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).

We study the expressive power of graph neural networks (GNNs) with mean as the aggregation function, with the following results. In the non-uniform setting, 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. In the uniform setting, the expressive power relative to MSO is exactly that of modal logic, and thus identical to the (absolute) expressive power of GNNs with max aggregation. The proof, however, depends on constructions that are not satisfactory from a practical perspective. This leads us to making the natural assumptions that combination functions are continuous and classification functions are thresholds. The resulting class of GNNs with mean aggregation turns out to be much less expressive: relative to MSO and in the uniform setting, it has the same expressive power as alternation-free modal logic. This is in contrast to the expressive power of GNNs with max and sum aggregation, which is not affected by these assumptions.

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 (CQs), and unions thereof (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 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.

Valued Constraint Satisfaction Problems (VCSPs) constitute a large class of computational optimization problems. It was recently shown that, over finite domains, every VCSP is in P or NP-complete, depending on the admitted cost functions. In this article, we study cost functions over countably infinite domains whose automorphisms form an oligomorphic permutation group. Our results include a hardness condition based on a generalization of pp-constructability as known from classical CSPs and a polynomial-time tractability condition based on the concept of fractional polymorphisms. We then observe that the resilience problem for Unions of Conjunctive Queries (UCQs) studied in database theory, under bag semantics, may be viewed as a special case of the VCSPs that we consider. We obtain a complexity dichotomy for the case of incidence-acyclic UCQs and exemplarily use our methods to determine the complexity of a conjunctive query that has been stated as an open problem in the literature. We conjecture that our hardness and tractability conditions match for resilience problems for UCQs. Further, we obtain a complete dichotomy for resilience problems for two-way regular path queries, under bag semantics.

We formalize and study the problem of repairing database queries based on user feedback in the form of a collection of labeled examples. We propose a framework based on the notion of a proximity pre-order, and we investigate and compare query repairs for conjunctive queries (CQs) using different such pre-orders. The proximity pre-orders we consider are based on query containment and on distance metrics for CQs.

We study a fitting problem inspired by ontology-mediated querying: given a collection of positive and negative examples of the form (A, q) with A an ABox and q a query, we seek an ontology O such that A ∪ O entails q for all positive examples (A, q) and A ∪ O does not entail q for all negative examples (A, q). We consider the description logics ALC and 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 coNP-complete for AQs and full CQs and 2ExpTime-complete for CQs and UCQs. These results hold for both ALC and ALCI.

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 EL and ELI 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 EL, ELI, guarded TGDs, and inclusion dependencies, they in general do not exist for full, frontier-guarded and frontier-one TGDs.

We study extensions of expressive decidable fragments of first-order logic with circumscription, considering 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 NExp to NExp^NP-complete, for GF it (remarkably!) increases from 2Exp to Tower-complete, and for C^2 it 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, 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 > 0 there is an ontology and query that are k-Exp-hard in data complexity.

We study extensions of description logics from the widely used EL family with operators that make it possible to speak about different levels of abstraction. We analyze the computational complexity of reasoning and show that often, this complexity is significantly lower than in the corresponding extension of the more expressive description logic ALC. By slightly varying the semantics, we also obtain a case that admits reasoning in polynomial time.

Valued constraint satisfaction problems (VCSPs) constitute a large class of computational optimisation problems. It was shown recently that, over finite domains, every VCSP is in P or NP-complete, depending on the admitted cost functions. In this article, we study cost functions over countably infinite domains whose automorphisms form an oligomorphic permutation group. Our results include a hardness condition based on a generalisation of pp-constructability as known from classical CSPs and a polynomial-time tractability condition based on the concept of fractional polymorphisms. We then observe that the resilience problem for unions of conjunctive queries (UCQs) studied in database theory, under bag semantics, may be viewed as a special case of the VCSPs that we consider. We obtain a complexity dichotomy for the case of incidence-acyclic UCQs and exemplarily use our methods to determine the complexity of a query that had remained open in the literature. Further, we conjecture that our hardness and tractability conditions match for resilience problems for UCQs.

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.

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.

We study the enumeration of answers to ontology-mediated queries when the ontology is formulated in a description logic that supports functional roles and the query is a CQ. In particular, we show that enumeration is possible with linear preprocessing and constant delay when a certain extension of the CQ (pertaining to functional roles) is acyclic and free-connex acyclic. This holds both for complete answers and for partial answers. We provide matching lower bounds for the case where the query is self-join free.

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 ELHr based on a SAT solver, and compare its performance to a state-of-the-art learner.

