We characterise the sentences in Monadic Second-Order Logic (MSO) that are over finite structures equivalent to a Datalog program, in terms of an existential pebble game. We also show that for every class \({\mathcal{C}}\) of finite structures that can be expressed in MSO and is closed under homomorphisms, and for all \(\ell,k\in{\mathbb{N}}\) , there exists a canonical Datalog program \(\Pi\) of width \((\ell,k)\) in the sense of Feder and Vardi. The same characterisations also hold for Guarded Second-Order Logic (GSO), which properly extends MSO. To prove our results, we show that every class \({\mathcal{C}}\) in GSO whose complement is closed under homomorphisms is a finite union of Constraint Satisfaction Problems (CSPs) of \(\omega\) -categorical structures. The intersection of MSO and Datalog is known to contain the class of nested monadically defined queries (Nemodeq) ; likewise, we show that the intersection of GSO and Datalog contains all problems that can be expressed by the more expressive language of nested guarded queries (GQ \({}^{+}\) ) . Yet, by exploiting our results, we can show that neither of the two query languages can serve as a characterisation, as we exhibit a CSP whose complement corresponds to a query in the intersection of MSO and Datalog that is not expressible in GQ \({}^{+}\) .

Standpoint extensions of KR formalisms have been recently introduced to incorporate multi-perspective modelling and reasoning capabilities. In such modal extensions, the integration of conceptual modelling and perspective annotations can be more or less tight, with monodic standpoint extensions striking a good balance as they enable advanced modelling while preserving good reasoning complexities. We consider the extension of C² – the counting two-variable fragment of first-order logic – by monodic standpoints. At the core of our treatise is a polytime translation of formulae in said formalism into standpoint-free C², requiring elaborate model-theoretic arguments. By virtue of this translation, the NEXPTIME-complete complexity of checking satisfiability in C² carries over to our formalism. As our formalism subsumes monodic S5 over C², our result also significantly advances the state of the art in research on first-order modal logics. As a practical consequence, the very expressive description logics ?ℋ?ℐ?ℬs and ?ℛ?ℐ?ℬs which subsume the popular W3C-standardized OWL 1 and OWL 2 ontology languages, are shown to allow for monodic standpoint extensions without any increase of standard reasoning complexity. We prove that NEXPTIME-hardness already occurs in much less expressive DLs as long as they feature both nominals and monodic standpoints. We also show that, with inverses, functionality, and nominals present, minimally lifting the monodicity restriction leads to undecidability.

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 propose a generic framework for establishing the decidability of a wide range of logical entailment problems (briefly called querying), based on the existence of countermodels that are structurally simple, gauged by certain types of width measures (with treewidth and cliquewidth as popular examples). As an important special case of our framework, we identify logics exhibiting width-finite finitely universal model sets, warranting decidable entailment for a wide range of homomorphism-closed queries, subsuming a diverse set of practically relevant query languages. As a particularly powerful width measure, we propose to employ Blumensath's partitionwidth, which subsumes various other commonly considered width measures and exhibits highly favorable computational and structural properties. Focusing on the formalism of existential rules as a popular showcase, we explain how finite partitionwidth sets of rules subsume other known abstract decidable classes but - leveraging existing notions of stratification - also cover a wide range of new rulesets. We expose natural limitations for fitting the class of finite unification sets into our picture and suggest several options for remedy.

We establish a generic, model-theoretic characterization of rational belief revision operators implementing the paradigm of minimal change according to the seminal work by Alchourrón, Gärdenfors, and Makinson (AGM). Our characterization applies to all Tarskian logics, that is, all logics with a classical model-theoretic semantics, and hence a wide variety of formalisms were used in knowledge representation and beyond, including many for which a model-theoretic characterization has hitherto been lacking. Our starting point is the approach by Katsuno and Mendelzon (K&M), who provided such a characterization for propositional logic over finite signatures. We generalize K&M’s approach to the setting of AGM-style revision over bases in arbitrary Tarskian logics, where base may refer to one of the various ways of representing an agent’s beliefs (such as belief sets, arbitrary or finite sets of sentences, or single sentences). Our first core result is a representation theorem providing a two-way correspondence between AGM-style revision operators and specific assignments : functions associating every base to a “preference” relation over interpretations, which must be total but is—in contrast to prior approaches—not always transitive. As our second core contribution, we provide a characterization of all logics for which our result can be strengthened to assignments producing transitive preference relations (as in K&M’s original work). Alongside these main contributions, we discuss diverse variants of our findings as well as ramifications for other areas of belief revision theory.

