Conjunctive Query Answering in Finitely-Valued Fuzzy Description Logics

Abstract

Fuzzy Description Logics (DLs) generalize crisp ones by providing membership degree semantics for concepts and roles. A popular technique for reasoning in fuzzy DL ontologies is by providing a reduction to crisp DLs and then employ reasoning in the crisp DL. In this paper we adopt this approach to solve conjunctive query (CQ) answering problems for fuzzy DLs. We give reductions for Gödel, and Łukasiewicz variants of fuzzy SROIQ and two kinds of fuzzy CQs. The correctness of the proposed reduction is proved and its complexity is studied for different fuzzy variants of SROIQ.

Answering Fuzzy Conjunctive Queries Over Finitely Valued Fuzzy Ontologies

“…-We prove that some of the previous crispification algorithms [7,9,27] are incorrect for qualified number restrictions (indicated by the letter Q in the name of DLs) by means of a counter-example (see Example 8). -We discuss a possible way to reduce such qualified number restrictions, but which depends on the presence of so-called Boolean role constructors [33] in the DL (see Section 3.1).…”

“…The reduction technique for the Łukasiewicz t-norm over SROIQ is studied in [9], and for arbitrary finite t-norms in [7,10]. It should be noted, as we show in Example 8, that the reduction of qualified number restrictions proposed in [7,9,27] is incorrect. This family of algorithms has been implemented in the DeLorean reasoner [4,5,8].…”

“…A preliminary version of this paper can be found in [27], where the incorrect reduction of number restrictions was still used. We also extend here that earlier paper by full proofs for the correctness of the crispification procedure, something which has not been done before in the literature.…”

“…However, a major drawback of existing approaches for finite chains using arbitrary t-norms (see [7,9,27]) is that this reduction introduces an exponential blow-up in the size of the fuzzy ontology. While this handicap can be remedied by our normalization step described in Section 3.2 (see also the experiments in Section 5.2), another obstacle needs to be addressed first: the reduction proposed in [7,9,27] is not correct for number restrictions. In the following, we describe this problem in detail and propose a (partial) solution.…”

Reasoning in ${\mathcal{ALC}}$ with Fuzzy Concrete Domains

“…This allows for a more fine-grained modelling for vague information as, for instance, in situation recognition in context-aware systems or even to model fuzzy spatial relations for image recognition. The combination of DLs and fuzzy concrete domains has been investigated already in a number of settings [11,3,9,8]. However, fuzzy DLs can easily turn out to be undecidable [4].…”