DL-Lite in the Light of First-Order Logic

Abstract

The use of ontologies in various application domains, such as Data Integration, the Semantic Web, or ontology-based data management, where ontologies provide the access to large amounts of data, is posing challenging requirements w.r.t. a trade-off between expressive power of a DL and efficiency of reasoning. The logics of the DL-Lite family were specifically designed to meet such requirements and optimized w.r.t. the data complexity of answering complex types of queries. In this paper we propose DL-Litebool, an extension of DLLite with full Booleans and number restrictions, and study the complexity of reasoning in DL-Litebool and its significant sub-logics. We obtain our results, together with useful insights into the properties of the studied logics, by a novel reduction to the one-variable fragment of first-order logic. We study the computational complexity of satisfiability and subsumption, and the data complexity of answering positive existential queries (which extend unions of conjunctive queries). Notably, we extend the LOGSPACE upper bound for the data complexity of answering unions of conjunctive queries in DL-Lite to positive queries and to the possibility of expressing also number restrictions, and hence local functionality in the TBox. Introduction Description Logics (DLs) provide the formal foundation for ontologies (http://owl1 1.cs.manchester.ac.uk/), and the tasks related to the use of ontologies in various application domains are posing new and challenging requirements w.r.t. a trade-off between expressive power of a DL and efficiency of reasoning over knowledge bases (KBs) expressed in the DL. On the one hand, it is expected that the DL provides the ability to express TBoxes without limitations. On the other hand, tractable reasoning is essential in a context where ontologies become large and/or are used to access large amounts of data. This is a scenario emerging, e.g., in Data Integration (Lenzerini 2002), the Semantic Web (Heflin & Hendler 2001), P2P data management (Bernstein et al. 2002; Calvanese et al. 2004; Franconi et al. 2004), ontology-based data access (Borgida et al. 1989; Calvanese et al. 2005b), and biological data management. These new requirements have led to the proposal of novel DLs with PTIME algorithms for reasoning Copyright c © 2007, Association for the Advancement of Artificial Intelligence (www.aaai.org). All rights reserved. over KBs (composed of a TBox storing intensional information, and an ABox representing the extensional data), such as those of the EL-family (Baader, Brandt, & Lutz 2005; Baader, Lutz, & Suntisrivaraporn 2005) and of the DL-Lite family (Calvanese et al. 2005a; 2006). The logics of the DL-Lite family, in addition to having inference that is polynomial in the size of the whole KB, have been designed with the aim of providing efficient access to large data repositories. The data that need to be accessed are assumed to be stored in a standard relational database (RDB), and one is interested in expressing, through the ontology, sufficiently complex queries to such data that go beyond the simple instance checking case (i.e., asking for instances of single concepts and roles). The logics of the DL-Lite family are tailored towards such a task. In other words, they are specifically optimized w.r.t. data complexity: for the various versions of DL-Lite , answering unions of conjunctive queries (UCQs) (Abiteboul, Hull, & Vianu 1995) can be done in LOGSPACE in data complexity (Calvanese et al. 2005a). Indeed, the aim of the original line of research on the DL-Lite family was precisely to establish the maximal subset of DLs constructs for which one can devise query answering techniques that leverage on RDB technology, and thus guarantee performance and scalability (see FOL-reducibility in (Calvanese et al. 2005a)). Clearly, a requirement for this is that the data complexity of query answering stays within LOGSPACE. In this paper, we pursue a similar objective and aim at providing useful insights for the investigation of the computational properties of the logics in the DL-Lite family. We extend the basic DL-Lite with full Booleans and number restrictions, obtaining the logic we call DL-Litebool, and introduce two sublanguages of it, DL-Litekrom and DL-Litehorn. Notably, the latter strictly extends basic DL-Lite with number restrictions, and hence local (as opposed to global) functionality. We then characterize the first-order logic nature of this class of newly introduced DLs by showing their strong connection with the one variable fragment QL of firstorder logic. The gained understanding allows us also to derive novel results on the computational complexity of inference for the newly introduced variants of DL-Lite . Specifically, we show that KB satisfiability (or subsumption w.r.t. a KB) is NLOGSPACE-complete for DL-Litekrom, P-complete for DL-Litehorn, and NP-complete (resp. CONPcomplete) for DL-Litebool. We prove that data complexity of both satisfiability and instance checking is in LOGSPACE for DL-Litebool. We then look into the data complexity of answering positive existential queries, which extend the wellknown class of UCQs by allowing for an unrestricted interaction of conjunction and disjunction. We extend the LOGSPACE upper bound already known for UCQs in DLLite to positive existential queries in DL-Litehorn. Due essentially to the presence of disjunction, the problem is CONP-hard for DL-Litekrom, and hence for DL-Litebool (Calvanese et al. 2006). The DL-Litebool family has been shown to be expressive enough to capture conceptual data models like UML and Extended ER (Artale et al. 2007). Such correspondence provided new complexity results for reasoning over various fragments of the Extended ER language. The rest of the paper is structured as follows. In the next section we introduce the three variants of DL-Lite mentioned above. Then we exhibit the translation to QL and derive the complexity results for satisfiability and subsumption. We proceed with the analysis of data complexity, and conclude with techniques and data complexity results for answering positive existential queries. (All proofs can be found at http://www.dcs.bbk.ac.uk/̃ roman.)

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@inproceedings{Artale2007DLLiteIT, title={DL-Lite in the Light of First-Order Logic}, author={Alessandro Artale and Diego Calvanese and Roman Kontchakov and Michael Zakharyaschev}, booktitle={AAAI}, year={2007} }