Carsten Lutz

Maurice Funk, Jean Christoph Jung, Carsten Lutz

We study ELI queries (ELIQs) in the presence of ontologies formulated in the description logic DL-Lite. For the dialect DL-LiteH, we show that ELIQs have a frontier (set of least general generalizations) that is of polynomial size and can be computed in polynomial time. In the dialect DL-LiteF, in contrast, frontiers may be infinite. We identify a natural syntactic restriction that enables the same positive results as for DL-LiteH. We use out results on frontiers to show that ELIQs are learnable in polynomial time in the presence of a DL-LiteH / restricted DL-LiteF ontology in Angluin's framework of exact learning with only membership queries.

Carsten Lutz, Quentin Manière, Robin Nolte

Circumscription is one of the main approaches for defining non-monotonic description logics (DLs). While the decidability and complexity of traditional reasoning tasks such as satisfiability of circumscribed DL knowledge bases (KBs) is well understood, for evaluating conjunctive queries (CQs) and unions thereof (UCQs), not even decidability had been established. In this paper, we prove decidability of (U)CQ evaluation on circumscribed DL KBs and obtain a rather complete picture of both the combined complexity and the data complexity, for DLs ranging from ALCHIO via EL to various versions of DL-Lite. We also study the much simpler atomic queries (AQs).

Balder ten Cate, Maurice Funk, Jean Christoph Jung et al.

This note serves three purposes: (i) we provide a self-contained exposition of the fact that conjunctive queries are not efficiently learnable in the Probably-Approximately-Correct (PAC) model, paying clear attention to the complicating fact that this concept class lacks the polynomial-size fitting property, a property that is tacitly assumed in much of the computational learning theory literature; (ii) we establish a strong negative PAC learnability result that applies to many restricted classes of conjunctive queries (CQs), including acyclic CQs for a wide range of notions of "acyclicity"; (iii) we show that CQs (and UCQs) are efficiently PAC learnable with membership queries.

Maurice Funk, Simon Hosemann, Jean Christoph Jung et al.

We present a method for automatically constructing a concept hierarchy for a given domain by querying a large language model. We apply this method to various domains using OpenAI's GPT 3.5. Our experiments indicate that LLMs can be of considerable help for constructing concept hierarchies.

Carsten Lutz, Lukas Schulze

Ontologies often require knowledge representation on multiple levels of abstraction, but description logics (DLs) are not well-equipped for supporting this. We propose an extension of DLs in which abstraction levels are first-class citizens and which provides explicit operators for the abstraction and refinement of concepts and roles across multiple abstraction levels, based on conjunctive queries. We prove that reasoning in the resulting family of DLs is decidable while several seemingly harmless variations turn out to be undecidable. We also pinpoint the precise complexity of our logics and several relevant fragments.

Carsten Lutz, Leif Sabellek, Lukas Schulze

We study the evaluation of ontology-mediated queries (OMQs) on databases of bounded cliquewidth from the viewpoint of parameterized complexity theory. As the ontology language, we consider the description logics $\mathcal{ALC}$ and $\mathcal{ALCI}$ as well as the guarded two-variable fragment GF$_2$ of first-order logic. Queries are atomic queries (AQs), conjunctive queries (CQs), and unions of CQs. All studied OMQ problems are fixed-parameter linear (FPL) when the parameter is the size of the OMQ plus the cliquewidth. Our main contribution is a detailed analysis of the dependence of the running time on the parameter, exhibiting several interesting effects.

Moritz Schönherr, Balder ten Cate, Maurice Funk et al.

The expressive limitations of message-passing Graph Neural Networks (GNNs) have motivated a wide range of more powerful graph learning architectures. We advocate Deep Homomorphism Networks (DHNs) as a model particularly well-suited for learning over relational databases, due to their close connection to important fragments of SQL such as conjunctive queries. We study the precise expressive power of DHNs by relating them to various natural fragments and extensions of first-order logic (FO). For DHNs with max, sum, and mean aggregations, we establish connections to the unary negation fragment (UNFO) and to the extensions of UNFO with counting quantifiers and with ratio quantifiers. We further relate sum-aggregation DHNs to the unary quantifier alternation fragment of FO and to an extension of FO with expressive counting. Through the classical correspondence between FO and SQL, these results also illuminate the relation between DHNs and SQL. They also enable us to study the decidability of two fundamental static analysis problems for DHNs, the emptiness problem and the subsumption problem. Finally, we confirm through experiments that the established differences in expressive power are reflected in the performance on suitable prediction tasks.

