Some Fundamental Properties of Multiple-Valued Kleenean Functions and Determination of Their Logic Formulas

Multiple-valued Kleenean functions that are models of a Kleene algebra and are logic functions expressed by logic formulas composed of variables, constants, and logic operations AND OR, and NOT are discussed. The set of Kleenean functions, is a model with the largest number of logic functions among existing models of a Kleene algebra, such as fuzzy logic functions, regular ternary logic functions, and B-ternary logic functions. Mainly, it is shown that any p-valued Kleenean function is derived from a monotonic ternary input functions and any p-valued unate function is derived from a unate binary input function. The mapping relations between them and the method to determine the logic formula of the Kleenean function and unate function from that of the monotonic ternary input function and unate binary input function, respectively, are classified. 7-or-less-valued Kleenean functions and unate functions of 3-or-fewer variables are enumerated. It is known that the number of p-valued Kleenean functions increases stepwise and that of unate functions increases smoothly as p becomes larger. >