Circumscription is one of the main approaches for defining non-monotonic description logics (DLs) and the decidability and complexity of traditional reasoning tasks, such as satisfiability of circumscribed DL knowledge bases (KBs) are well understood. For evaluating conjunctive queries (CQs) and unions thereof (UCQs), in contrast, 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).

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.

The fitting problem for conjunctive queries (CQs) is the problem to construct a CQ that fits a given set of labeled data examples. When a fitting CQ exists, it is in general not unique. This leads us to proposing natural refinements of the notion of a fitting CQ, such as most-general fitting CQ, most-specific fitting CQ, and unique fitting CQ. We give structural characterizations of these notions in terms of (suitable refinements of) homomorphism dualities, frontiers, and direct products, which enable the construction of the refined fitting CQs when they exist. We also pinpoint the complexity of the associated existence and verification problems, and determine the size of fitting CQs. We study the same problems for UCQs and for the more restricted class of tree CQs.

We study the complexity of answer counting for ontology-mediated queries and for querying under constraints, considering conjunctive queries and unions thereof (UCQs) as the query language and guarded TGDs as the ontology and constraint language, respectively. Our main result is a classification according to whether answer counting is fixed-parameter tractable (FPT), W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent, lifting a recent classification for UCQs without ontologies and constraints due to Dell et al. The classification pertains to various structural measures, namely treewidth, contract treewidth, starsize, and linked matching number. Our results rest on the assumption that the arity of relation symbols is bounded by a constant and, in the case of ontology-mediated querying, that all symbols from the ontology and query can occur in the data (so-called full data schema). We also study the meta-problems for the mentioned structural measures, that is, to decide whether a given ontology-mediated query or constraint-query specification is equivalent to one for which the structural measure is bounded.

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 our 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.

We study the enumeration of answers to ontology-mediated queries (OMQs) where the ontology is a set of guarded TGDs or formulated in the description logic ELI and the query is a conjunctive query (CQ). In addition to the traditional notion of an answer, we propose and study two novel notions of partial answers that can take into account nulls generated by existential quantifiers in the ontology. Our main result is that enumeration of the traditional complete answers and of both kinds of partial answers is possible with linear-time preprocessing and constant delay for OMQs that are both acyclic and free-connex acyclic. We also provide partially matching lower bounds. Similar results are obtained for the related problems of testing a single answer in linear time and of testing multiple answers in constant time after linear time preprocessing. In both cases, the border between tractability and intractability is characterized by similar, but slightly different acyclicity properties.

We study the complexity of answer counting for ontology-mediated queries and for querying under constraints, considering conjunctive queries and unions thereof (UCQs) as the query language and guarded TGDs as the ontology and constraint language, respectively. Our main result is a classification according to whether answer counting is fixed-parameter tractable (FPT), W[1]-equivalent, #W[1]-equivalent, #W[2]-hard, or #A[2]-equivalent, lifting a recent classification for UCQs without ontologies and constraints due to Dell et al. [Holger Dell et al., 2019]. The classification pertains to various structural measures, namely treewidth, contract treewidth, starsize, and linked matching number. Our results rest on the assumption that the arity of relation symbols is bounded by a constant and, in the case of ontology-mediated querying, that all symbols from the ontology and query can occur in the data (so-called full data schema). We also study the meta-problems for the mentioned structural measures, that is, to decide whether a given ontology-mediated query or constraint-query specification is equivalent to one for which the structural measure is bounded.

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.

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.

We study the separation of positive and negative data examples in terms of description logic concepts in the presence of an ontology. In contrast to previous work, we add a signature that specifies a subset of the symbols that can be used for separation, and we admit individual names in that signature. We consider weak and strong versions of the resulting problem that differ in how the negative examples are treated and we distinguish between separation with and without helper symbols. Within this framework, we compare the separating power of different languages and investigate the complexity of deciding separability. While weak separability is shown to be closely related to conservative extensions, strongly separating concepts coincide with Craig interpolants, for suitably defined encodings of the data and ontology. This enables us to transfer known results from those fields to separability. Conversely, we obtain original results on separability that can be transferred backward. For example, rather surprisingly, conservative extensions and weak separability in ALCO are both 3ExpTime-complete.