We propose $ω$MSO$\Join$BAPA, an expressive logic for describing countable structures, which subsumes and transcends both Counting Monadic Second-Order Logic (CMSO) and Boolean Algebra with Presburger Arithmetic (BAPA). We show that satisfiability of $ω$MSO$\Join$BAPA is decidable over the class of labeled infinite binary trees, whereas it becomes undecidable even for a rather mild relaxations. The decidability result is established by an elaborate multi-step transformation into a particular normal form, followed by the deployment of Parikh-Muller Tree Automata, a novel kind of automaton for infinite labeled binary trees, integrating and generalizing both Muller and Parikh automata while still exhibiting a decidable (in fact PSpace-complete) emptiness problem. By means of MSO-interpretations, we lift the decidability result to all tree-interpretable classes of structures, including the classes of finite/countable structures of bounded treewidth/cliquewidth/partitionwidth. We generalize the result further by showing that decidability is even preserved when coupling width-restricted $ω$MSO$\Join$BAPA with width-unrestricted two-variable logic with advanced counting. A final showcase demonstrates how our results can be leveraged to harvest decidability results for expressive $μ$-calculi extended by global Presburger constraints.

Standpoint logic is a recently proposed modal logic framework that is well-suited for multiperspective reasoning and ontology integration. For this reason, combinations of standpoint logic with description logics (DLs) are of special interest. Prior work has shown that it is possible to add standpoints to numerous decidable fragments of first-order logics - including very expressive DLs up to SROIQbs - while preserving their reasoning complexity, so long as standpoint modalities are limited to the axiom level. A more expressive tighter modal integration, where standpoint modalities are also allowed to occur in concept expressions, has so far only been investigated for the much less expressive DL EL+. In this paper, we push this line of research showing that the DL SHIQ allows for a tight modal integration with standpoints without compromising its ExpTime reasoning complexity. The core insight toward this result is that any satisfiable knowledge base admits a model with only polynomially many worlds, an argument which requires a rather elaborate model-theoretic construction. This allows us to establish a polynomial equisatisfiable translation into plain SHIQ which, beyond showing the theoretical result, enables us to use highly optimised OWL reasoners to provide practical reasoning support for ontology languages extended by standpoint modelling. We complement our findings with the observation that our techniques would fail upon adding the modelling feature of nominals to the underlying DL.

Extensive research in the field of ontology-based query answering has led to the identification of numerous fragments of existential rules (also known as tuple-generating dependencies) that exhibit decidable answering of atomic and conjunctive queries. Motivated by the increased theoretical and practical interest in navigational queries, this paper considers the question for which of these fragments decidability of querying extends to regular path queries (RPQs). In fact, decidability of RPQs has recently been shown to generally hold for the comprehensive family of all fragments that come with the guarantee of universal models being reasonably well-shaped (that is, being of finite cliquewidth). Yet, for the second major family of fragments, known as finite unification sets (short: fus), which are based on first-order-rewritability, corresponding results have been largely elusive so far. We complete the picture by showing that RPQ answering over arbitrary fus rulesets is undecidable. On the positive side, we establish that the problem is decidable for the prominent fus subclass of sticky rulesets, with the caveat that a very mild extension of the RPQ formalism turns the problem undecidable again.

In our pursuit of generic criteria for decidable ontology-based querying, we introduce finite-cliquewidth sets (fcs) of existential rules, a model-theoretically defined class of rule sets, inspired by the cliquewidth measure from graph theory. By a generic argument, we show that fcs ensures decidability of entailment for a sizable class of queries (dubbed DaMSOQs) subsuming conjunctive queries (CQs). The fcs class properly generalizes the class of finite-expansion sets (fes), and for signatures of arity ≤ 2, the class of bounded-treewidth sets (bts). For higher arities, bts is only indirectly subsumed by fcs by means of reification. Despite the generality of fcs, we provide a rule set with decidable CQ entailment (by virtue of first-order-rewritability) that falls outside fcs, thus demonstrating the incomparability of fcs and the class of finite-unification sets (fus). In spite of this, we show that if we restrict ourselves to single-headed rule sets over signatures of arity ≤ 2, then fcs subsumes fus.

The tractability of the lightweight description logic EL has allowed for the construction of large and widely used ontologies that support semantic interoperability. However, comprehensive domains with a broad user base are often at odds with strong axiomatisations otherwise useful for inferencing, since these are usually context dependent and subject to diverging perspectives. In this paper we introduce Standpoint EL, a multi-modal extension of EL that allows for the integrated representation of domain knowledge relative to diverse, possibly conflicting standpoints (or contexts), which can be hierarchically organised and put in relation to each other. We establish that Standpoint EL still exhibits EL's favourable PTime standard reasoning, whereas introducing additional features like empty standpoints, rigid roles, and nominals makes standard reasoning tasks intractable.