Nofar Carmeli, Carsten Lutz, Marcin Przybyłko

We study the limits of linear time evaluation of conjunctive queries under constraints expressed as tuple-generating dependencies (TGDs), across several modes of query evaluation: single-testing, all-testing, counting, lexicographic direct access, and enumeration. While full classifications seem far beyond reach, we propose an approach that, for some evaluation modes and classes of TGDs, makes it possible to lift known dichotomies from the unconstrained setting. In particular, our approach applies to all mentioned evaluation modes except enumeration, when the constraints fall into one of two classes: non-recursive sets of TGDs in which every TGD uses at most binary relation symbols in the head or has at most two frontier variables; and frontier-guarded full TGDs. We further provide a collection of examples showcasing the challenges that arise for enumeration and for less restrictive classes of TGDs.

Carsten Lutz, Quentin Manière

We study extensions of expressive decidable fragments of first-order logic with circumscription, in particular the two-variable fragment FO$^2$, its extension C$^2$ with counting quantifiers, and the guarded fragment GF. We prove that if only unary predicates are minimized (or fixed) during circumscription, then decidability of logical consequence is preserved. For FO$^2$ the complexity increases from $\textrm{coNexp}$ to $\textrm{coNExp}^\textrm{NP}$-complete, for GF it (remarkably!) increases from $\textrm{2Exp}$ to $\textrm{Tower}$-complete, and for C$^2$ the complexity remains open. We also consider querying circumscribed knowledge bases whose ontology is a GF sentence, showing that the problem is decidable for unions of conjunctive queries, $\textrm{Tower}$-complete in combined complexity, and elementary in data complexity. Already for atomic queries and ontologies that are sets of guarded existential rules, however, for every $k \geq 0$ there is an ontology and query that are $k$-$\textrm{Exp}$-hard in data complexity.

Marvin Grosser, Carsten Lutz

We study the problem of fitting a description logic (DL) ontology to a given set of positive and negative examples that take the form of an ABox and a Boolean query. While previous work has investigated this problem for the expressive DLs ALC and ALCI, we here focus on the Horn DLs EL and ELI, as well as their extensions with the bottom concept. As the query language, we consider atomic queries (AQs), conjunctive queries (rooted CQs), and unions thereof (rooted UCQs). We provide characterization of the existence of a fitting ontology based on simulations, use them to develop decision procedures, and clarify the exact computational complexity. For AQs, the problem is in PTime for both EL and ELI. For rooted CQs and UCQ, it is Sigma_P^2-complete for EL and ExpTime-complete for ELI. Adding the bottom concept does not change any of these complexities. Interestingly, moving from ALC and ALCI to EL and ELI introduces additional technical challenges rather than simplifying the matter.

Veeti Ahvonen, Damian Heiman, Antti Kuusisto et al.

In pioneering work from 2019, Barceló and coauthors identified logics that precisely match the expressive power of constant iteration-depth graph neural networks (GNNs) relative to properties definable in first-order logic. In this article, we give exact logical characterizations of recurrent GNNs in two scenarios: (1) in the setting with floating-point numbers and (2) with reals. For floats, the formalism matching recurrent GNNs is a rule-based modal logic with counting, while for reals we use a suitable infinitary modal logic, also with counting. These results give exact matches between logics and GNNs in the recurrent setting without relativising to a background logic in either case, but using some natural assumptions about floating-point arithmetic. Applying our characterizations, we also prove that, relative to graph properties definable in monadic second-order logic (MSO), our infinitary and rule-based logics are equally expressive. This implies that recurrent GNNs with reals and floats have the same expressive power over MSO-definable properties and shows that, for such properties, also recurrent GNNs with reals are characterized by a (finitary!) rule-based modal logic. In the general case, in contrast, the expressive power with floats is weaker than with reals. In addition to logic-oriented results, we also characterize recurrent GNNs, with both reals and floats, via distributed automata, drawing links to distributed computing models.

