete logic levels is not confined to two, has been the subject of much research over many years. The practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation. In this tutorial/survey paper we will review the historical developments in this field, both in circuit realizations and in methods of handling multiple-valued design data, and consider the present state-of-the-art and future expectations.

Multiple-Valued Logic&#8212;its Status and its Future

A survey is presented of the developments in algebras and techniques for the realisation of 3-valued switching functions. Digital arithmetic, ternary codes, composition algebras, minimisation, circuits and sequential circuit design are discussed. The feasibility of 3-valued digital systems is demonstrated

Three-valued digital systems

The relationship between multivalued switching algebra and Boolean algebra is presented by introducing different definitions for the complements of multivalued variables. For every definition introduced, the paper points out which Boolean algebra theorems are valid for multivalued cases, which are invalid, and gives proofs to substantiate the claim. It is shown that DeMorgan's theorem holds for all four definitions of complement given in this paper. One definition allows us to map the multivalued variables into binary variables. Under this definition, all axioms and theorems of Boolean algebra are satisfied and can be used for minimization of any multivalued switching function f. Illustrative examples for minimizing f and its complement f are given.

The Relationship Between Multivalued Switching Algebra and Boolean Algebra Under Different Definitions of Complement

Some new procedures for synthetizing three-input majority gate networks have been described. Both a geometrical approach on the map of the assigned function and an algebraic procedure have been discussed. The geometrical methods give a number of different solutions and may lead to the simplest networks, but their use requires ingenuity. Therefore, some of the solutions suggested by the analysis of the maps have been translated into algebraic expressions, which may be easily used for automatic computation. The techniques described are applicable to any and all Boolean functions and lead to simple solutions. In fact, every function of n variables can be implemented by a network having (n-1) levels of gates, if n is odd, or n levels, if n is even, at the most. All the procedures are relatively simple and can be extended to the synthesis of networks of any type (m, t).

Majority Gate Networks

Free spectra of 3-element algebras

A new logical algebra, ternary threshold logic, is defined and developed. The system is shown to be capable of representing all three-valued functions, and two methods of synthesizing these functions from their truth tables are given. One of the methods produces a normal form that is analogous to the disjunctive normal form of Boolean algebra. A list of single-threshold-operator equivalents of certain normal forms is given. It is pointed out that magnetic film parametrons may be made to exhibit the logical properties of the operators of the system. Designs for ternary adder and comparator stages are given that could be constructed out of magnetic film parametrons.

W. H. Hanson

Papers

Ternary Threshold Logic

Threshold-Logic Synthesis by Algebraic Methods

Ternary threshold logic