We consider existential rules (aka Datalog+) as a formalism for specifying ontologies. In recent years, many classes of existential rules have been exhibited for which conjunctive query (CQ) entailment is decidable. However, most of these classes cannot express transitivity of binary relations, a frequently used modelling construct. In this paper, we address the issue of whether transitivity can be safely combined with decidable classes of existential rules. First, we prove that transitivity is incompatible with one of the simplest decidable classes, namely aGRD (acyclic graph of rule dependencies), which clarifies the landscape of `finite expansion sets' of rules. Second, we show that transitivity can be safely added to linear rules (a subclass of guarded rules, which generalizes the description logic DL-Lite-R) in the case of atomic CQs, and also for general CQs if we place a minor syntactic restriction on the rule set. This is shown by means of a novel query rewriting algorithm that is specially tailored to handle transitivity rules. Third, for the identified decidable cases, we pinpoint the combined and data complexities of query entailment.

Description Logic (DL) ontologies and non-monotonic rules are two prominent Knowledge Representation (KR) formalisms with complementary features that are essential for various applications. Nonmonotonic Description Logic (DL) programs combine these formalisms thus providing support for rule-based reasoning on top of DL ontologies using a well-defined query interface represented by so-called DL-atoms. Unfortunately, interaction of the rules and the ontology may incur inconsistencies such that a DL-program lacks answer sets (i.e., models), and thus yields no information. This issue is addressed by recently defined repair answer sets, for computing which an effective practical algorithm was proposed for DL-Lite A ontologies that reduces a repair computation to constraint matching based on so-called support sets. However, the algorithm exploits particular features of DL-Lite A and can not be readily applied to repairing DL-programs over other prominent DLs like EL. compared to DL-Lite A , in EL support sets may neither be small nor only few support sets might exist, and completeness of the algorithm may need to be given up when the support information is bounded. We thus provide an approach for computing repairs for DL-programs over EL ontologies based on partial (incomplete) support families. The latter are constructed using datalog query rewriting techniques as well as ontology approximation based on logical difference between EL-terminologies. We show how the maximal size and number of support sets for a given DL-atom can be estimated by analyzing the properties of a support hypergraph, which characterizes a relevant set of TBox axioms needed for query derivation. We present a declarative implementation of the repair approach and experimentally evaluate it on a set of benchmark problems; the promising results witness practical feasibility of our repair approach.

Two essential tasks in managing description logic knowledge bases are eliminating problematic axioms and incorporating newly formed ones. Such elimination and incorporation are formalised as the operations of contraction and revision in belief change. In this paper, we deal with contraction and revision for the DL-Lite family through a model-theoretic approach. Standard description logic semantics yields an infinite number of models for DL-Lite knowledge bases, thus it is difficult to develop algorithms for contraction and revision that involve DL models. The key to our approach is the introduction of an alternative semantics called type semantics which can replace the standard semantics in characterising the standard inference tasks of DL-Lite.

Two essential tasks in managing description logic knowledge bases are eliminating problematic axioms and incorporating newly formed ones. Such elimination and incorporation are formalised as the operations of contraction and revision in belief change. In this paper, we deal with contraction and revision for the DL-Lite family through a model-theoretic approach. Standard description logic semantics yields an infinite number of models for DL-Lite knowledge bases, thus it is difficult to develop algorithms for contraction and revision that involve DL models. The key to our approach is the introduction of an alternative semantics called type semantics which can replace the standard semantics in characterising the standard inference tasks of DL-Lite. Type semantics has several advantages over the standard one. It is more succinct and importantly, with a finite signature, the semantics always yields a finite number of models. We then define model-based contraction and revision functions for DL-Lite knowledge bases under type semantics and provide representation theorems for them. Finally, the finiteness and succinctness of type semantics allow us to develop tractable algorithms for instantiating the functions.

We investigate a higher-order extension of the description logic (DL) SROIQ that provides a fixedly interpreted role semantically coupled with instantiation. It is useful to express interesting meta-level constraints on the modelled ontology. We provide a model-theoretic characterization of the semantics, and we show the decidability by means of reduction.

In this paper we show that the problem of deciding the consistency of a knowledge base in the Description Logic ALCM is ExpTime-complete. The M stands for meta-modelling as defined by Motz, Rohrer and Severi. To show our main result, we define an ExpTime Tableau algorithm as an extension of an algorithm for ALC by Nguyen and Szalas.

Some applications of Description Logic (DL) ontologies combine complete information (e.g., stemming from relational databases) with incomplete, open-world knowledge. Several research efforts in the last years have advocated closed predicates, which are predicates whose extension is interpreted as complete, as a suitable way to leverage partial completeness within the standard open-world semantics of DLs. These works have also studied the data complexity of query answering in the presence of closed predicates, which is generally intractable. In this paper we contribute to the understanding the combined complexity of the problem, by establishing tight complexity results for a range of DLs and query answering problems. In summary, our results show that consistency testing and instance query answering in the presence of closed predicates are feasible in NP even for rich dialects of the DL-Lite family; this is the lowest complexity that could be expected. For EL, in contrast, they are EXPTIME-complete, thus as hard as for ALC and some of its extensions. If unions of conjunctive queries (UCQs) are considered, the picture is even bleaker: we can show 2EXPTIME-hardness even for DL-Lite_R and EL. This is in sharp contrast to the NP-upper bound in the standard setting without closed predicates, and coincides with known upper bounds for much richer DLs. We note that our results imply 2EXPTIME-hardness of query answering in ALCO for the standard setting, where all predicates are interpreted under the open-world semantics. This singles out nominals as a previously unidentified source of complexity when answering queries over expressive DLs. Despite these negative results, we can still identify several useful classes of queries for which the increase in hardness is not as drastic, and the combined complexity of query answering remains between NP and coNEXPTIME.

We study anti-unification for the description logic EL and introduce thenotion of least general generalisation, which generalises simultaneously leastcommon subsumer and concept matching. The idea of generalisation of twoconcepts is to detect maximal similarities between them, and to abstract overtheir differences uniformly. We demonstrate that a finite minimal complete setof generalisations for ELconcepts always exists and establish complexitybounds for computing them. Wepresent an anti-unification algorithm that computes generalisations with afixed skeleton, study its properties and report on preliminary experimental evaluation.

We study query containment in three closely related formalisms: monadic disjunctive Datalog (MDDLog), MMSNP (a logical generalization of constraint satisfaction problems), and ontology-mediated queries (OMQs) based on expressive description logics and unions of conjunctive queries. Containment in MMSNP was known to be decidable due to a result by Feder and Vardi, but its exact complexity has remained open. We prove 2NExpTime-completeness and extend this result to monadic disjunctive Datalog and to OMQs.

The problem The need to equip reasoning systems with explanation services of querying such KBs using database-style queries (in is widely acknowledged by the DL community (see particular, conjunctive queries) has been a major focus of Section 6 for discussion and references), and such facilities recent DL research. Since scalability is a key concern, much are all the more essential when using inconsistency-tolerant of the work has focused on lightweight DLs for which query semantics, as recently argued in (Arioua et al. 2014). Indeed, answering can be performed in polynomial time w.r.t. the the brave, AR, and IAR semantics allow one to classify size of the ABox. The DL-Lite family of lightweight DLs query answers into three categories of increasing reliability, (Calvanese et al. 2007) is especially popular due to the fact and a user may naturally wonder why a given tuple was assigned that query answering can be reduced, via query rewriting, to to, or excluded from, one of these categories.