Moritz Schönherr, Carsten Lutz

We study the expressive power of graph neural networks (GNNs) with mean as the aggregation function. In the non-uniform setting, we show that such GNNs have exactly the same expressive power as ratio modal logic, which has modal operators expressing that at least a certain ratio of the successors of a vertex satisfies a specified property. The non-uniform expressive power of mean GNNs is thus higher than that of GNNs with max aggregation, but lower than for sum aggregation--the latter are characterized by modal logic and graded modal logic, respectively. In the uniform setting, we show that the expressive power relative to MSO is exactly that of alternation-free modal logic, under the natural assumptions that combination functions are continuous and classification functions are thresholds. This implies that, relative to MSO and in the uniform setting, mean GNNs are strictly less expressive than sum GNNs and max GNNs. When any of the assumptions is dropped, the expressive power increases.

Simon Hosemann, Jean Christoph Jung, Carsten Lutz et al.

We study the problem of fitting ontologies and constraints to positive and negative examples that take the form of a finite relational structure. As ontology and constraint languages, we consider the description logics $\mathcal{E\mkern-2mu L}$ and $\mathcal{E\mkern-2mu LI}$ as well as several classes of tuple-generating dependencies (TGDs): full, guarded, frontier-guarded, frontier-one, and unrestricted TGDs as well as inclusion dependencies. We pinpoint the exact computational complexity, design algorithms, and analyze the size of fitting ontologies and TGDs. We also investigate the related problem of constructing a finite basis of concept inclusions / TGDs for a given set of finite structures. While finite bases exist for $\mathcal{E\mkern-2mu L}$, $\mathcal{E\mkern-2mu LI}$, guarded TGDs, and inclusion dependencies, they in general do not exist for full, frontier-guarded and frontier-one TGDs.

Maurice Funk, Marvin Grosser, Carsten Lutz

We study a fitting problem inspired by ontology-mediated querying: given a collection of positive and negative examples of the form $(\mathcal{A},q)$ with $\mathcal{A}$ an ABox and $q$ a Boolean query, we seek an ontology $\mathcal{O}$ that satisfies $\mathcal{A} \cup \mathcal{O} \vDash q$ for all positive examples and $\mathcal{A} \cup \mathcal{O}\not\vDash q$ for all negative examples. We consider the description logics $\mathcal{ALC}$ and $\mathcal{ALCI}$ as ontology languages and a range of query languages that includes atomic queries (AQs), conjunctive queries (CQs), and unions thereof (UCQs). For all of the resulting fitting problems, we provide effective characterizations and determine the computational complexity of deciding whether a fitting ontology exists. This problem turns out to be ${\scriptsize CO}NP$ for AQs and full CQs and $2E{\scriptsize XP}T{\scriptsize IME}$-complete for CQs and UCQs. These results hold for both $\mathcal{ALC}$ and $\mathcal{ALCI}$.

Veeti Ahvonen, Maurice Funk, Damian Heiman et al.

Transformers are the basis of modern large language models, but relatively little is known about their precise expressive power on graphs. We study the expressive power of graph transformers (GTs) by Dwivedi and Bresson (2020) and GPS-networks by Rampásek et al. (2022), both under soft-attention and average hard-attention. Our study covers two scenarios: the theoretical setting with real numbers and the more practical case with floats. With reals, we show that in restriction to vertex properties definable in first-order logic (FO), GPS-networks have the same expressive power as graded modal logic (GML) with the global modality. With floats, GPS-networks turn out to be equally expressive as GML with the counting global modality. The latter result is absolute, not restricting to properties definable in a background logic. We also obtain similar characterizations for GTs in terms of propositional logic with the global modality (for reals) and the counting global modality (for floats).

Balder ten Cate, Maurice Funk, Jean Christoph Jung et al.

We propose bounded fitting as a scheme for learning description logic concepts in the presence of ontologies. A main advantage is that the resulting learning algorithms come with theoretical guarantees regarding their generalization to unseen examples in the sense of PAC learning. We prove that, in contrast, several other natural learning algorithms fail to provide such guarantees. As a further contribution, we present the system SPELL which efficiently implements bounded fitting for the description logic $\mathcal{ELH}^r$ based on a SAT solver, and compare its performance to a state-of-the-art learner.

