On the Failure of the Finite Model Property in some Fuzzy Description Logics

Description Logics (DLs) [2] are a logical reconstruction of the so-called frame-based knowledge representation languages, with the aim of providing a simple well-established Tarski-style declarative semantics to capture the meaning of the most popular features of structured representation of knowledge. Nowadays, DLs have gained even more popularity due to their application in 1 the context of the Semantic Web [4]. For example, the current standard language for specifying ontologies, the Web Ontology Language OWL is based on Description Logics. It is very natural to extend DLs to the fuzzy case in order to manage fuzzy/vague/imprecise pieces of knowledge for which a clear and precise definition is not possible. For a good and recent survey on the advances in the field of fuzzy DLs, we refer the reader to [14]. One of the challenges of the research in this community is the fact that different families of fuzzy operators (or fuzzy logics) lead to fuzzy DLs with different properties. In fuzzy logic, there are a lot of families of fuzzy operators (or fuzzy logics). Table 1 shows the connectives involved in what are considered the main four families. The most famous families correspond to the three basic continuous t-norms (i.e., Lukasiewicz, Gödel and Product [10]) together with an R-implication