Description Logics Go Second-Order -- Extending EL with Universally Quantified Concepts

The study of Description Logics have been historically mostly focused on features that can be translated to decidable fragments of first-order logic. In this paper, we leave this restriction behind and look for useful and decidable extensions outside first-order logic. We introduce universally quantified concepts, which take the form of variables that can be replaced with arbitrary concepts, and define two semantics of this extension. A schema semantics allows replacements of concept variables only by concepts from a particular language, giving us axiom schemata similar to modal logics. A second-order semantics allows replacement of concept variables with arbitrary subsets of the domain, which is similar to quantified predicates in second-order logic. To study the proposed semantics, we focus on the extension of the description logic $\mathcal{EL}$. We show that for a useful fragment of the extension, the conclusions entailed by the different semantics coincide, allowing us to use classical $\mathcal{EL}$ reasoning algorithms even for the second-order semantics. For a slightly smaller, but still useful, fragment, we were also able to show polynomial decidability of the extension. This fragment, in particular, can express a generalized form of role chain axioms, positive self restrictions, and some forms of (local) role-value-maps from KL-ONE, without requiring any additional constructors

Application of Definability to Query Answering over Knowledge Bases

Answering object queries (i.e. instance retrieval) is a central task in ontology based data access (OBDA). Performing this task involves reasoning with respect to a knowledge base K (i.e. ontology) over some description logic (DL) dialect L. As the expressive power of L grows, so does the complexity of reasoning with respect to K. Therefore, eliminating the need to reason with respect to a knowledge base K is desirable. In this work, we propose an optimization to improve performance of answering object queries by eliminating the need to reason with respect to the knowledge base and, instead, utilizing cached query results when possible. In particular given a DL dialect L, an object query C over some knowledge base K and a set of cached query results S={S1, ..., Sn} obtained from evaluating past queries, we rewrite C into an equivalent query D, that can be evaluated with respect to an empty knowledge base, using cached query results S' = {Si1, ..., Sim}, where S' is a subset of S. The new query D is an interpolant for the original query C with respect to K and S. To find D, we leverage a tool for enumerating interpolants of a given sentence with respect to some theory. We describe a procedure that maps a knowledge base K, expressed in terms of a description logic dialect of first order logic, and object query C into an equivalent theory and query that are input into the interpolant enumerating tool, and resulting interpolants into an object query D that can be evaluated over an empty knowledge base. We show the efficacy of our approach through experimental evaluation on a Lehigh University Benchmark (LUBM) data set, as well as on a synthetic data set, LUBMMOD, that we created by augmenting an LUBM ontology with additional axioms

Optimal Fixed-Premise Repairs of EL TBoxes: Extended Version

Reasoners can be used to derive implicit consequences from an ontology. Sometimes unwanted consequences are revealed, indicating errors or privacy-sensitive information, and the ontology needs to be appropriately repaired. The classical approach is to remove just enough axioms such that the unwanted consequences vanish. However, this is often too rough since mere axiom deletion also erases many other consequences that might actually be desired. The goal should not be to remove a minimal number of axioms but to modify the ontology such that only a minimal number of consequences is removed, including the unwanted ones. Specifically, a repair should rather be logically entailed by the input ontology, instead of being a subset. To this end, we introduce a framework for computing fixed-premise repairs of $\mathcal{EL}$ TBoxes. In the first variant the conclusions must be generalizations of those in the input TBox, while in the second variant no such restriction is imposed. In both variants, every repair is entailed by an optimal one and, up to equivalence, the set of all optimal repairs can be computed in exponential time. A prototypical implementation is provided. In addition, we show new complexity results regarding gentle repairs.This is an extended version of an article accepted at the 45th German Conference on Artificial Intelligence (KI 2022)

A survey of large-scale reasoning on the Web of data

As more and more data is being generated by sensor networks, social media and organizations, the Webinterlinking this wealth of information becomes more complex. This is particularly true for the so-calledWeb of Data, in which data is semantically enriched and interlinked using ontologies. In this large anduncoordinated environment, reasoning can be used to check the consistency of the data and of asso-ciated ontologies, or to infer logical consequences which, in turn, can be used to obtain new insightsfrom the data. However, reasoning approaches need to be scalable in order to enable reasoning over theentire Web of Data. To address this problem, several high-performance reasoning systems, whichmainly implement distributed or parallel algorithms, have been proposed in the last few years. Thesesystems differ significantly; for instance in terms of reasoning expressivity, computational propertiessuch as completeness, or reasoning objectives. In order to provide afirst complete overview of thefield,this paper reports a systematic review of such scalable reasoning approaches over various ontologicallanguages, reporting details about the methods and over the conducted experiments. We highlight theshortcomings of these approaches and discuss some of the open problems related to performing scalablereasoning