Standpoint EL is a multi-modal extension of the popular description logic EL that allows for the integrated representation of domain knowledge relative to diverse standpoints or perspectives. Advantageously, its satisfiability problem has recently been shown to be in PTime, making it a promising framework for large-scale knowledge integration. In this paper, we show that we can further push the expressivity of this formalism, arriving at an extended logic, called Standpoint EL+, which allows for axiom negation, role chain axioms, self-loops, and other features, while maintaining tractability. This is achieved by designing a satisfiability-checking deduction calculus, which at the same time addresses the need for practical algorithms. We demonstrate the feasibility of our calculus by presenting a prototypical Datalog implementation of its deduction rules.

The core chase, a popular algorithm for answering conjunctive queries (CQs) over existential rules, is guaranteed to terminate and compute a finite universal model whenever one exists, leading to the equivalence of the universal-model-based and the chase-based definitions of finite expansion sets (fes) -- a class of rulesets featuring decidable CQ entailment. In case of non-termination, however, it is non-trivial to define a ''result'' of the core chase, due to its non-monotonicity. This causes complications when dealing with advanced decidability criteria based on the existence of (universal) models of finite treewidth. For these, sufficient chase-based conditions have only been established for weaker, monotonic chase variants. This paper investigates the -- prima facie plausible -- hypothesis that the existence of a treewidth-bounded universal model and the existence of a treewidth-bounded core-chase sequence coincide -- which would conveniently entail decidable CQ entailment whenever the latter holds. Perhaps surprisingly, carefully crafted examples show that both directions of this hypothesized correspondence fail. On a positive note, we are still able to define an aggregation scheme for the infinite core chase that preserves treewidth bounds and produces a finitely universal model, i.e., one that satisfies exactly the entailed CQs. This allows us to prove that the existence of a treewidth-bounded core-chase sequence does warrant decidability of CQ entailment (yet, on other grounds than expected). Hence, for the first time, we are able to define a chase-based notion of bounded treewidth sets of rules that subsumes fes.

It is commonly known that the conjunctive query entailment problem for certain extensions of (the well-known ontology language) ALC is computationally harder than their knowledge base satisfiability problem while for others the complexities coincide, both under the standard and the finite-model semantics. We expose a uniform principle behind this divide by identifying a wide class of (finitely) locally-forward description logics, for which we prove that (finite) query entailment problem can be solved by a reduction to exponentially many calls of the (finite) knowledge base satisfiability problem. Consequently, our algorithm yields tight ExpTime upper bounds for locally-forward logics with ExpTime-complete knowledge base satisfiability problem, including logics between ALC and µALCHbregQ (and more), as well as ALCSCC with global cardinality constraints, for which the complexity of querying remained open. Moreover, to make our technique applicable in future research, we provide easy-to-check sufficient conditions for a logic to be locally-forward based several versions of the on model-theoretic notion of unravellings. Together with existing results, this provides a nearly complete classification of the “benign” vs. “malign” primitive modelling features extending ALC, missing out only the Self operator. We then show a rather counter-intuitive result, namely that the conjunctive entailment problem for ALCSelf is exponentially harder than for ALC. This places the seemingly innocuous Self operator among the “malign” modelling features, like inverses, transitivity or nominals.

Abstract We give an in-depth account of compositional matrix-space models (CMSMs), a type of generic models for natural language, wherein compositionality is realized via matrix multiplication. We argue for the structural plausibility of this model and show that it is able to cover and combine various common compositional natural language processing approaches. Then, we consider efficient task-specific learning methods for training CMSMs and evaluate their performance in compositionality prediction and sentiment analysis.

In logic-based knowledge representation, query answering has essentially replaced mere satisfiability checking as the inferencing problem of primary interest. For knowledge bases in the basic description logic ALC, the computational complexity of conjunctive query (CQ) answering is well known to be EXPTIME-complete and hence not harder than satisfiability. This does not change when the logic is extended by certain features (such as counting or role hierarchies), whereas adding others (inverses, nominals or transitivity together with role-hierarchies) turns CQ answering exponentially harder. We contribute to this line of results by showing the surprising fact that even extending ALC by just the Self operator – which proved innocuous in many other contexts – increases the complexity of CQ entailment to 2EXPTIME. As common for this type of problem, our proof establishes a reduction from alternating Turing machines running in exponential space, but several novel ideas and encoding tricks are required to make the approach work in that specific, restricted setting.

