Answering Fuzzy Conjunctive Queries Over Finitely Valued Fuzzy Ontologies

Abstract

Fuzzy Description Logics (DLs) provide a means for representing vague knowledge about an application domain. In this paper, we study fuzzy extensions of conjunctive queries (CQs) over the DL SROIQ based on finite chains of degrees of truth. To answer such queries, we extend a well-known technique that reduces the fuzzy ontology to a classical one, and use classical DL reasoners as a black box. We improve the complexity of previous reduction techniques for finitely valued fuzzy DLs, which allows us to prove tight complexity results for answering certain kinds of fuzzy CQs. We conclude with an experimental evaluation of a prototype implementation, showing the feasibility of our approach.

The complexity of fuzzy EL under the Łukasiewicz T-norm

“…Matching upper bounds have been shown for more expressive fuzzy DLs [10,17,35]. Recall that every finite chain that does not contain a Łukasiewicz component must be a finite Gödel chain, which has only idempotent elements; for these structures, the complexity of reasoning in EL ++ is known to remain the same as in the classical case; that is, subsumption can be decided in polynomial time [31,35]. The key insight for the hardness proofs is that the Łukasiewicz t-norm is powerful enough to simulate the (two-valued) disjunction constructor.…”

“…Under this restriction, reasoning in even very expressive DLs, in particular EL extended with complex role inclusions, disjunction, inverse roles, nominals, and bottom is in 2-ExpTime [37]. This upper bound can be transferred to the corresponding finitely valued FDLs using generic (polynomial) reductions from fuzzy to classical TBoxes, such as the one described in [35], which is based on the ideas first presented in [17]. Since these reductions introduce disjunctions, they cannot be applied to derive a polynomial upper bound for Ł n -EL + from the known polynomial complexity of classical EL + .…”

Fuzzy Description Logics – A Survey

“…The ontology resulting from this translation is consistent in the classical sense iff the original fuzzy ontology is consistent. The first reductions for general finitely valued semantics included an exponential blowup [13,15,16,20,22,75], which can however be avoided by preprocessing the ontology [35]; again, this problem does not occur when using the Zadeh semantics [11,12,97]. Based on these polynomial translations, it can be shown that deciding consistency in finitely valued FDLs has the same complexity as in the underlying classical DLs.…”

“…In particular, consistency in L-NALC with finite L is ExpTime-complete. It was pointed out recently [35] that some reductions [16,20,75] are incorrect for so-called number restrictions, which allow to restrict the number of r-successors of a particular type; unfortunately, no alternative reduction has been found so far. The crispification approach has been implemented in the FDL reasoner DeLorean [14].…”

“…In FDLs this problem becomes more involved, since individuals may satisfy the query to some intermediate truth degree. This has motivated different approaches and solutions to CQ answering over fuzzy ontologies, for example by crispification or adaptation of classical CQ answering techniques such as query rewriting [35,75,78,84,99].…”