Non-convex Fuzzy Truth Values and De Morgan Bisemilattices

Many types of fuzzy truth values, numerical, interval, triangular, and trapezoid truth values, have been proposed and studied about their mathematical properties. It is the characteristic that they are convex fuzzy truth values. Recently, a new type of fuzzy truth values, which we call multi-interval truth values, have been proposed. A characteristic feature of multi-interval truth values is that some of them are not convex. The conventional operations min, max and x \mapsto 1-x$ on the unit interval [0, 1] can be expanded into those on the set of multi-interval truth values. These operations are denoted as $\land$, $\sqcup$, $\bar{~~}$, respectively. Then, this paper first shows that $(S, \land, \sqcup, \bar{~~}, \mathbf{0}, \mathbf{1})$ is a de Morgan bisemilattice. Next, this paper focuses on functions that are expressed by logic formulas, where a logic formula is composed of variables on multi-interval truth values, and the operations $\sqcup$, $\land$ and $\bar{~~}$. Necessary conditions for a function on multi-interval truth values to be expressed by a logic formula are clarified.