Symmetric Ternary Switching Functions

Abstract

This paper develops a theory of symmetric ternary switching functions and presents systematic methods for their detection, identification and synthesis. Shannon's theory of binary symmetric functions is extended to ternary functions by defining a set of five ``priming'' operations which, together with the ``permutation'' operations, form a group. Algebraic characterizations of totally and partially symmetric ternary functions are discussed. The method of detection and identification is based on a set of simple rules derived in terms of equalities of the residual functions of the given function with respect to pairs of variables of symmetry in a cyclic order. The notions of fundamental and simple symmetric ternary functions have been introduced and their algebraic properties have been studied. These concepts are then applied to develop a synthesis procedure which uses two basic 3-valued electronic gates recognizing the ``maximum'' or the ``minimum'' of the inputs. Possible generalizations of the results derived in this paper to arbitrarily many-valued functions are indicated.