Jean Christoph Jung, Carsten Lutz, Jerzy Marcinkowski

We study the problem to decide, given sets T1,T2 of tuple-generating dependencies (TGDs), also called existential rules, whether T2 is a conservative extension of T1. We consider two natural notions of conservative extension, one pertaining to answers to conjunctive queries over databases and one to homomorphisms between chased databases. Our main results are that these problems are undecidable for linear TGDs, undecidable for guarded TGDs even when T1 is empty, and decidable for frontier-one TGDs.

Anneke Haga, Carsten Lutz, Leif Sabellek et al.

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.

Maurice Funk, Jean Christoph Jung, Carsten Lutz

We consider the problem to learn a concept or a query in the presence of an ontology formulated in the description logic ELr, in Angluin's framework of active learning that allows the learning algorithm to interactively query an oracle (such as a domain expert). We show that the following can be learned in polynomial time: (1) EL-concepts, (2) symmetry-free ELI-concepts, and (3) conjunctive queries (CQs) that are chordal, symmetry-free, and of bounded arity. In all cases, the learner can pose to the oracle membership queries based on ABoxes and equivalence queries that ask whether a given concept/query from the considered class is equivalent to the target. The restriction to bounded arity in (3) can be removed when we admit unrestricted CQs in equivalence queries. We also show that EL-concepts are not polynomial query learnable in the presence of ELI-ontologies.

Jean Christoph Jung, Carsten Lutz, Mauricio Martel et al.

We investigate the decidability and computational complexity of conservative extensions and the related notions of inseparability and entailment in Horn description logics (DLs) with inverse roles. We consider both query conservative extensions, defined by requiring that the answers to all conjunctive queries are left unchanged, and deductive conservative extensions, which require that the entailed concept inclusions, role inclusions, and functionality assertions do not change. Upper bounds for query conservative extensions are particularly challenging because characterizations in terms of unbounded homomorphisms between universal models, which are the foundation of the standard approach to establishing decidability, fail in the presence of inverse roles. We resort to a characterization that carefully mixes unbounded and bounded homomorphisms and enables a decision procedure that combines tree automata and a mosaic technique. Our main results are that query conservative extensions are 2ExpTime-complete in all DLs between ELI and Horn-ALCHIF and between Horn-ALC and Horn-ALCHIF, and that deductive conservative extensions are 2ExpTime-complete in all DLs between ELI and ELHIF_\bot. The same results hold for inseparability and entailment.

Meghyn Bienvenu, Peter Hansen, Carsten Lutz et al.

We study FO-rewritability of conjunctive queries in the presence of ontologies formulated in a description logic between EL and Horn-SHIF, along with related query containment problems. Apart from providing characterizations, we establish complexity results ranging from ExpTime via NExpTime to 2ExpTime, pointing out several interesting effects. In particular, FO-rewriting is more complex for conjunctive queries than for atomic queries when inverse roles are present, but not otherwise.

Pablo Barcelo, Gerald Berger, Carsten Lutz et al.

We focus on ontology-mediated queries (OMQs) based on (frontier-)guarded existential rules and (unions of) conjunctive queries, and we investigate the problem of FO-rewritability, i.e., whether an OMQ can be rewritten as a first-order query. We adopt two different approaches. The first approach employs standard two-way alternating parity tree automata. Although it does not lead to a tight complexity bound, it provides a transparent solution based on widely known tools. The second approach relies on a sophisticated automata model, known as cost automata. This allows us to show that our problem is 2ExpTime-complete. In both approaches, we provide semantic characterizations of FO-rewritability that are of independent interest.

Carsten Lutz, Johannes Marti, Leif Sabellek

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 $Π^p_2$-complete in DL-Lite, coNExpTime-complete between EL and ELHI when source queries are rooted, and 2ExpTime-complete for unrestricted source queries.

Cristina Feier, Carsten Lutz, Frank Wolter

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.

Pierre Bourhis, Carsten Lutz

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.

Jean Christoph Jung, Carsten Lutz, Hadrien Pulcini et al.

