In this paper we consider the most common ABox reasoning services for the description logic DL 4LQS R,× (D) (DL 4,× D , for short) and prove their decidability via a reduction to the satisfiability problem for the set-theoretic fragment 4LQS R . The description logic DL 4,× D is very expressive, as it admits various concept and role constructs, and data types, that allow one to represent rule-based languages such as SWRL. Decidability results are achieved by defining a generalization of the conjunctive query answering problem, called HOCQA (Higher Order Conjunctive Query Answering), that can be instantiated to the most widespread ABox reasoning tasks. We also present a KE-tableau based procedure for calculating the answer set from DL 4,× D knowledge bases and higher order DL 4,× D conjunctive queries, thus providing means for reasoning on several well-known ABox reasoning tasks. Our calculus extends a previously introduced KE-tableau based decision procedure for the CQA problem.

In this paper we use results from Computable Set Theory as a means to represent and reason about description logics and rule languages for the semantic web. Specifically, we introduce the description logic DL 4LQS R (D)-admitting features such as min/max cardinality constructs on the left-hand/righthand side of inclusion axioms, role chain axioms, and datatypes-which turns out to be quite expressive if compared with SROIQ(D), the description logic underpinning the Web Ontology Language OWL. Then we show that the consistency problem for DL 4LQS R (D)-knowledge bases is decidable by reducing it, through a suitable translation process, to the satisfiability problem of the stratified fragment 4LQS R of set theory, involving variables of four sorts and a restricted form of quantification. We prove also that, under suitable not very restrictive constraints, the consistency problem for DL 4LQS R (D)-knowledge bases is NP-complete. Finally, we provide a 4LQS R -translation of rules belonging to the Semantic Web Rule Language (SWRL).

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