We study two forms of least general generalizations in description logic, the least common subsumer (LCS) and most specific concept (MSC). While the LCS generalizes from examples that take the form of concepts, the MSC generalizes from individuals in data. Our focus is on the complexity of existence and verification, the latter meaning to decide whether a candidate concept is the LCS or MSC. We consider cases with and without a background TBox and a target signature. Our results range from coNP-complete for LCS and MSC verification in the description logic εℒ without TBoxes to undecidability of LCS and MSC verification and existence in εℒI with TBoxes. To obtain results in the presence of a TBox, we establish a close link between the problems studied in this paper and concept learning from positive and negative examples. We also give a way to regain decidability in εℒI with TBoxes and study single example MSC as a special case.

We study complete approximations of an ontology formulated in a non-Horn description logic (DL) such as ALC in a Horn DL such as EL. We provide concrete approximation schemes that are necessarily infinite and observe that in the ELU-to-EL case finite approximations tend to exist in practice and are guaranteed to exist when the source ontology is acyclic. In contrast, neither of this is the case for ELU_bot-to-EL_bot and for ALC-to-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.

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 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.

We consider fragments of the description logic SHOIF extended with regular expressions on roles. Our main result is that satisfiability and finite satisfiability are decidable in two fragments SHOIF^1 and SHOIF^2, NExpTime-complete for the former and in 2NExpTime for the more expressive latter fragment. Both fragments impose restrictions on regular role expressions of the form r*. SHOIF^1 encompasses the extension of SHOIF with transitive closure of roles (when functional roles have no subroles) and the modal logic of linear orders and successor, with converse. Consequently, these logics are also decidable and NExpTime-complete.

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, focusing 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.

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.

We study ontology-mediated querying in the case where ontologies are formulated in the guarded fragment of first-order logic (GF) or extensions thereof with counting and where the actual queries are (unions of) conjunctive queries. Our aim is to classify the data complexity and Datalog rewritability of query evaluation depending on the ontology O , where query evaluation w.r.t. O is in PT ime (resp. Datalog rewritable) if all queries can be evaluated in PT ime w.r.t. O (resp. rewritten into Datalog under O ), and co NP-hard if at least one query is co NP-hard w.r.t. O . We identify several fragments of GF that enjoy a dichotomy between Datalog-rewritability (which implies PT ime ) and co NP-hardness as well as several other fragments that enjoy a dichotomy between PT ime and co NP-hardness, but for which PT ime does not imply Datalog-rewritability. For the latter, we establish and exploit a connection to constraint satisfaction problems. We also identify fragments for which there is no dichotomy between PT ime and co NP. To prove this, we establish a non-trivial variation of Ladner’s theorem on the existence of NP-intermediate problems. Finally, we study the decidability of whether a given ontology enjoys PT ime query evaluation, presenting both positive and negative results, depending on the fragment.

We study the approximation of a description logic (DL) ontology in a less expressive DL, focusing on the case of Horn DLs. It is common to construct such approximations in an ad hoc way in practice and the resulting incompleteness is typically neither analyzed nor understood. In this paper, we show how to construct complete approximations. These are typically infinite or of excessive size and thus cannot be used directly in applications, but our results provide an important theoretical foundation that enables informed decisions when constructing incomplete approximations in practice.

Learning description logic (DL) concepts from positive and negative examples given in the form of labeled data items in a KB has received significant attention in the literature. We study the fundamental question of when a separating DL concept exists and provide useful model-theoretic characterizations as well as complexity results for the associated decision problem. For expressive DLs such as ALC and ALCQI, our characterizations show a surprising link to the evaluation of ontology-mediated conjunctive queries. We exploit this to determine the combined complexity (between ExpTime and NExpTime) and data complexity (second level of the polynomial hierarchy) of separability. For the Horn DL EL, separability is ExpTime-complete both in combined and in data complexity while for its modest extension ELI it is even undecidable. Separability is also undecidable when the KB is formulated in ALC and the separating concept is required to be in EL or ELI.

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⊥, we provide a characterization of the classes of OMQs that are fixed-parameter tractable. For its fragment ELH⊥dr, 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, unlike εLHI⊥, admit functional roles.

We study rewritability of monadic disjunctive Datalog programs, (the complements of) MMSNP sentences, and ontology-mediated queries (OMQs) based on expressive description logics of the ALC family and on conjunctive queries. We show that rewritability into FO and into monadic Datalog (MDLog) are decidable, and that rewritability into Datalog is decidable when the original query satisfies a certain condition related to equality. We establish 2NExpTime-completeness for all studied problems except rewritability into MDLog for which there remains a gap between 2NExpTime and 3ExpTime. We also analyze the shape of rewritings, which in the MMSNP case correspond to obstructions, and give a new construction of canonical Datalog programs that is more elementary than existing ones and also applies to formulas with free variables.

