Enumeration of multiple‐valued kleenean functions and some related properties

Abstract

AbstractThe set of multiple‐valued Kleenean functions define a model of a Kleene algebra (a fuzzy algebra) suitable for treating ambiguity. This paper enumerates the Kleenean functions exactly, using the relation that the mapping from p‐valued Kleenean functions to monotonic ternary input p‐valued output functions is a bijection. Thus, the number pn of input vectors that should be searched is 3n for p‐valued n‐variable Kleenean functions. In this paper, we show how to obtain this number for 4‐ to 8‐valued Kleenean functions with 3 or fewer variables. We show that the number increases stepwise as the radix becomes larger and that the number of even (2m)‐valued Kleenean functions is uniquely determined by the number of odd (2m ‐ 1)‐valued Kleenean functions. These results show that the essence of Kleenean algebra is principally revealed by its odd‐valued functions. Moreover, the number of input variables increases exponentially as the variable becomes larger, and increases logarithmically as the radix becomes larger. Thus, increasing the number of variables has more of an effect on increasing this number than does incrementing the radix.