Existential rules are a very popular ontology-mediated query language for which the chase represents a generic computational approach for query answering. It is straightforward that existential rule queries exhibiting chase termination are decidable and can only recognize properties that are preserved under homomorphisms. This paper is an extended abstract of our eponymous publication at KR 2021 where we show the converse: every decidable query that is closed under homomorphism can be expressed by an existential rule set for which the standard chase universally terminates. Membership in this fragment is not decidable, but we show via a diagonalisation argument that this is unavoidable.

We consider (first-order) query rewritability in the context of theory-mediated query answering. The starting point of our journey is the FUS/FES conjecture, which states that any theory that is a finite expansion set (FES) and admits query rewriting (BDD, FUS) must be uniformly bounded. We show that this conjecture holds for a large class of BDD theories, which we call "local". Upon investigating how "non-local" BDD theories can actually get, we discover unexpected phenomena that, we think, are at odds with prevailing intuitions about BDD theories.

Abstract The importance of taking individual, potentially conflicting perspectives into account when dealing with knowledge has been widely recognised. Many existing ontology management approaches fully merge knowledge perspectives, which may require weakening in order to maintain consistency; others represent the distinct views in an entirely detached way. As an alternative, we proposeStandpoint Logic, a simple, yet versatile multi-modal logic “add-on” for existing KR languages intended for the integrated representation of domain knowledge relative to diverse, possibly conflictingstandpoints, which can be hierarchically organised, combined, and put in relation with each other. Starting from the generic framework ofFirst-Order Standpoint Logic(FOSL), we subsequently focus our attention on the fragment ofsententialformulas, for which we provide a polytime translation into the standpoint-free version. This result yields decidability and favourable complexities for a variety of highly expressive decidable fragments of first-order logic. Using some elaborate encoding tricks, we then establish a similar translation for the very expressive description logic $$\mathcal {SROIQ}b_s$$ SROIQbs underlying the OWL 2 DL ontology language. By virtue of this result, existing highly optimised OWL reasoners can be used to provide practical reasoning support for ontology languages extended by standpoint modelling.

Ontologies and knowledge bases encode, to a certain extent, the standpoints or perspectives of their creators. As differences and conflicts between standpoints should be expected in multi-agent scenarios, this will pose challenges for shared creation and usage of knowledge sources. Our work pursues the idea that, in some cases, a framework that can handle diverse and possibly conflicting standpoints is more useful and versatile than forcing their unification, and avoids common compromises required for their merge. Moreover, in analogy to the notion of family resemblance concepts, we propose that a collection of standpoints can provide a simpler yet more faithful and nuanced representation of some domains. To this end, we present standpoint logic, a multi-modal framework that is suitable for expressing information with semantically heterogeneous vocabularies, where a standpoint is a partial and acceptable interpretation of the domain. Standpoints can be organised hierarchically and combined, and complex correspondences can be established between them. We provide a formal syntax and semantics, outline the complexity for the propositional case, and explore the representational capacities of the framework in relation to standard techniques in ontology integration, with some examples in the Bio-Ontology domain.

Existential rules are a very popular ontology-mediated query language for which the chase represents a generic computational approach for query answering. It is straightforward that existential rule queries exhibiting chase termination are decidable and can only recognize properties that are preserved under homomorphisms. In this paper, we show the converse: every decidable query that is closed under homomorphism can be expressed by an existential rule set for which the standard chase universally terminates. Membership in this fragment is not decidable, but we show via a diagonalisation argument that this is unavoidable.

Predicate logic is the premier choice for specifying classes of relational structures. Homomorphisms are key to describing correspondences between relational structures. Questions concerning the interdependencies between these two means of characterizing (classes of) structures are of fundamental interest and can be highly non-trivial to answer. We investigate several problems regarding the homomorphism closure (homclosure) of the class of all (finite or arbitrary) models of logical sentences: membership of structures in a sentence's homclosure; sentence homclosedness; homclosure characterizability in a logic; normal forms for homclosed sentences in certain logics. For a wide variety of fragments of first- and second-order predicate logic, we clarify these problems' computational properties.

The Triguarded Fragment (TGF) is among the most expressive decidable fragments of first-order logic, subsuming both its two-variable and guarded fragments without equality. We show that the TGF has the finite model property (providing a tight doubly exponential bound on the model size) and hence finite satisfiability coincides with satisfiability known to be N2ExpTime-complete. Using similar constructions, we also establish 2ExpTime-completeness for finite satisfiability of the constant-free (tri)guarded fragment with transitive guards.