In the context of ontology-mediated querying with description logics (DLs), we study the data complexity of queries in which selected predicates can be closed (OMQCs). We provide a non-uniform analysis, aiming at a classification of the complexity into tractable and non-tractable for ontologies in the lightweight DLs DL-Lite and EL, and the expressive DL ALCHI. At the level of ontologies, we prove a dichotomy between FO-rewritable and coNP-complete for DL-Lite and between PTime and coNP-complete for EL. The meta problem of deciding tractability is proved to be in PTime. At the level of OMQCs, we show that there is no dichotomy (unless NP equals PTime) if both concept and role names can be closed. If only concept names can be closed, we tightly link the complexity of query evaluation to the complexity of surjective CSPs. We also identify a class of OMQCs based on ontologies formulated in DL-Lite that are guaranteed to be tractable and even FO-rewritable.

The unary negation fragment of first-order logic (UNFO) has recently been proposed as a generalization of modal logic that shares many of its good computational and model-theoretic properties. It is attractive from the perspective of database theory because it can express conjunctive queries (CQs) and ontologies formulated in many description logics (DLs). Both are relevant for ontology-mediated querying and, in fact, CQ evaluation under UNFO ontologies (and thus also under DL ontologies) can be `expressed' in UNFO as a satisfiability problem. In this paper, we consider the natural extension of UNFO with regular expressions on binary relations. The resulting logic UNFOreg can express (unions of) conjunctive two-way regular path queries (C2RPQs) and ontologies formulated in DLs that include transitive roles and regular expressions on roles. Our main results are that evaluating C2RPQs under UNFOreg ontologies is decidable, 2ExpTime-complete in combined complexity, and coNP-complete in data complexity, and that satisfiability in UNFOreg is 2ExpTime-complete, thus not harder than in UNFO.

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.

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 (in other words: the complement of a monadic disjunctive Datalog program) is equivalent to a constraint satisfaction problem.

In ontology-mediated querying with an expressive description logic L, two desirable properties of a TBox T are (1) being able to replace T with a TBox formulated in the Horn-fragment of L without affecting the answers to conjunctive queries, and (2) that every conjunctive query can be evaluated in PTime w.r.t. T. We investigate in which cases (1) and (2) are equivalent, finding that the answer depends on whether the unique name assumption (UNA) is made, on the description logic under consideration, and on the nesting depth of quantifiers in the TBox. We also clarify the relationship between query evaluation with and without UNA and consider natural variations of property (1).

It was recently shown by van den Broeck at al. that the symmetric weighted first-order model counting problem (WFOMC) for sentences of two-variable logic FO2 is in polynomial time, while it is #P1-complete for some FO3-sentences. We extend the result for FO2 in two independent directions: to sentences of the form φ∧∀x∃=1 y ψ (x, y) with φ and ψ formulated in FO2 and to sentences of the uniform one-dimensional fragment U1 of FO, a recently introduced extension of two-variable logic with the capacity to deal with relation symbols of all arities. We note that the former generalizes the extension of FO2 with a functional relation symbol. We also identify a complete classification of first-order prefix classes according to whether WFOMC is in polynomial time or #P1-complete.

We investigate the decidability and computational complexity of (deductive) conservative extensions in fragments of first-order logic (FO), with a focus on the two-variable fragment FO$^2$ and the guarded fragment GF. We prove that conservative extensions are undecidable in any FO fragment that contains FO$^2$ or GF (even the three-variable fragment thereof), and that they are decidable and 2\ExpTime-complete in the intersection GF$^2$ of FO$^2$ and GF.

We study rewritability of monadic disjunctive Datalog programs, (the complements of) MMSNP sentences, and ontology-mediated queries (OMQs) based on expressive description logics of the ALC family and on conjunctive queries. We show that rewritability into FO and into monadic Datalog (MDLog) are decidable, and that rewritability into Datalog is decidable when the original query satisfies a certain condition related to equality. We establish 2NExpTime-completeness for all studied problems except rewritability into MDLog for which there remains a gap between 2NExpTime and 3ExpTime. We also analyze the shape of rewritings, which in the MMSNP case correspond to obstructions, and give a new construction of canonical Datalog programs that is more elementary than existing ones and also applies to non-Boolean queries.

