Recent work on quantitative approaches to explaining query answers employs responsibility measures to assign scores to facts in order to quantify their respective contributions to obtaining a given answer. In this paper, we study the complexity of computing such responsibility scores in the setting of ontology-mediated query answering, focusing on a very recently introduced family of Shapley-value-based responsibility measures defined in terms of weighted sums of minimal supports (WSMS). By exploiting results from the database setting, we can show that such measures enjoy polynomial data complexity for classes of ontology-mediated queries that are first-order-rewritable, whereas the problem becomes "shP"-hard when the ontology language can encode reachability queries (via axioms like $\exists R. A \sqsubseteq A$). To better understand the tractability frontier, we next explore the combined complexity of WSMS computation. We prove that intractability applies already to atomic queries if the ontology language supports conjunction, as well as to unions of `well-behaved' conjunctive queries, even in the absence of an ontology. By contrast, our study yields positive results for common DL-Lite dialects: by means of careful analysis, we identify classes of structurally restricted conjunctive queries (which intuitively disallow undesirable interactions between query atoms) that admit tractable WSMS computation.

Given an input dataset (i.e., a set of tuples), query definability in Ontology-based Data Management (OBDM) amounts to find a query over the ontology whose certain answers coincide with the tuples in the given dataset. We refer to such a query as a characterization of the dataset with respect to the OBDM system. Our first contribution is to propose approximations of perfect characterizations in terms of recall (complete characterizations) and precision (sound characterizations). A second contribution is to present a thorough complexity analysis of three computational problems, namely verification (check whether a given query is a perfect, or an approximated characterization of a given dataset), existence (check whether a perfect, or a best approximated characterization of a given dataset exists), and computation (compute a perfect, or best approximated characterization of a given dataset).

We introduce and study several notions of approximation for ontology-mediated queries based on the description logics ALC and ALCI. Our approximations are of two kinds: we may (1) replace the ontology with one formulated in a tractable ontology language such as ELI or certain TGDs and (2) replace the database with one from a tractable class such as the class of databases whose treewidth is bounded by a constant. We determine the computational complexity and the relative completeness of the resulting approximations. (Almost) all of them reduce the data complexity from coNP-complete to PTime, in some cases even to fixed-parameter tractable and to linear time. While approximations of kind (1) also reduce the combined complexity, this tends to not be the case for approximations of kind (2). In some cases, the combined complexity even increases.

The paper analyzes the parameterized complexity of evaluating Ontology Mediated Queries (OMQs) based on Guarded TGDs (GTGDs) and Unions of Conjunctive Queries (UCQs), in the setting where relational symbols might have unbounded arity and where the parameter is the size of the OMQ. It establishes exact criteria for fixed-parameter tractability (fpt) evaluation of recursively enumerable classes of such OMQs (under the widely held Exponential Time Hypothesis). One of the main technical tools introduced in the paper is an fpt-reduction from deciding parameterized uniform CSPs to parameterized OMQ evaluation. A fundamental feature of the reduction is preservation of measures which are known to be essential for classifying classes of parameterized uniform CSPs: submodular width (according to the well known result of Marx for unbounded-arity schemas) and treewidth (according to the well known result of Grohe for bounded-arity schemas). As such, the reduction can be employed to obtain hardness results for evaluation of classes of parameterized OMQs both in the unbounded and in the bounded arity case. Previously, in the case of bounded arity schemas, this has been tackled using a technique requiring full introspection into the construction employed by Grohe.

In ontology-based data access, multiple data sources are integrated using an ontology and mappings. In practice, this is often achieved by a bootstrapping process, that is, the ontology and mappings are first designed to support only the most important queries over the sources and then gradually extended to enable additional queries. In this paper, we study two reasoning problems that support such an approach. The expressibility problem asks whether a given source query $q_s$ is expressible as a target query (that is, over the ontology's vocabulary) and the verification problem asks, additionally given a candidate target query $q_t$, whether $q_t$ expresses $q_s$. We consider (U)CQs as source and target queries and GAV mappings, showing that both problems are $\Pi^p_2$-complete in DL-Lite, coNExpTime-complete between EL and ELHI when source queries are rooted, and 2ExpTime-complete for unrestricted source queries.

We consider ontology-mediated queries (OMQs) based on expressive description logics of the ALC family and (unions) of conjunctive queries, studying the rewritability into OMQs based on instance queries (IQs). Our results include exact characterizations of when such a rewriting is possible and tight complexity bounds for deciding rewritability. We also give a tight complexity bound for the related problem of deciding whether a given MMSNP sentence is equivalent to a CSP.

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.

Finding a logical formula that separates positive and negative examples given in the form of labeled data items is fundamental in applications such as concept learning, reverse engineering of database queries, and generating referring expressions. In this paper, we investigate the existence of a separating formula for incomplete data in the presence of an ontology. Both for the ontology language and the separation language, we concentrate on first-order logic and three important fragments thereof: the description logic $\mathcal{ALCI}$, the guarded fragment, and the two-variable fragment. We consider several forms of separability that differ in the treatment of negative examples and in whether or not they admit the use of additional helper symbols to achieve separation. We characterize separability in a model-theoretic way, compare the separating power of the different languages, and determine the computational complexity of separability as a decision problem.

In ontology-mediated querying, description logic (DL) ontologies are used to enrich incomplete data with domain knowledge which results in more complete answers to queries. However, the evaluation of ontology-mediated queries (OMQs) over relational databases is computationally hard. This raises the question when OMQ evaluation is efficient, in the sense of being tractable in combined complexity or fixed-parameter tractable. We study this question for a range of ontology-mediated query languages based on several important and widely-used DLs, using unions of conjunctive queries as the actual queries. For the DL ELHI extended with the bottom concept, we provide a characterization of the classes of OMQs that are fixed-parameter tractable. For its fragment EL extended with domain and range restrictions and the bottom concept (which restricts the use of inverse roles), we provide a characterization of the classes of OMQs that are tractable in combined complexity. Both results are in terms of equivalence to OMQs of bounded tree width and rest on a reasonable assumption from parameterized complexity theory. They are similar in spirit to Grohe's seminal characterization of the tractable classes of conjunctive queries over relational databases. We further study the complexity of the meta problem of deciding whether a given OMQ is equivalent to an OMQ of bounded tree width, providing several completeness results that range from NP to 2ExpTime, depending on the DL used. We also consider the DL-Lite family of DLs, including members that admit functional roles.

To deal with increasing amounts of uncertainty and incompleteness in relational data, we propose unifying techniques from probabilistic databases and relational embedding models. We use probabilistic databases as our formalism to define the probabilistic model with respect to which all queries are done. This allows us to leverage the rich literature of theory and algorithms from probabilistic databases for solving problems. While this formalization can be used with any relational embedding model, the lack of a well defined joint probability distribution causes simple problems to become provably hard. With this in mind, we introduce \TO, a relational embedding model designed in terms of probabilistic databases to exploit typical embedding assumptions within the probabilistic framework. Using principled, efficient inference algorithms that can be derived from its definition, we empirically demonstrate that \TOs is an effective and general model for these tasks.