Abstract dialectical frameworks (ADFs) are a recently introduced powerful generalization of Dung’s popular abstract argumentation frameworks (AFs). Inspired by similar work for AFs, we introduce a decomposition scheme for ADFs, which proceeds along the ADF’s strongly connected components. We find that, for several semantics, the decompositionbased version coincides with the original semantics, whereas for others, it gives rise to a new semantics. These new semantics allow us to deal with pertinent problems such as odd-length negative cycles in a more general setting, that for instance also encompasses logic programs. We perform an exhaustive analysis of the computational complexity of these new, so-called naive-based semantics. The results are quite interesting, for some of them involve little-known classes of the so-called Boolean hierarchy (another hierarchy in between classes of the polynomial hierarchy). Furthermore, in credulous and sceptical entailment, the complexity can be different depending on whether we check for truth or falsity of a specific statement.

We introduce and investigate the expressive description logic (DL) ALCSCC++, in which the global and local cardinality constraints introduced in previous papers can be mixed. On the one hand, we prove that this does not increase the complexity of satisfiability checking and other standard inference problems. On the other hand, the satisfiability problem becomes undecidable if inverse roles are added to the languages. In addition, even without inverse roles, conjunctive query entailment in this DL turns out to be undecidable. We prove that decidability of querying can be regained if global and local constraints are not mixed and the global constraints are appropriately restricted. The latter result is based on a locally-acyclic model construction, and it reduces query entailment to ABox consistency in the restricted setting, i.e., to ABox consistency w.r.t. restricted cardinality constraints in ALCSCC, for which we can show an ExpTime upper bound.

The chase has become a staple of modern database theory with applications in data integration, query optimisation, data exchange, ontology-based query answering, and many other areas. Most application scenarios and implementations require the chase to terminate and produce a finite universal model, and a large arsenal of sufficient termination criteria is available to guarantee this (generally undecidable) condition. In this invited tutorial, we therefore ask about the expressive power of logical theories for which the chase terminates. Specifically, which database properties can be recognised by such theories, i.e., which Boolean queries can they realise? For the skolem (semi-oblivious) chase, and almost any known termination criterion, this expressivity is just that of plain Datalog. Surprisingly, this limitation of most prior research does not apply to the chase in general. Indeed, we show that standard - chase terminating theories can realise queries with data complexities ranging from PTime to non-elementary that are out of reach for the terminating skolem chase. A "Datalog-first" standard chase that prioritises applications of rules without existential quantifiers makes modelling simpler - and we conjecture: computationally more efficient. This is one of the many open questions raised by our insights, and we conclude with an outlook on the research opportunities in this area.

Among the most expressive knowledge representation formalisms are the description logics of the Z family. For well-behaved fragments of ZOIQ, entailment of positive two-way regular path queries is well known to be 2EXPTIME-complete under the proviso of unary encoding of numbers in cardinality constraints. We show that this assumption can be dropped without an increase in complexity and EXPTIME-completeness can be achieved when bounding the number of query atoms, using a novel reduction from query entailment to knowledge base satisfiability. These findings allow to strengthen other results regarding query entailment and query containment problems in very expressive description logics. Our results also carry over to GC2, the two-variable guarded fragment of first-order logic with counting quantifiers, for which hitherto only conjunctive query entailment has been investigated.

Conjunctive query answering over databases with constraints – also known as (tuple-generating) dependencies – is considered a central database task. To this end, several versions of a construction called chase have been described. Given a set Sigma of dependencies, it is interesting to ask which constraints not contained in Sigma that are initially satisfied in a given database instance are preserved when computing a chase over Sigma. Such constraints are an example for the more general class of incidental constraints, which when added to Sigma as new dependencies do not affect certain answers and might even speed up query answering. After formally introducing incidental constraints, we show that deciding incidentality is undecidable for tuple-generating dependencies, even in cases for which query entailment is decidable. For dependency sets with a finite universal model, the core chase can be used to decide incidentality. For the infinite case, we propose the stable chase, which generalises the core chase, and study its relation to incidental constraints.

