Kleene-Stone logic functions

Kleene algebra has correspondence with fuzzy sets or fuzzy logic and has recently been studied as an algebraic system treating ambiguity or fuzziness. In contrast, Stone algebra, which has connections with modality, has properties different from Kleene algebra. Kleene-Stone algebra has been proposed as an algebra that is both a Kleene algebra and a Stone algebra. A set of Kleene-Stone logic functions is one of the models of Kleene-Stone algebra. Fundamental properties, such as a quantization theorem for Kleene-Stone logic functions in which logic functions are determined by n-tuple vector spaces over (0, 1/4, 2/4, 3/4, 1), is clarified. The authors define a partial-order relation over (0, 1/4, 2/4, 3/4, 1), and then they show that any Kleene-Stone logic function satisfies the monotonicity for the partial-order relation. A canonical disjunctive form that enables them to represent any Kleene-Stone logic function uniquely is introduced.<>