Reasoning in Fuzzy Description Logics with General Concept Inclusion Axioms

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Description logics (DLs) are a family of logic-based knowledge representation languages that are tailored towards representing terminological knowledge, by allowing the knowledge engineer to define the relevant concepts of an application domain within this logic and then reason about these definitions using terminating inference algorithms. In order to deal with applications where the boundaries between members and non-members of concepts (e.g., “tall man,” “high blood pressure,” or “heavy network load”) are blurred, DLs have been combined with fuzzy logics, resulting in fuzzy description logics (fuzzy DLs). Considering the literature on fuzzy description logics of the last 20 years, one could get the impression that, from an algorithmic point of view, fuzzy DLs behave very similarly to their crisp counterparts: for fuzzy DLs based on simple t-norms such as G¨odel, black-box procedures that call reasoners for the corresponding crisp DLs can be used, whereas fuzzy DLs based on more complicated t-norms (such as product and Lukasiewicz) can be dealt with by appropriately modifying the tableau-based reasoners for the crisp DLs. However, it has recently turned out that, in the presence of so-called general concept inclusion axioms (GCIs), the published extensions of tableaubased reasoners to fuzzy DLs do not work correctly. In fact, we were able to show that GCIs can cause undecidability for certain fuzzy DLs based on product t-norm. However, for most fuzzy DLs, the decidability status of reasoning w.r.t. GCIs is still open. The purpose of this project is to investigate the border between decidability and undecidability for fuzzy DLs with GCIs. On the one hand, we will try to show more undecidability results for specific fuzzy DLs, and then attempt to derive from these results general criteria that imply undecidability. On the other hand, we will try to determine decidable special cases, by extending tableau- and automatabased decision procedures for DLs to the fuzzy case, and also looking at other reasoning approaches for inexpressive DLs.