Abstract. We present a novel combination of disjunctive logic programs under the answer set semantics with description logics for the Semantic Web. The combination is based on a well-balanced interface between disjunctive logic programs and description logics, which guarantees the decidability of the resulting formalism without assuming syntactic restrictions. We show that the new formalism has very nice semantic properties. In particular, it faithfully extends both disjunctive programs and description logics. Furthermore, we describe algorithms for reasoning in the new formalism, and we give a precise picture of its computational complexity. We also provide a special case with polynomial data complexity. 1

The realization of the Semantic Web vision, in which computational logic has a prominent role, has stimulated a lot of research on combining rules and ontologies, which are formulated in different formalisms, into a framework that is more useful for describing semantic content. In particular, combining logic programming with the Web Ontology Language (OWL), which is a standard based on description logics, emerged as an important issue for linking the Rules and Ontology Layers of the Semantic Web. Non-monotonic description logic programs (or dl-programs) were introduced for such a combination, in which a pair (L,P) of a description logic knowledge base L and a set of rules P with negation as failure is given a model-based semantics that generalizes the answer set semantics of logic programs. In this paper, we reconsider dl-programs and present a well-founded semantics for them as an analog for the other main semantics of logic programs. It generalizes the canonical definition of the well-founded semantics based on unfounded sets, and, as we show, lifts many of the well-known properties from ordinary logic programs to dl-programs. Among these properties: our semantics amounts to a partial model approximating the answer set semantics, which yields for positive and stratified dl-programs a total model coinciding with the answer set semantics; it has polynomial data complexity provided the access to the description logic

Abstract. With the advent of the Semantic Web, the question becomes important how to best combine open-world based ontology languages, like OWL, with closed-world rules paradigms. One of the most mature proposals for this combination is known as Hybrid MKNF knowledge bases [11], which is based on an adaptation of the stable model semantics to knowledge bases consisting of ontology axioms and rules. In this paper, we propose a well-founded semantics for such knowledge bases which promises to provide better efficiency of reasoning, which is compatible both with the OWL-based semantics and the traditional well-founded semantics for logic programs, and which surpasses previous proposals for such a well-founded semantics by avoiding some issues related to inconsistency handling. 1

Abstract. In the ongoing discussion about combining rules and Ontologies on the Semantic Web a recurring issue is how to combine first-order classical logic with nonmonotonic rule languages. Whereas several modular approaches to define a combined semantics for such hybrid knowledge bases focus mainly on decidability issues, we tackle the matter from a more general point of view. In this paper we show how Quantified Equilibrium Logic (QEL) can function as a unified framework which embraces classical logic as well as disjunctive logic programs under the (open) answer set semantics. In the proposed variant of QEL we relax the unique names assumption, which was present in earlier versions of QEL. Moreover, we show that this framework elegantly captures the existing modular approaches for hybrid knowledge bases in a unified way. 1

Abstract. Recently, the logics of minimal knowledge and negation as failure MKNF [12] was used to introduce hybrid MKNF knowledge bases [14], a powerful formalism for combining open and closed world reasoning for the Semantic Web. We present an extension based on a new three-valued framework including an alternating fixpoint, the well-founded MKNF model. This approach, the well-founded MKNF semantics, derives its name from the very close relation to the corresponding semantics known from logic programming. We show that the well-founded MKNF model is the least model among all (three-valued) MKNF models, thus soundly approximating also the two-valued MKNF models from [14]. Furthermore, its computation yields better complexity results (up to polynomial) than the original semantics where models usually have to be guessed. 1

Reasoning in systems integrating Description Logics (DL) ontologies and Datalog rules is a very hard task, and previous studies have shown undecidability of reasoning in systems integrating (even very simple) DL ontologies with recursive Datalog. However, the results obtained so far constitute a very partial picture of the computational properties of systems combining DL ontologies and Datalog rules. The aim of this paper is to contribute to complete this picture, extending the computational analysis of reasoning in systems integrating ontologies and Datalog rules. More precisely, we first provide a set of decidability and complexity results for reasoning in systems combining ontologies specified in DLs and rules specified in nonrecursive Datalog (and its extensions with inequality and negation): such results identify, from the viewpoint of the expressive abilities of the two formalisms, minimal combinations of Description Logics and Datalog in which reasoning is undecidable. Then, we present new results on the decidability and complexity of the so-called restricted (or safe) integration of DL ontologies and Datalog rules. Our results show that: (1) the unrestricted interaction between DLs and Datalog is computationally very hard even in the absence of recursion in rules; (2) surprisingly, the various ”safeness” restrictions, which have been defined to regain decidability of reasoning in the interaction between DLs and recursive Datalog, appear as necessary restrictions even when rules are not recursive.

Abstract. WSML presents a framework encompassing different language variants, rooted in Description Logics and (F-)Logic Programming. So far, the precise relationships between these variants have not been investigated. We take the nonmonotonic first-order autoepistemic logic, which generalizes both Description Logics and Logic Programming, and extend it with frames and concrete domains, to capture all features of WSML; we call this novel formalism FF-AEL. We consider two forms of language layering for WSML, namely loose and strict layering, where the latter enforces additional restrictions on the use of certain language constructs in the rule-based language variants, in order to give additional guarantees about the layering. Finally, we demonstrate that each WSML variant semantically corresponds to its target formalism, i.e. WSML-DL corresponds to SHIQ(D), WSML-Rule to the Stable Model Semantics for Logic Programs (the Well-Founded Semantics can be seen as an approximation), and WSML-Core to

Abstract. In [10], hybrid MKNF knowledge bases have been proposed for combining open and closed world reasoning within the logics of minimal knowledge and negation as failure ([8]). For this powerful framework, we define a three-valued semantics and provide an alternating fixpoint construction for nondisjunctive hybrid MKNF knowledge bases. We thus provide a well-founded semantics which is a sound approximation of the cautious MKNF model semantics, and which also features improved computational properties. We also show that whenever the DL knowledge base part is empty, then the alternating fixpoint coincides with the classical well-founded model.

In the ongoing discussion about rule extensions for Ontology languages on the Semantic Web a recurring issue is how to combine first-order classical logic with nonmonotonic rule languages. Whereas several modular approaches to define a combined semantics for such hybrid knowledge bases focus mainly on decidability issues, we tackle the matter from a more general point of view. In this paper we show how Quantified Equilibrium Logic (QEL) can function as a unified framework that embraces classical logic as well as disjunctive logic programs under the (open) answer set semantics. In the proposed variant of QEL we relax the unique names assumption from earlier versions. Moreover, we show that this framework elegantly captures several modular approaches to nonmonotonic semantics for hybrid knowledge bases. 1

Abstract. Description logics (DLs) and rules are formalisms that emphasize different aspects of knowledge representation: whereas DLs are focused on specifying and reasoning about conceptual knowledge, rules are focused on nonmonotonic inference. Many applications, however, require features of both DLs and rules. Developing a formalism that integrates DLs and rules would be a natural outcome of a large body of research in knowledge representation and reasoning of the last two decades; however, achieving this goal is very challenging and the approaches proposed thus far have not fully reached it. In this paper, we present a hybrid formalism of MKNF + knowledge bases, which integrates DLs and rules in a coherent semantic framework. Achieving seamless integration is nontrivial, since DLs use an open-world assumption, while the rules are based on a closed-world assumption. We overcome this discrepancy by basing the semantics of our formalism on the logic of minimal knowledge and negation as failure (MKNF) by Lifschitz. We present several algorithms for reasoning with MKNF + knowledge bases, each suitable to different kinds of rules, and establish tight complexity bounds.