It is well-known that satisfiability (and hence validity) in the minimal classical modal logic is a PSPACE-complete problem. In this paper we consider the satisfiability and validity problems (here they are not dual, although mutually reducible) for the minimal modal logic over a finite Lukasiewicz chain, and show that they also are PSPACE-complete. This result is also true when adding either the Delta operator or truth constants in the language, i.e. in all these cases it is PSPACE-complete.

Description Logics (DLs) are knowledge representation languages built on the basis of classical logic. DLs allow the creation of knowledge bases and provide ways to reason on the contents of these bases. Fuzzy Description Logics (FDLs) are natural extensions of DLs for dealing with vague concepts, commonly present in real applications. Following the ideas of Hájek in [17] and García-Cerdaña et al. in [15] we develop a family of FDLs whose underlying logic is the fuzzy logic of a finite linearly ordered residuated lattice, that is, an n-graded fuzzy logic defined by a divisible finite t-norm over a finite chain. Moreover, the role of the constructor of implication in the languages for FDLs is discussed, and a hierarchy of AL-languages adapted to the behavior of the connectives in the fuzzy logics underlying these description languages is proposed. Finally, we deal with reasoning tasks within the framework of finitely valued DLs.