We investigate the decidability and computational complexity of query conservative extensions in Horn description logics (DLs) with inverse roles. This is more challenging than without inverse roles because characterizations in terms of unbounded homomorphisms between universal models fail, blocking the standard approach to establishing decidability. We resort to a combination of automata and mosaic techniques, proving that the problem is 2EXPTIME-complete in Horn-ALCHIF (and also in Horn-ALC and in ELI). We obtain the same upper bound for deductive conservative extensions, for which we also prove a coNEXPTIME lower bound.

We consider ontology-mediated queries (OMQs) based on an EL ontology and an atomic query (AQ), provide an ultimately fine-grained analysis of data complexity and study rewritability into linear Datalog-aiming to capture linear recursion in SQL. 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, show that deciding linear Datalog rewritability (as well as the mentioned complexities) is ExpTime-complete, give a way to construct linear Datalog rewritings when they exist, and prove that there is no constant bound on the arity of IDB relations in linear Datalog rewritings.

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.

We propose a family of probabilistic description logics (DLs) that are derived in a principled way from Halpern's probabilistic first-order logic. The resulting probabilistic DLs have a two-dimensional semantics similar to temporal DLs and are well-suited for representing subjective probabilities. We carry out a detailed study of reasoning in the new family of logics, concentrating on probabilistic extensions of the DLs ALC and EL, and showing that the complexity ranges from PTime via ExpTime and 2ExpTime to undecidable.

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.

Data sets that have been collected from multiple sources or extracted from the web or often highly incomplete and heterogeneous, which makes them hard to process and query. One way to address this challenge is to use ontologies, which provide a way to assign a semantics to the data, to enrich it with domain knowledge, and to provide an enriched and uniform vocabulary for querying. The combination of a traditional database query with an ontology is called an ontology-mediated query (OMQ). The aim of this talk is to survey fundamental properties of OMQs such as their complexity, expressive power, descriptive strength, and rewritability into traditional query languages such as SQL and Datalog. A central observation is that there is a close and fruitful connection between OMQs and constraint satisfaction problems (CSPs) as well as related fragments of monadic NP, which puts OMQs into a more general perspective and gives raise to a number of interesting results.

In ontology-based data access (OBDA), database querying is enriched with an ontology that provides domain knowledge and additional vocabulary for query formulation. We identify query emptiness and predicate emptiness as two central reasoning services in this context. Query emptiness asks whether a given query has an empty answer over all databases formulated in a given vocabulary. Predicate emptiness is defined analogously, but quantifies universally over all queries that contain a given predicate. In this paper, we determine the computational complexity of query emptiness and predicate emptiness in the EL, DL-Lite, and ALC-families of description logics, investigate the connection to ontology modules, and perform a practical case study to evaluate the new reasoning services.

Recently, Fontaine has pointed out a connection between consistent query answering (CQA) and constraint satisfaction problems (CSP) [Fontaine, LICS 2013]. We investigate this connection more closely, identifying classes of CQA problems based on denial constraints and GAV constraints that correspond exactly to CSPs in the sense that a complexity classification of the CQA problems in each class is equivalent (up to FO-reductions) to classifying the complexity of all CSPs. We obtain these classes by admitting only monadic relations and only a single variable in denial constraints/GAVs and restricting queries to hypertree UCQs. We also observe that dropping the requirement of UCQs to be hypertrees corresponds to transitioning from CSP to its logical generalization MMSNP and identify a further relaxation that corresponds to transitioning from MMSNP to GMSNP (also know as MMSNP_2). Moreover, we use the CSP connection to carry over decidability of FO-rewritability and Datalog-rewritability to some of the identified classes of CQA problems.

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 article, 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 article are threefold. First, we show that popular ontology-mediated query languages have the same expressive power as natural fragments of disjunctive datalog, and we study the relative succinctness of ontology-mediated queries and disjunctive datalog queries. 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.

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.

Most of the research on temporalized Description Logics (DLs) has concentrated on the case where temporal operators can be applied to concepts, and sometimes additionally to TBox axioms and ABox assertions. The aim of this article is to study temporalized DLs where temporal operators on TBox axioms and ABox assertions are available, but temporal operators on concepts are not. While the main application of existing temporalized DLs is the representation of conceptual models that explicitly incorporate temporal aspects, the family of DLs studied in this article addresses applications that focus on the temporal evolution of data and of ontologies. Our results show that disallowing temporal operators on concepts can significantly decrease the complexity of reasoning. In particular, reasoning with rigid roles (whose interpretation does not change over time) is typically undecidable without such a syntactic restriction, whereas our logics are decidable in elementary time even in the presence of rigid roles. We analyze the effects on computational complexity of dropping rigid roles, dropping rigid concepts, replacing temporal TBoxes with global ones, and restricting the set of available temporal operators. In this way, we obtain a novel family of temporalized DLs whose complexity ranges from 2- ExpTime-complete via NExpTime-complete to ExpTime-complete.

