Extended regular ternary logic functions and majority functions capable of synthesizing any ternary logic function

Abstract

AbstractMultiple‐valued logic functions using only AND, OR, and NOT as operators are called Kleenean functions. They were actively studied in Japan beginning in the early 1990s. Previously, regular ternary logic functions corresponding to the ambiguous state in which the truth value 1/2 is neither 0 nor 1 were defined in ternary logic; but regular ternary logic functions are equivalent to ternary Kleenean functions. Regular ternary logic functions represent only a small fraction of all ternary logic functions, and only a part of the ternary logic functions can be synthesized. In this paper we define regular ternary logic functions of the zeroth to third kinds by introducing the cyclical operation. We then define majority functions of the zeroth to third kinds, which are a subset of the extended regular ternary logic functions, and show that any ternary logic function can be synthesized by means of these three majority functions. Finally, using a genetic algorithm as the base algorithm, we present a technique for synthesizing any ternary logic function by means of extended ternary majority elements, and give illustrative examples. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn, 35(1): 79–90, 2004; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/scj.10226