Both OWL-DL and function-free Horn rules are decidable fragments of first-order logic with interesting, yet orthogonal expressive power. A combination of OWL-DL and rules is desirable for the Semantic Web; however, it might easily lead to the undecidability of interesting reasoning problems. Here, we present a decidable such combination where rules are required to be DL-safe: each variable in the rule is required to occur in a non-DL-atom in the rule body. We discuss the expressive power of such a combination and present an algorithm for query answering in the related logic SHIQ extended with DL-safe rules, based on a reduction to disjunctive programs.

Conjunctive queries play an important role as an expressive query language for Description Logics (DLs). Although modern DLs usually provide for transitive roles, it was an open problem whether conjunctive query answering over DL knowledge bases is decidable if transitive roles are admitted in the query. In this paper, we consider conjunctive queries over knowledge bases formulated in the popular DL SHIQ and allow transitive roles in both the query and the knowledge base. We show that query answering is decidable and establish the following complexity bounds: regarding combined complexity, we devise a deterministic algorithm for query answering that needs time single exponential in the size of the KB and double exponential in the size of the query. Regarding data complexity, we prove co-NP-completeness. 1

Description Logics (DLs) are the formal foundations of the standard web ontology languages OWL-DL and OWL-Lite. In the Semantic Web and other domains, ontologies are increasingly seen also as a mechanism to access and query data repositories. This novel context poses an original combination of challenges that has not been addressed before: (i) sufficient expressive power of the DL to capture common data modeling constructs; (ii) well established and flexible query mechanisms such as Conjunctive Queries (CQs); (iii) optimization of inference techniques with respect to data size, which typically dominates the size of ontologies. This calls for investigating data complexity of query answering in expressive DLs. While the complexity of DLs has been studied extensively, data complexity has been characterized only for answering atomic queries, and was still open for answering CQs in expressive DLs. We tackle this issue and prove a tight CONP upper bound for the problem in SHIQ, as long as no transitive roles occur in the query. We thus establish that for a whole range of DLs from AL to SHIQ, answering CQs with no transitive roles has CONP-complete data complexity. We obtain our result by a novel tableaux-based algorithm for checking query entailment, inspired by the one in [19], but which manages the technical challenges of simultaneous inverse roles and number restrictions (which leads to a DL lacking the finite model property).

Expressive Description Logics (DLs) have been advocated as formalisms for modeling the domain of interest in various application areas. An important requirement is the ability to answer complex queries beyond instance retrieval, taking into account constraints expressed in a knowledge base. We consider this task for positive existential path queries (which generalize conjunctive queries and unions thereof), whose atoms are regular expressions over the roles (and concepts) of a knowledge base in the expressive DL ALCQIbreg. Using techniques based on two-way tree-automata, we first provide an elegant characterization of TBox and ABox reasoning, which gives us also a tight EXPTIME bound. We then prove decidability (more precisely, a 2EXPTIME upper bound) of query answering, thus significantly pushing the decidability frontier, both with respect to the query language and the considered DL. We also show that query answering is EXP-SPACE-hard already in rather restricted settings.

The novel context of accessing and querying large data repositories through ontologies that are formalized in terms of expressive DLs requires on the one hand to consider query answering as the primary inference technique, and on the other hand to optimize it with respect to the size of the data, which dominates the size of ontologies. While the complexity of DLs has been studied extensively, data complexity in expressive DLs has been characterized only for answering atomic queries, and was still open for more expressive query languages, such as unions of conjunctive queries (UCQs). In this paper we advocate the need for studying this problem, and provide a significant technical contribution in this direction. Specifically, we prove a tight coNP upper bound for answering UCQs over SHIQ knowledge bases, for the case where the queries do not contain transitive roles. We thus establish that for a whole range of DLs from AL to SHIQ, answering such UCQs has coNP-complete data complexity. We obtain our result by a novel tableaux-based algorithm for checking query entailment, inspired by the one in [20], but which manages the technical challenges of simultaneous inverse roles and number restrictions (which leads to a DL lacking the finite model property). 1

Abstract. Despite the success of the Web Ontology Language OWL, the development of expressive means for querying OWL knowledge bases is still an open issue. In this paper, we investigate how a very natural and desirable form of queries—namely conjunctive ones—can be used in conjunction with OWL such that one of the major design criteria of the latter—namely decidability—can be retained. More precisely, we show that querying the tractable fragmentEL ++ of OWL 1.1 is decidable. We also provide a complexity analysis and show that querying unrestrictedEL ++ is undecidable. 1

Abstract. We present a resolution-based decision procedure for the description logic SHOIQ—the logic underlying the Semantic Web ontology language OWL-DL. Our procedure is goal-oriented, and it naturally extends a similar procedure for SHIQ, which has proven itself in practice. Applying existing techniques for deriving saturation-based decision procedures to SHOIQ is not straightforward due to nominals, number restrictions, and inverse roles—a combination known to cause termination problems. We overcome this difficulty by using the basic superposition calculus, extended with custom simplification rules. 1

Abstract. As applications of description logics proliferate, efficient reasoning with knowledge bases containing many assertions becomes ever more important. For such cases, we developed a novel reasoning algorithm that reduces a SHIQ knowledge base to a disjunctive datalog program while preserving the set of ground consequences. Queries can then be answered in the resulting program while reusing existing and practically proven optimization techniques of deductive databases, such as join-order optimizations or magic sets. Moreover, we use our algorithm to derive precise data complexity bounds: we show that SHIQ is data complete for NP, and we identify an expressive fragment of SHIQ with polynomial data complexity.

Abstract. An important reasoning task, in addition to the standard DL reasoning services, is conjunctive query answering. In this paper, we present a decision procedure for conjunctive query entailment in the expressive Description Logic SHOQ. This is, to the best of our knowledge, the first decision procedure for conjunctive query entailment in a logic that allows for nominals. We achieve this by combining the techniques used in the conjunctive query entailment procedure for SHIQ with the techniques proposed for a restricted class of conjunctive queries in SHOQ. 1

In this paper we present an overview of our recent work on the relationship between description logics and disjunctive datalog. In particular, we reduce satisfiability and instance checking in SHIQ to corresponding problems in disjunctive datalog. This allows us to apply practically successful deductive database optimization techniques, such as magic sets. Interestingly, the reduction also allows us to obtain novel theoretical results on description logics. In particular, we show that the data complexity of reasoning in SHIQ is in NP, and we define a fragment called Horn-SHIQ for which the data complexity is in P. Finally, the reduction provides a basis for query answering in an extension of SHIQ with so-called DL-safe rules.