The data-complexity of both satisfiability and finite satisfiability for the two-variable fragment with counting is NP-complete; the data-complexity of both query-answering and finite query-answering for the two-variable guarded fragment with counting is co-NP-complete.

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.

We study the computational complexity of conjunctive query answering w.r.t. ontologies formulated in fragments of the description logic SHIQ. Our main result is the identification of two new sources of complexity: the combination of transitive roles and role hierarchies which results in 2-EXPTIMEhardness, and transitive roles alone which result in CO-NEXPTIME-hardness. These bounds complement the existing result that inverse roles make query answering in SHIQ 2-EXPTIME-hard. We also show that conjunctive query answering with transitive roles, but without inverse roles and role hierarchies, remains in EXPTIME if the ABox is tree-shaped.

Abstract. Ontologies provide a conceptualization of a domain of interest. Nowadays, they are typically represented in terms of Description Logics (DLs), and are seen as the key technology used to describe the semantics of information at various sites. The idea of using ontologies as a conceptual view over data repositories is becoming more and more popular, but for it to become widespread in standard applications, it is fundamental that the conceptual layer through which the underlying data layer is accessed does not introduce a significant overhead in dealing with the data. Based on these observations, in recent years a family of DLs, called DL-Lite, has been proposed, which is specifically tailored to capture basic ontology and conceptual data modeling languages, while keeping low complexity of reasoning and of answering complex queries, in particular when the complexity is measured w.r.t. the size of the data. In this article, we present a detailed account of the major results that have been achieved for the DL-Lite family. Specifically, we concentrate on DL-LiteA,id, an expressive member of this family, present algorithms for reasoning and query answering over DL-LiteA,id ontologies,

Answering conjunctive queries (CQs) has been recognized as a key task for the usage of Description Logics (DLs) in a number of applications. The problem has been studied by many authors, who developed a number of different techniques for it. We present a novel method for CQ answering based on knots, which are schematic trees of depth ≤ 1. It yields an algorithm for CQ answering that works in exponential time for ALCH and for large classes of CQs in SH. This improves over previous algorithms which require double exponential time and is worst-case optimal, as already satisfiability testing in ALC is EXPTIMEcomplete. Our result reconfirms Lutz’s result that inverse roles cause an exponential jump in complexity, being the problem 2EXPTIME-complete for ALCI. The algorithm is CONP, and hence also worst-case optimal, under data complexity.

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Answering queries over ontologies is an important issue for the Semantic Web. Aggregate queries were widely studied for relational databases but almost no results are known for aggregate queries over ontologies. In this work we investigate the latter problem. We propose syntax and semantics for epistemic aggregate queries over ontologies and study query answering for MAX, MIN, COUNT, CNTD, SUM, AVG queries for the ontology language DL-LiteA.

Answering conjunctive queries (CQs) has been recognized as a key task for the usage of Description Logics (DLs) in a number of applications, and has thus been studied by many authors. In this paper, we present an algorithm for this problem in the DL ALCH which works in exponential time. It improves over previous algorithms which require double exponential time and is worst-case optimal, as already satisfiability testing in ALC is EXPTIME-complete. Furthermore, it shows that inverse roles cause an exponential jump in complexity; as recently shown, the problem is 2EXPTIMEcomplete for ALCI. The algorithm is based on a technique that compiles knowledge bases into sets of trees of depth 1. It is in CONP under data complexity (i.e., if the taxonomy part and the query are fixed), thus worst-case optimal. An extension from ALCH to DLs with further constructs is possible.

www.dis.uniroma1.it/∼lastname Abstract. We investigate an extension of Description Logics with higher-order capabilities, based on Henkin-style semantics. Our study starts from the observation that the various possibilities of adding higher-order constructs to a DL form a spectrum of increasing expressive power, including domain metamodeling, i.e., using concepts and roles as predicate arguments, and full metamodeling, providing the ability of using the language constructors and operators as predicate arguments, in the style of RDF. We argue that higher-order features of the former type are sufficiently rich and powerful for the modeling requirements arising in many relevant situations, and therefore we carry out an investigation of the computational complexity of reasoning in DLs extended with such features. In particular, we show that adding domain metamodeling capabilities to expressive DLs has no impact on the complexity of the various reasoning tasks. 1

We investigate an extension of Description Logics (DL) with higher-order capabilities, based on Henkin-style semantics. Our study starts from the observation that the various possibilities of adding higher-order constructs to a DL form a spectrum of increasing expressive power, including domain metamodeling, i.e., using concepts and roles as predicate arguments. We argue that higher-order features of this type are sufficiently rich and powerful for the modeling requirements arising in many relevant situations, and therefore we carry out an investigation of the computational complexity of satisfiability and conjunctive query answering in DLs extended with such higher-order features. In particular, we show that adding domain metamodeling capabilities to SHIQ (the core of OWL 2) has no impact on the complexity of the various reasoning tasks. This is also true for DL-LiteR (the core of OWL 2 QL) under suitable restrictions on the queries.