We study probabilistic variants of the description logic EL. For the case where probabilities apply only to concepts, we provide a careful analysis of the borderline between tractability and ExpTime-completeness. One outcome is that any probability value except zero and one leads to intractability in the presence of general TBoxes, while this is not the case for classical TBoxes. For the case where probabilities can also be applied to roles, we show PSpace-completeness. This result is (positively) surprising as the best previously known upper bound was 2-ExpTime and there were reasons to believe in completeness for this class.

We characterize the expressive power of descrip-tion logic (DL) TBoxes, both for expressive DLs such as ALC and ALCQIO and lightweight DLs such as DL-Lite and EL. Our characterizations are relative to first-order logic, based on a wide range of semantic notions such as bisimulation, equisim-ulation, disjoint union, and direct product. We ex-emplify the use of the characterizations by a first study of the following novel family of decision problems: given a TBox T formulated in a DL L, decide whether T can be equivalently rewritten as a TBox in the fragment L ′ of L. 1

We study uniform interpolation and forgetting in the description logic ALC. Our main results are model-theoretic characterizations of uniform interpolants and their existence in terms of bisimulations, tight complexity bounds for deciding the existence of uniform interpolants, an approach to computing interpolants when they exist, and tight bounds on their size. We use a mix of modeltheoretic and automata-theoretic methods that, as a by-product, also provides characterizations of and decision procedures for conservative extensions. 1

XPath is a prominent W3C standard for navigating XML documents that has stimulated a lot of research into query answering and static analysis. In particular, query containment has been studied extensively for fragments of the 1.0 version of this standard, whereas little is known about query containment in (fragments of) the richer language XPath 2.0. In this article, we consider extensions of CoreXPath, the navigational core of XPath 1.0, with operators that are part of or inspired by XPath 2.0: path intersection, path equality, path complementation, for-loops, and transitive closure. For each combination of these operators, we determine the complexity of query containment, both with and without DTDs. It turns out to range from ExpTime (for extensions with path equality) and 2-ExpTime (for extensions with path intersection) to non-elementary (for extensions with path complementation or for-loops). In almost all cases, adding transitive closure on top has no further impact on the complexity. We also investigate the effect of dropping the upward and/or sibling axes, and show that this sometimes leads to a reduction in complexity. Since the languages we study include negation and conjunction in filters, our complexity results can equivalently be stated in terms of satisfiability. We also analyze the above languages in terms of succinctness.

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.

Abstract We study satisfiability and infinite-state model checking in ICPDL, which extends Propositional Dynamic Logic (PDL) with intersection and converse operators on programs. The two main results of this paper are that (i) satisfiability is in 2ΕΧΡΤΙΜΕ, thus 2ΕΧΡΤΙΜΕ-complete by an existing lower bound, and (ii) infinite-state model checking of basic process algebras and pushdown systems is also 2ΕΧΡΤΙΜΕ-complete. Both upper bounds are obtained by polynomial time computable reductions to ω-regular tree satisfiability in ICPDL, a reasoning problem that we introduce specifically for this purpose. This problem is then reduced to the emptiness problem for alternating two-way automata on infinite trees. Our approach to (i) also provides a shorter and more elegant proof of Danecki's difficult result that satisfiability in IPDL is in 2ΕΧΡΤΙΜΕ. We prove the lower bound(s) for infinite-state model checking using an encoding of alternating Turing machines.

We perform an exhaustive study of the complexity of subsumption in the ℰℒ family of lightweight description logics w.r.t. acyclic and cyclic TBoxes. It turns out that there are interesting members of this family for which subsumption w.r.t. cyclic TBoxes is tractable, whereas it is EXPTIME-complete w.r.t. general TBoxes. For other extensions that are intractable w.r.t. general TBoxes, we establish intractability already for acyclic and cyclic TBoxes.

The aim of this paper is to study semantic notions of modularity in description logic (DL) terminologies and reasoning problems that are relevant for modularity. We define two notions of a module whose independence is formalised in a model-theoretic way. Focusing mainly on the DLs ℰℒ and 𝒜ℒ𝒞, we then develop algorithms for module extraction, for checking whether a part of a terminology is a module, and for a number of related problems. We also analyse the complexity of these problems, which ranges from tractable to undecidable. Finally, we provide an experimental evaluation of our module extraction algorithms based on the large-scale terminology SNOMED CT.

