GCIs Make Reasoning in Fuzzy DL with the Product T-norm Undecidable

Fuzzy variants of Description Logics (DLs) were introduced in order to deal with applications where not all concepts can be defined in a precise way. A great variety of fuzzy DLs have been investigated in the literature [12,8]. In fact, compared to crisp DLs, fuzzy DLs offer an additional degree of freedom when defining their expressiveness: in addition to deciding which concept constructors (like conjunction, disjunction, existential restriction) and which TBox formalism (like no TBox, acyclic TBox, general concept inclusions) to use, one must also decide how to interpret the concept constructors by appropriate functions on the domain of fuzzy values [0, 1]. For example, conjunction can be interpreted by different t-norms (such as Gödel, Łukasiewicz, and product) and there are also different options for how to interpret negation (such as involutive negation and residual negation). In addition, one can either consider all models or only so-called witnessed models [10] when defining the semantics of fuzzy DLs. Decidability of fuzzy DLs is often shown by adapting the tableau-based algorithms for the corresponding crisp DL to the fuzzy case. This was first done for the case of DLs without general concept inclusion axioms (GCIs) [19,17,14,6], but then also extended to GCIs [16,15,18,4,5]. Usually, these tableau algorithms reason w.r.t. witnessed models.1 It should be noted, however, that in the presence of GCIs there are different ways of extending the notion of witnessed models from [10], depending on whether the witnessed property is required to apply also to GCIs (in which case we talk about strongly witnessed models) or not (in which case we talk about witnessed models). The paper [4] considers the case of reasoning w.r.t. fuzzy GCIs in the setting of a logic with product t-norm and involutive negation. More precisely, the tableau algorithm introduced in that paper is supposed to check whether an ontology consisting of fuzzy GCIs and fuzzy ABox assertions expressed in this DL has a strongly witnessed model or not.2 Actually, the proof of correctness of this algorithm given in [4] implies that, whenever such an ontology has a strongly witnessed model, then it has a finite model. However, it was recently shown in [2]