Past research into decidable fragments of first-order logic (FO) has produced two very prominent fragments: the guarded fragment GF, and the two-variable fragment FO2. These fragments are of crucial importance because they provide significant insights into decidabil- ity and expressiveness of other (computational) logics like Modal Logics (MLs) and various Description Logics (DLs), which play a central role in Verification, Knowledge Represen- tation, and other areas. In this paper, we take a closer look at GF and FO2, and present a new fragment that subsumes them both. This fragment, called the triguarded fragment (denoted TGF), is obtained by relaxing the standard definition of GF: quantification is required to be guarded only for subformulae with three or more free variables. We show that, in the absence of equality, satisfiability in TGF is N2ExpTime-complete, but becomes NExpTime-complete if we bound the arity of predicates by a constant (a natural assumption in the context of MLs and DLs). Finally, we observe that many natural extensions of TGF, including the addition of equality, lead to undecidability.

After decades of fruitful research, description logics (DLs) have evolved into a de facto standard in logic-based knowledge representation. In particular, they serve as the formal basis of the standardized and very popular web ontology language (OWL), which also comes with the advantage of readily available user-friendly modeling tools and optimized reasoning engines. In the course of the wide-spread adoption of OWL and DLs, situations have been observed where logically less skilled practitioners are (ab)using these formalisms as constraint languages adopting a closed-world assumption, contrary to the open-world semantics imposed by the classical definitions and the standards. To provide a clear theoretical basis and inferencing support for this often practically reasonable “off-label use” we propose an alternative formal semantics reflecting the intuitive understanding of such scenarios. To that end, we introduce the fixed-domain semantics and argue that this semantics gives rise to an interesting new inferencing task: model enumeration. We describe how the new semantics can be axiomatized in very expressive DLs. We thoroughly investigate the complexities for standard reasoning as well as query answering under the fixed-domain semantics for a wide range of DLs. Further, we present an implementation of a fixed-domain DL reasoner based on a translation into answer set programming (ASP) which is competitive with alternative approaches for standard reasoning tasks and provides the added functionality of model enumeration.

Ontology-Based Data Access (OBDA) is a recent paradigm aiming at enhancing data access by taking ontological knowledge into account. When using existential rules as ontological language, query answering is an undecidable problem, whence numerous decidable classes of ontologies have been defined, ranging from classes with very good computational complexities (AC0 in data complexity) to classes with much larger expressivity. However, actually implementable algorithms have been proposed only for very restricted classes (typically those coinciding with lightweight description logics). The aim of this paper is to show how to deal with more expressive ontologies by proposing an algorithm that performs both materialization and rewriting and is applicable for a significant generalization of lightweight description logics. To this end, we first modify an existing algorithm previously proposed for a very generic class of rules, namely greedy bounded treewidth sets of rules. We then exhibit a special case, called pattern oblivious rule sets, which significantly generalizes the ℰℒℋdrdescription logic, which underlies the OWL 2 EL ontology standard, while keeping the beneficial worst-case computational complexity. We last define a subclass of pattern oblivious rules that is recognizable in polynomial time.

The so-called existential rules have recently gained attention, mainly due to their adequate expressiveness for ontological query answering. Several decidable fragments of such rules have been introduced, employing restriction such as various forms of guardedness to ensure decidability. Some of the more well-known languages in this arena are (weakly) guarded and (weakly) frontier-guarded fragments of existential rules. In this paper, we explore their relative and absolute expressiveness. In particular, we provide a new proof that queries expressed via frontier-guarded and guarded rules can be translated into plain Datalog queries. Since the converse translations are impossible, we develop generalizations of frontier-guarded and guarded rules to nearly frontier-guarded and nearly guarded rules, respectively, which have exactly the expressive power of Datalog. We further show that weakly frontier-guarded rules can be translated into weakly guarded rules, and thus, weakly frontier-guarded and weakly guarded rules have exactly the same expressive power. Such rules cannot be translated into Datalog since their query answering problem is ExpTime-complete in data complexity. We strengthen this result by showing that on ordered databases and with input negation available, weakly guarded rules capture all queries decidable in exponential time. We then show that weakly guarded rules extended with stratified negation are expressive enough to capture all database queries decidable in exponential time, without any assumptions about input databases. Finally, we note that the translations of this paper are, in general, exponential in size, but lead to worst-case optimal algorithms for query answering with considered languages.