We survey temporal description logics that are based on standard temporal logics such as LTL and CTL. In particular, we concentrate on the computational complexity of the satisfiability problem and algorithms for deciding it.

Conjunctive queries play an important role as an expressive query language for Description Logics (DLs). Although modern DLs usually provide for transitive roles, conjunctive query answering over DL knowledge bases is only poorly understood if transitive roles are admitted in the query. In this paper, we consider unions of conjunctive queries over knowledge bases formulated in the prominent DL SHIQ and allow transitive roles in both the query and the knowledge base. We show decidability of query answering in this setting and establish two tight complexity bounds: regarding combined complexity, we prove that there is a deterministic algorithm for query answering that needs time single exponential in the size of the KB and double exponential in the size of the query, which is optimal. Regarding data complexity, we prove containment in co-NP.

The fully enriched μ-calculus is the extension of the propositional μ-calculus with inverse programs, graded modalities, and nominals. While satisfiability in several expressive fragments of the fully enriched μ-calculus is known to be decidable and ExpTime-complete, it has recently been proved that the full calculus is undecidable. In this paper, we study the fragments of the fully enriched μ-calculus that are obtained by dropping at least one of the additional constructs. We show that, in all fragments obtained in this way, satisfiability is decidable and ExpTime-complete. Thus, we identify a family of decidable logics that are maximal (and incomparable) in expressive power. Our results are obtained by introducing two new automata models, showing that their emptiness problems are ExpTime-complete, and then reducing satisfiability in the relevant logics to these problems. The automata models we introduce are two-way graded alternating parity automata over infinite trees (2GAPTs) and fully enriched automata (FEAs) over infinite forests. The former are a common generalization of two incomparable automata models from the literature. The latter extend alternating automata in a similar way as the fully enriched μ-calculus extends the standard μ-calculus.

Query containment has been studied extensively for fragments of XPath 1.0. For instance, the problem is known to be ExpTime-complete for CoreXPath, the navigational core of XPath 1.0. Much less is known about query containment in (fragments of) the richer language XPath 2.0. In this paper, we consider extensions of CoreXPath with the following operators, which are all part of XPath 2.0 (except the last): path intersection, path equality, path complementation, for-loops, and transitive closure. For each combination of these operators, we determine the complexity of query containment, both with and without DTDs. It turns out to range from ExpTime (for extensions with path equality) and 2-ExpTime (for extensions with path intersection) to non-elementary (for extensions with path complementation or for-loops). In almost all cases, adding transitive closure on top has no further impact on the complexity. We also investigate the effect of dropping the upward and/or sibling axes, and show that this sometimes leads to a reduction in complexity.Since the languages we study include negation and conjunction infilters, our complexity results can equivalently be stated in terms ofsatisfiability.We also analyze the above languages in terms of succinctness.

It is known that for temporal languages, such as first-order LTL, reasoning about constant (time-independent) relations is almost always undecidable. This applies to temporal description logics as well: constant binary relations together with general concept subsumptions in combinations of LTL and the basic description logic ALC cause undecidability. In this paper, we explore temporal extensions of two recently introduced families of 'weak' description logics known as DL-Lite and EL. Our results are twofold: temporalisations of even rather expressive variants of DL-Lite turn out to be decidable, while the temporalisation of EL with general concept subsumptions and constant relations is undecidable.

Logical formalisms for reasoning about relations between spatial regions play a fundamental role in geographical information systems, spatial and constraint databases, and spatial reasoning in AI. In analogy with Halpern and Shoham's modal logic of time intervals based on the Allen relations, we introduce a family of modal logics equipped with eight modal operators that are interpreted by the Egenhofer-Franzosa (or RCC8) relations between regions in topological spaces such as the real plane. We investigate the expressive power and computational complexity of logics obtained in this way. It turns out that our modal logics have the same expressive power as the two-variable fragment of first-order logic, but are exponentially less succinct. The complexity ranges from (undecidable and) recursively enumerable to highly undecidable, where the recursively enumerable logics are obtained by considering substructures of structures induced by topological spaces. As our undecidability results also capture logics based on the real line, they improve upon undecidability results for interval temporal logics by Halpern and Shoham. We also analyze modal logics based on the five RCC5 relations, with similar results regarding the expressive power, but weaker results regarding the complexity.

