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)-allowing features such as min/max cardinality constructs on the left-hand/right-hand side of inclusion axioms, role chain axioms, and datatypes-which turn out to be quite expressive if compared with SROTQ(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 ALQS~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~Rl-translation of rules belonging to the Semantic Web Rule Language (SWRL).
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