EL is a popular description logic, used as a core formalism in large existing knowledge bases. Uniform interpolants of knowledge bases are of high interest, e.g. in scenarios where a knowledge base is supposed to be partially reused. However, to the best of our knowledge no procedure has yet been proposed that computes uniform EL interpolants of general EL terminologies. Up to now, also the bound on the size of uniform EL interpolants has remained unknown. In this article, we propose an approach to computing a finite uniform interpolant for a general EL terminology if it exists. To this end, we develop a quadratic representation of EL TBoxes as regular tree grammars. Further, we show that, if a finite uniform EL interpolant exists, then there exists one that is at most triple exponential in the size of the original TBox, and that, in the worst case, no smaller interpolants exist, thereby establishing tight worst-case bounds on their size. Beyond showing these bounds, the notions and results established in this paper also provide useful insights for designing efficient ontology reformulation algorithms, for instance, within the context of module extraction.

OWL 2 EL is a popular ontology language that supports role inclusions---that is, axioms that capture compositional properties of roles. Role inclusions closely correspond to context-free grammars, which was used to show that answering conjunctive queries (CQs) over OWL 2 EL knowledge bases with unrestricted role inclusions is undecidable. However, OWL 2 EL inherits from OWL 2 DL the syntactic regularity restriction on role inclusions, which ensures that role chains implying a particular role can be described using a finite automaton (FA). This is sufficient to ensure decidability of CQ answering; however, the FAs can be worst-case exponential in size so the known approaches do not provide a tight upper complexity bound. In this paper, we solve this open problem and show that answering CQs over OWL 2 EL knowledge bases is PSPACE-complete in combined complexity (i.e., the complexity measured in the total size of the input). To this end, we use a novel encoding of regular role inclusions using bounded-stack pushdown automata---that is, FAs extended with a stack of bounded size. Apart from theoretical interest, our encoding can be used in practical tableau algorithms to avoid the exponential blowup due to role inclusions. In addition, we sharpen the lower complexity bound and show that the problem is PSPACE-hard even if we consider only role inclusions as part of the input (i.e., the query and all other parts of the knowledge base are fixed). Finally, we turn our attention to navigational queries over OWL 2 EL knowledge bases, and we show that answering positive, converse-free conjunctive graph XPath queries is PSPACE-complete as well; this is interesting since allowing the converse operator in queries is known to make the problem EXPTIME-hard. Thus, in this paper we present several important contributions to the landscape of the complexity of answering expressive queries over description logic knowledge bases.

We introduce monadically defined queries (MODEQs) and nested monadically defined queries (NEMODEQs), two querying formalisms that extend conjunctive queries, conjunctive two-way regular path queries, and monadic Datalog queries. Both can be expressed as Datalog queries and in monadic second-order logic, yet they have a decidable query containment problem and favorable query answering complexities: a data complexity of P, and a combined complexity of NP (MODEQs) and PSpace (NEMODEQs).

We propose the use of a structure index for RDF. It can be used for querying RDF data for which the schema is incomplete or not available. More importantly, we leverage it for a structure-oriented approach to RDF data partitioning and query processing. Based on information captured by the structure index, similarly structured data elements are physically grouped and stored contiguously on disk. At querying time, the index is used for "structure-level" processing to identify the groups of data that match the query structure. Structure-level processing is then combined with standard "data-level" operations that involve retrieval and join procedures executed against the data. In the experiment, our solution provides several times faster performance than a state-of-the-art technique for data partitioning and query processing, and compares favorably with full-fledged RDF stores.

Description logics (DLs) have become a prominent paradigm for representing knowledge in a variety of application areas, partly due to their ability to achieve a favourable balance between expressivity of the logic and performance of reasoning. Horn description logics are obtained, roughly speaking, by disallowing all forms of disjunctions. They have attracted attention since their (worst-case) data complexities are in general lower than those of their non-Horn counterparts, which makes them attractive for reasoning with large sets of instance data (ABoxes). It is therefore natural to ask whether Horn DLs also provide advantages for schema (TBox) reasoning, that is, whether they also feature lower combined complexities. This article settles this question for a variety of Horn DLs. An example of a tractable Horn logic is the DL underlying the ontology language OWL RL, which we characterize as the Horn fragment of the description logic SROIQ without existential quantifiers. If existential quantifiers are allowed, however, many Horn DLs become intractable. We find that Horn- ALC already has the same worst-case complexity as ALC , that is, ExpTime , but we also identify various DLs for which reasoning is PSpace -complete. As a side effect, we derive simplified syntactic definitions of Horn DLs for which we exploit suitable normal form transformations.

Epistemic extensions of description logics (DLs) have been introduced several years ago in order to enhance expressivity and querying capabilities of these logics by knowledge base introspection. We argue that unintended effects occur when imposing the traditionally employed semantics on the very expressive DLs that underly the OWL 1 and OWL 2 standards. Consequently, we suggest a revised semantics that behaves more intuitively in these cases and coincides with the traditional semantics of less expressive DLs. Moreover, we introduce a way of answering epistemic queries to OWL knowledge bases by a reduction to standard OWL reasoning. We provide an implementation of our approach and present first evaluation results.