The alternating-time temporal logic (ATL) of Alur, Henzinger and Kupferman is being increasingly widely applied in the specification and verification of open distributed systems and game-like multi-agent systems. In this article, we investigate the computational complexity of the satisfiability problem for ATL. For the case where the set of agents is fixed in advance, this problem was settled at ExpTime-complete in a result of van Drimmelen. If the set of agents is not fixed in advance, then van Drimmelen's construction yields a 2ExpTime upper bound. In this article, we focus on the latter case and define three natural variations of the satisfiability problem. Although none of these variations fixes the set of agents in advance, we are able to prove containment in ExpTime for all of them by means of a type elimination construction—thus improving the existing 2ExpTime upper bound to a tight ExpTime one.

Often, the addition of metric operators to qualitative temporal logics leads to an increase of the complexity of satisfiability by at least one exponential. In this paper, we exhibit a number of metric extensions of qualitative temporal logics of the real line that do not lead to an increase in computational complexity. We show that the language obtained by extending since/until logic of the real line with the operators 'sometime within n time units', n coded in binary, is PSpace-complete even without the finite variability assumption. Without qualitative temporal operators the complexity of this language turns out to depend on whether binary or unary coding of parameters is assumed: it is still PSpace-hard under binary coding but in NP under unary coding.

Many description logics (DLs) combine knowledge representation on an abstract, logical level with an interface to 'concrete' domains like numbers and strings with built-in predicates such as >, +, and prefix-of. These hybrid DLs have turned out to be useful in several application areas, such as reasoning about conceptual database models. We propose to further extend such DLs with key constraints that allow the expression of statements like 'US citizens are uniquely identified by their social security number'. Based on this idea, we introduce a number of natural description logics and perform a detailed analysis of their decidability and computational complexity. It turns out that naive extensions with key constraints easily lead to undecidability, whereas more careful extensions yield NExpTime-complete DLs for a variety of useful concrete domains.

Abstract In 1984. Danecki proved that satisfiability in IPDL, i.e., Propositional Dynamic Logic (PDL) extended with an intersection operator on programs, is decidabie in deterministic double exponential time. Since then, the exact complexity of IPDL has remained an open problem: the best known lower bound was the ExpTime one stemming from plain PDL until, in 2004. the first author established ExpSpace-hardness. In this paper, we finally close the gap and prove that IPDL is hard for 2-ExpTime. thus 2-ExpTime-complete. We then sharpen our lower bound, showing that it even applies to IPDL without the test operator interpreted on tree structures.

Concrete domains are an extension of Description Logics (DLs) that allow one to integrate reasoning about conceptual knowledge with reasoning about "concrete qualities" of real-world entities such as their sizes, weights, and durations. In this article, we are concerned with the complexity of Description Logics providing for concrete domains: starting from the complexity result established in Lutz [2002b], which states that reasoning with the basic propositionally closed DL with concrete domains ALC(D) is PSpace-complete (provided that some weak conditions are satisfied), we perform an in-depth analysis of the complexity of extensions of this logic. More precisely, we consider five natural and seemingly "harmless" extensions of ALC(D) and prove that, for all five extensions, reasoning is NExpTime-complete (again if some weak conditions are satisfied). Thus, we show that the PSpace upper bound for reasoning with ALC(D) cannot be considered robust with respect to extensions of the language.

Fusions are a simple way of combining logics. For normal modal logics, fusions have been investigated in detail. In particular, it is known that, under certain conditions, decidability transfers from the component logics to their fusion. Though description logics are closely related to modal logics, they are not necessarily normal. In addition, ABox reasoning in description logics is not covered by the results from modal logics. In this paper, we extend the decidability transfer results from normal modal logics to a large class of description logics. To cover different description logics in a uniform way, we introduce abstract description systems, which can be seen as a common generalization of description and modal logics, and show the transfer results in this general setting.

This article presents the description logic ALCRP(D) with concrete domains and a role-forming predicate operator as its prominent aspects. We demonstrate the feasibility of ALCRP(D) for reasoning about spatial objects and their qualitative spatial relationships and provide an appropriate concrete domain for spatial objects. The general significance of ALCRP(D) is demonstrated by adding temporal reasoning to spatial and terminological reasoning using a combined concrete domain. The theory is motivated as a basis for knowledge representation and query processing in the domain of geographic information systems. In contrast to existing work in this domain, which mainly focuses either on conceptual reasoning or on reasoning about qualitative spatial relations, we integrate reasoning about spatial information with terminological reasoning. Key words: Description logic, spatial reasoning, spatio temporal reasoning, theoretical foundations for GIS.