Well-Founded Semantics for Description Logic Programs in the Semantic Web

In previous work, towards the integration of rules and ontologies in the Semantic Web, we have proposed a combination of logic programming under the answer set semantics with the description logics ${\cal SHIF}({\mathbf{D}})$ and ${\cal SHOIN}({\mathbf{D}})$, which underly the Web ontology languages OWL Lite and OWL DL, respectively. More precisely, we have introduced description logic programs (or dl-programs), which consist of a description logic knowledge base L and a finite set of description logic rules P, and we have defined their answer set semantics. In this paper, we continue this line of research. Here, as a central contribution, we present the well-founded semantics for dl-programs, and we analyze its semantic properties. In particular, we show that it generalizes the well-founded semantics for ordinary normal programs. Furthermore, we show that in the general case, the well-founded semantics of dl-programs is a partial model that approximates the answer set semantics, whereas in the positive and the stratified case, it is a total model that coincides with the answer set semantics. Finally, we also provide complexity results for dl-programs under the well-founded semantics.