International audience

The high computational complexity of the expressive Description Logics (DLs) that underlie the OWL standard has motivated the study of their Horn fragments, which are usually tractable in data complexity and can also have lower combined complexity, particularly for query answering. In this paper we provide algorithms for answering conjunctive 2-way regular path queries (2CRPQs), a nontrivial generalization of plain conjunctive queries, in the Horn fragments of the DLs SHOIQ and SROIQ underlying OWL 1 and OWL 2. We show that the combined complexity of the problem is ExpTime-complete for Horn-SHOIQ and 2ExpTimecomplete for the more expressive Horn-SROIQ, butisPTime-complete in data complexity for both. In contrast, even decidability of plain conjunctive queries is still open for full SHOIQ and SROIQ. These are the first completeness results for query answering in DLs with inverses, nominals, and counting, and show that for the considered logics the problem is not more expensive than standard reasoning.

Streams of events appear increasingly today in various Web applications such as blogs, feeds, sensor data streams, geospatial information, on-line financial data, etc. Event Processing (EP) is concerned with timely detection of compound events within streams of simple events. State-of-the-art EP provides on-the-fly analysis of event streams, but cannot combine streams with background knowledge and cannot perform reasoning tasks. On the other hand, semantic tools can effectively handle background knowledge and perform reasoning thereon, but cannot deal with rapidly changing data provided by event streams.

Description Logics are knowledge representation formalisms that provide, for example, the logical underpinning of the W3C OWL standards. Conjunctive queries, the standard query language in databases, have recently gained significant attention as an expressive formalism for querying Description Logic knowledge bases. Several different techniques for deciding conjunctive query entailment are available for a wide range of DLs. Nevertheless, the combination of nominals, inverse roles, and number restrictions in OWL 1 and OWL 2 DL causes unsolvable problems for the techniques hitherto available. We tackle this problem and present a decidability result for entailment of unions of conjunctive queries in the DL ALCHOIQb that contains all three problematic constructors simultaneously. Provided that queries contain only simple roles, our result also shows decidability of entailment of (unions of) conjunctive queries in the logic that underpins OWL 1 DL and we believe that the presented results will pave the way for further progress towards conjunctive query entailment decision procedures for the Description Logics underlying the OWL standards.

Keyword queries enjoy widespread usage as they represent an intuitive way of specifying information needs. Recently, answering keyword queries on graph-structured data has emerged as an important research topic. The prevalent approaches build on dedicated indexing techniques as well as search algorithms aiming at finding substructures that connect the data elements matching the keywords. In this paper, we introduce a novel keyword search paradigm for graph-structured data, focusing in particular on the RDF data model. Instead of computing answers directly as in previous approaches, we first compute queries from the keywords, allowing the user to choose the appropriate query, and finally, process the query using the underlying database engine. Thereby, the full range of database optimization techniques can be leveraged for query processing. For the computation of queries, we propose a novel algorithm for the exploration of top-k matching subgraphs. While related techniques search the best answer trees, our algorithm is guaranteed to compute all k subgraphs with lowest costs, including cyclic graphs. By performing exploration only on a summary data structure derived from the data graph, we achieve promising performance improvements compared to other approaches.

We introduce description logic (DL) rules as a new rule-based formalism for knowledge representation in DLs. As a fragment of the Semantic Web Rule Language SWRL, DL rules allow for a tight integration with DL knowledge bases. In contrast to SWRL, however, the combination of DL rules with expressive description logics remains decidable, and we show that the DL 𝒮ℛ𝒪ℐ𝒬 – the basis for the ongoing standardisation of OWL 2 – can completely internalise DL rules. On the other hand, DL rules capture many expressive features of 𝒮ℛ𝒪ℐ𝒬 that are not available in simpler DLs yet. While reasoning in 𝒮ℛ𝒪ℐ𝒬 is highly intractable, it turns out that DL rules can be introduced to various lightweight DLs without increasing their worst-case complexity. In particular, DL rules enable us to significantly extend the tractable DLs ℰℒ++and DLP.

In this paper, we propose a systematic, reasoner-aided approach to Web ontology acquisition and refinement. It complements methods for acquiring expressive ontology axioms from textual definitions with methodic knowledge exploration techniques based on formal concept analysis. We demonstrate the practical relevance of our approach by means of a real-world example.