Fuzzy Ontologies over Lattices with T-norms

In some knowledge domains, a correct handling of vagueness and imprecision is fundamental for adequate knowledge representation and reasoning. For example, when trying to diagnose a disease, medical experts need to confront symptoms described by the patient, which are by definition subjective, and hence vague. Moreover, a single malady may present a diversity of clinical manifestations in different patients, which leads to imprecise (partial) diagnoses. Fuzzy logic [15] is a prominent approach for dealing with imprecise knowledge. It is based on the notion of fuzzy sets [25], where elements are assigned a membership degree from the real interval [0, 1]. So-called t-norms are used to define the interpretation of the logical connectives. The notion of membership degrees and the operators used can be generalized to lattices, giving rise to L-fuzzy sets [13] and lattice-based t-norms [26, 12]. During the last two decades, several fuzzy DLs have been defined by enriching classical DLs first with fuzzy set semantics [24, 20, 19] and then t-norms [16, 7, 11]. Attempts have also been made at using L-fuzzy set semantics [21, 17]. However, all these approaches either disregard the terminological knowledge, or allow only for a limited class of TBoxes. In fact, it is still unknown whether standard reasoning in fuzzy DLs with general TBoxes is decidable [5, 3]. To the best of our knowledge, the only approaches capable of dealing with full fuzzy TBoxes are based on a finite total order with the Lukasiewicz t-norm [6, 8] or finite De Morgan lattices with the minimum t-norm [9]. In this paper we introduce the lattice-based fuzzy DL ALCL, where L is a complete De Morgan lattice equipped with a t-norm operator. We show that satisfiability in this logic is undecidable if L is infinite. Undecidability holds even if L is a countable, residuated total order. On the other hand, if L is finite, then satisfiability becomes decidable and, under some conditions on the lattice and the t-norm, ExpTime-complete, i.e. not harder than satisfiability in crisp ALC. Our reasoning procedure is in fact general enough to handle any kind of truth-functional semantics, as long as the functions defining the connectives are computable.