We study the separation of positive and negative data examples in terms of description logic (DL) concepts and formulas of decidable FO fragments, in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols from the data and ontology that can be used for separation. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated. Our main results are that (a projective form of) the weak version is decidable in $\mathcal{ALCI}$ while it is undecidable in the guarded fragment GF, the guarded negation fragment GNF, and the DL $\mathcal{ALCFIO}$, and that strong separability is decidable in $\mathcal{ALCI}$, GF, and GNF. We also provide (mostly tight) complexity bounds.

Jean Christoph Jung, Carsten Lutz, Hadrien Pulcini et al.

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, generating referring expressions, and entity comparison in knowledge graphs. In this paper, we investigate the existence of a separating formula for data in the presence of an ontology. Both for the ontology language and the separation language, we concentrate on first-order logic and the following important fragments thereof: the description logic $\mathcal{ALCI}$, the guarded fragment, the two-variable fragment, and the guarded negation fragment. For separation, we also consider (unions of) conjunctive queries. 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. Our main results are model-theoretic characterizations of (all variants of) separability, the comparison of the separating power of different languages, and the investigation of the computational complexity of deciding separability.

Angela Bonifati, Giovanna Guerrini, Carsten Lutz et al.

The joint EDBT/ICDT conference (International Conference on Extending Database Technology/International Conference on Database Theory) is a well established conference series on data management, with annual meetings in the second half of March that attract 250 to 300 delegates. Three weeks before EDBT/ICDT 2020 was planned to take place in Copenhagen, the rapidly developing Covid-19 pandemic led to the decision to cancel the face-to-face event. In the interest of the research community, it was decided to move the conference online while trying to preserve as much of the real-life experience as possible. As far as we know, we are one of the first conferences that moved to a fully synchronous online experience due to the COVID-19 outbreak. With fully synchronous, we mean that participants jointly listened to presentations, had live Q&A, and attended other live events associated with the conference. In this report, we share our decisions, experiences, and lessons learned.

Pablo Barcelo, Cristina Feier, Carsten Lutz et al.

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.

Anneke Haga, Carsten Lutz, Johannes Marti et al.

We study complete approximations of an ontology formulated in a non-Horn description logic (DL) such as $\mathcal{ALC}$ in a Horn DL such as~$\mathcal{EL}$. We provide concrete approximation schemes that are necessarily infinite and observe that in the $\mathcal{ELU}$-to-$\mathcal{EL}$ case finite approximations tend to exist in practice and are guaranteed to exist when the original ontology is acyclic. In contrast, neither of this is the case for $\mathcal{ELU}_\bot$-to-$\mathcal{EL}_\bot$ and for $\mathcal{ALC}$-to-$\mathcal{EL}_\bot$ approximations. We also define a notion of approximation tailored towards ontology-mediated querying, connect it to subsumption-based approximations, and identify a case where finite approximations are guaranteed to exist.

Pablo Barcelo, Victor Dalmau, Cristina Feier et al.

Ontology-mediated querying and querying in the presence of constraints are two key database problems where tuple-generating dependencies (TGDs) play a central role. In ontology-mediated querying, TGDs can formalize the ontology and thus derive additional facts from the given data, while in querying in the presence of constraints, they restrict the set of admissible databases. In this work, we study the limits of efficient query evaluation in the context of the above two problems, focussing on guarded and frontier-guarded TGDs and on UCQs as the actual queries. We show that a class of ontology-mediated queries (OMQs) based on guarded TGDs can be evaluated in FPT iff the OMQs in the class are equivalent to OMQs in which the actual query has bounded treewidth, up to some reasonable assumptions. For querying in the presence of constraints, we consider classes of constraint-query specifications (CQSs) that bundle a set of constraints with an actual query. We show a dichotomy result for CQSs based on guarded TGDs that parallels the one for OMQs except that, additionally, FPT coincides with PTime combined complexity. The proof is based on a novel connection between OMQ and CQS evaluation. Using a direct proof, we also show a similar dichotomy result, again up to some reasonable assumptions, for CQSs based on frontier-guarded TGDs with a bounded number of atoms in TGD heads. Our results on CQSs can be viewed as extensions of Grohe's well-known characterization of the tractable classes of CQs (without constraints). Like Grohe's characterization, all the above results assume that the arity of relation symbols is bounded by a constant. We also study the associated meta problems, i.e., whether a given OMQ or CQS is equivalent to one in which the actual query has bounded treewidth.

Carsten Lutz, Leif Sabellek

We provide an ultimately fine-grained analysis of the data complexity and rewritability of ontology-mediated queries (OMQs) based on an EL ontology and a conjunctive query (CQ). Our main results are that every such OMQ is in AC0, NL-complete, or PTime-complete and that containment in NL coincides with rewritability into linear Datalog (whereas containment in AC0 coincides with rewritability into first-order logic). We establish natural characterizations of the three cases in terms of bounded depth and (un)bounded pathwidth, and show that every of the associated meta problems such as deciding wether a given OMQ is rewritable into linear Datalog is ExpTime-complete. We also give a way to construct linear Datalog rewritings when they exist and prove that there is no constant Datalog rewritings.

Elena Botoeva, Carsten Lutz, Vladislav Ryzhikov et al.

We investigate the problem whether two ALC ontologies are indistinguishable (or inseparable) by means of queries in a given signature, which is fundamental for ontology engineering tasks such as ontology versioning, modularisation, update, and forgetting. We consider both knowledge base (KB) and TBox inseparability. For KBs, we give model-theoretic criteria in terms of (finite partial) homomorphisms and products and prove that this problem is undecidable for conjunctive queries (CQs), but 2ExpTime-complete for unions of CQs (UCQs). The same results hold if (U)CQs are replaced by rooted (U)CQs, where every variable is connected to an answer variable. We also show that inseparability by CQs is still undecidable if one KB is given in the lightweight DL EL and if no restrictions are imposed on the signature of the CQs. We also consider the problem whether two ALC TBoxes give the same answers to any query over any ABox in a given signature and show that, for CQs, this problem is undecidable, too. We then develop model-theoretic criteria for Horn-ALC TBoxes and show using tree automata that, in contrast, inseparability becomes decidable and 2ExpTime-complete, even ExpTime-complete when restricted to (unions of) rooted CQs.

Elena Botoeva, Boris Konev, Carsten Lutz et al.

The question whether an ontology can safely be replaced by another, possibly simpler, one is fundamental for many ontology engineering and maintenance tasks. It underpins, for example, ontology versioning, ontology modularization, forgetting, and knowledge exchange. What safe replacement means depends on the intended application of the ontology. If, for example, it is used to query data, then the answers to any relevant ontology-mediated query should be the same over any relevant data set; if, in contrast, the ontology is used for conceptual reasoning, then the entailed subsumptions between concept expressions should coincide. This gives rise to different notions of ontology inseparability such as query inseparability and concept inseparability, which generalize corresponding notions of conservative extensions. We survey results on various notions of inseparability in the context of description logic ontologies, discussing their applications, useful model-theoretic characterizations, algorithms for determining whether two ontologies are inseparable (and, sometimes, for computing the difference between them if they are not), and the computational complexity of this problem.

Peter Hansen, Carsten Lutz

A prominent approach to implementing ontology-mediated queries (OMQs) is to rewrite into a first-order query, which is then executed using a conventional SQL database system. We consider the case where the ontology is formulated in the description logic EL and the actual query is a conjunctive query and show that rewritings of such OMQs can be efficiently computed in practice, in a sound and complete way. Our approach combines a reduction with a decomposed backwards chaining algorithm for OMQs that are based on the simpler atomic queries, also illuminating the relationship between first-order rewritings of OMQs based on conjunctive and on atomic queries. Experiments with real-world ontologies show promising results.

Andre Hernich, Carsten Lutz, Fabio Papacchini et al.

We study the complexity of ontology-mediated querying when ontologies are formulated in the guarded fragment of first-order logic (GF). Our general aim is to classify the data complexity on the level of ontologies where query evaluation w.r.t. an ontology O is considered to be in PTime if all (unions of conjunctive) queries can be evaluated in PTime w.r.t. O and coNP-hard if at least one query is coNP-hard w.r.t. O. We identify several large and relevant fragments of GF that enjoy a dichotomy between PTime and coNP, some of them additionally admitting a form of counting. In fact, almost all ontologies in the BioPortal repository fall into these fragments or can easily be rewritten to do so. We then establish a variation of Ladner's Theorem on the existence of NP-intermediate problems and use this result to show that for other fragments, there is provably no such dichotomy. Again for other fragments (such as full GF), establishing a dichotomy implies the Feder-Vardi conjecture on the complexity of constraint satisfaction problems. We also link these results to Datalog-rewritability and study the decidability of whether a given ontology enjoys PTime query evaluation, presenting both positive and negative results.

Boris Konev, Carsten Lutz, Ana Ozaki et al.

We study the problem of learning description logic (DL) ontologies in Angluin et al.'s framework of exact learning via queries. We admit membership queries ("is a given subsumption entailed by the target ontology?") and equivalence queries ("is a given ontology equivalent to the target ontology?"). We present three main results: (1) ontologies formulated in (two relevant versions of) the description logic DL-Lite can be learned with polynomially many queries of polynomial size; (2) this is not the case for ontologies formulated in the description logic EL, even when only acyclic ontologies are admitted; and (3) ontologies formulated in a fragment of EL related to the web ontology language OWL 2 RL can be learned in polynomial time. We also show that neither membership nor equivalence queries alone are sufficient in cases (1) and (3).

Carsten Lutz, Frank Wolter

We analyze the data complexity of ontology-mediated querying where the ontologies are formulated in a description logic (DL) of the ALC family and queries are conjunctive queries, positive existential queries, or acyclic conjunctive queries. Our approach is non-uniform in the sense that we aim to understand the complexity of each single ontology instead of for all ontologies formulated in a certain language. While doing so, we quantify over the queries and are interested, for example, in the question whether all queries can be evaluated in polynomial time w.r.t. a given ontology. Our results include a PTime/coNP-dichotomy for ontologies of depth one in the description logic ALCFI, the same dichotomy for ALC- and ALCI-ontologies of unrestricted depth, and the non-existence of such a dichotomy for ALCF-ontologies. For the latter DL, we additionally show that it is undecidable whether a given ontology admits PTime query evaluation. We also consider the connection between PTime query evaluation and rewritability into (monadic) Datalog.

Piero A. Bonatti, Carsten Lutz, Frank Wolter

As fragments of first-order logic, Description logics (DLs) do not provide nonmonotonic features such as defeasible inheritance and default rules. Since many applications would benefit from the availability of such features, several families of nonmonotonic DLs have been developed that are mostly based on default logic and autoepistemic logic. In this paper, we consider circumscription as an interesting alternative approach to nonmonotonic DLs that, in particular, supports defeasible inheritance in a natural way. We study DLs extended with circumscription under different language restrictions and under different constraints on the sets of minimized, fixed, and varying predicates, and pinpoint the exact computational complexity of reasoning for DLs ranging from ALC to ALCIO and ALCQO. When the minimized and fixed predicates include only concept names but no role names, then reasoning is complete for NExpTime^NP. It becomes complete for NP^NExpTime when the number of minimized and fixed predicates is bounded by a constant. If roles can be minimized or fixed, then complexity ranges from NExpTime^NP to undecidability.

Meghyn Bienvenu, Balder ten Cate, Carsten Lutz et al.

Ontology-based data access is concerned with querying incomplete data sources in the presence of domain-specific knowledge provided by an ontology. A central notion in this setting is that of an ontology-mediated query, which is a database query coupled with an ontology. In this paper, we study several classes of ontology-mediated queries, where the database queries are given as some form of conjunctive query and the ontologies are formulated in description logics or other relevant fragments of first-order logic, such as the guarded fragment and the unary-negation fragment. The contributions of the paper are three-fold. First, we characterize the expressive power of ontology-mediated queries in terms of fragments of disjunctive datalog. Second, we establish intimate connections between ontology-mediated queries and constraint satisfaction problems (CSPs) and their logical generalization, MMSNP formulas. Third, we exploit these connections to obtain new results regarding (i) first-order rewritability and datalog-rewritability of ontology-mediated queries, (ii) P/NP dichotomies for ontology-mediated queries, and (iii) the query containment problem for ontology-mediated queries.