On a notion of abduction and relevance for first-order logic clause sets

I propose techniques to help with explaining entailment and non-entailment in first-order logic respectively relying on deductive and abductive reasoning. First, given an unsatisfiable clause set, one could ask which clauses are necessary for any possible deduction (\emph{syntactically relevant}), usable for some deduction (\emph{syntactically semi-relevant}), or unusable (\emph{syntactically irrelevant}). I propose a first-order formalization of this notion and demonstrate a lifting of this notion to the explanation of an entailment w.r.t some axiom set defined in some description logic fragments. Moreover, it is accompanied by a semantic characterization via \emph{conflict literals} (contradictory simple facts). From an unsatisfiable clause set, a pair of conflict literals are always deducible. A \emph{relevant} clause is necessary to derive any conflict literal, a \emph{semi-relevant} clause is necessary to derive some conflict literal, and an \emph{irrelevant} clause is not useful in deriving any conflict literals. It helps provide a picture of why an explanation holds beyond what one can get from the predominant notion of a minimal unsatisfiable set. The need to test if a clause is (syntactically) semi-relevant leads to a generalization of a well-known resolution strategy: resolution equipped with the set-of-support strategy is refutationally complete on a clause set $N$ and SOS $M$ if and only if there is a resolution refutation from $N\cup M$ using a clause in $M$. This result non-trivially improves the original formulation. Second, abductive reasoning helps find extensions of a knowledge base to obtain an entailment of some missing consequence (called observation). Not only that it is useful to repair incomplete knowledge bases but also to explain a possibly unexpected observation. I particularly focus on TBox abduction in \EL description logic (still first-order logic fragment via some model-preserving translation scheme) which is rather lightweight but prevalent in practice. The solution space can be huge or even infinite. So, different kinds of minimality notions can help sort the chaff from the grain. I argue that existing ones are insufficient, and introduce \emph{connection minimality}. This criterion offers an interpretation of Occam's razor in which hypotheses are accepted only when they help acquire the entailment without arbitrarily using axioms unrelated to the problem at hand. In addition, I provide a first-order technique to compute the connection-minimal hypotheses in a sound and complete way. The key technique relies on prime implicates. While the negation of a single prime implicate can already serve as a first-order hypothesis, a connection-minimal hypothesis which follows \EL syntactic restrictions (a set of simple concept inclusions) would require a combination of them. Termination by bounding the term depth in the prime implicates is provable by only looking into the ones that are also subset-minimal. I also present an evaluation on ontologies from the medical domain by implementing a prototype with SPASS as a prime implicate generation engine.Ich schlage Techniken vor, die bei der Erklärung von Folgerung und Nichtfolgerung in der Logik erster Ordnung helfen, die sich jeweils auf deduktives und abduktives Denken stützen. Erstens könnte man bei einer gegebenen unerfüllbaren Klauselmenge fragen, welche Klauseln für eine mögliche Deduktion notwendig (\emph{syntaktisch relevant}), für eine Deduktion verwendbar (\emph{syntaktisch semi-relevant}) oder unbrauchbar (\emph{syntaktisch irrelevant}). Ich schlage eine Formalisierung erster Ordnung dieses Begriffs vor und demonstriere eine Anhebung dieses Begriffs auf die Erklärung einer Folgerung bezüglich einer Reihe von Axiomen, die in einigen Beschreibungslogikfragmenten definiert sind. Außerdem wird sie von einer semantischen Charakterisierung durch \emph{Konfliktliteral} (widersprüchliche einfache Fakten) begleitet. Aus einer unerfüllbaren Klauselmenge ist immer ein Konfliktliteralpaar ableitbar. Eine \emph{relevant}-Klausel ist notwendig, um ein Konfliktliteral abzuleiten, eine \emph{semi-relevant}-Klausel ist notwendig, um ein Konfliktliteral zu generieren, und eine \emph{irrelevant}-Klausel ist nicht nützlich, um Konfliktliterale zu generieren. Es hilft, ein Bild davon zu vermitteln, warum eine Erklärung über das hinausgeht, was man aus der vorherrschenden Vorstellung einer minimalen unerfüllbaren Menge erhalten kann. Die Notwendigkeit zu testen, ob eine Klausel (syntaktisch) semi-relevant ist, führt zu einer Verallgemeinerung einer bekannten Resolutionsstrategie: Die mit der Set-of-Support-Strategie ausgestattete Resolution ist auf einer Klauselmenge $N$ und SOS $M$ widerlegungsvollständig, genau dann wenn es eine Auflösungswiderlegung von $N\cup M$ unter Verwendung einer Klausel in $M$ gibt. Dieses Ergebnis verbessert die ursprüngliche Formulierung nicht trivial. Zweitens hilft abduktives Denken dabei, Erweiterungen einer Wissensbasis zu finden, um eine implikantion einer fehlenden Konsequenz (Beobachtung genannt) zu erhalten. Es ist nicht nur nützlich, unvollständige Wissensbasen zu reparieren, sondern auch, um eine möglicherweise unerwartete Beobachtung zu erklären. Ich konzentriere mich besonders auf die TBox-Abduktion in dem leichten, aber praktisch vorherrschenden Fragment der Beschreibungslogik \EL, das tatsächlich ein Logikfragment erster Ordnung ist (mittels eines modellerhaltenden Übersetzungsschemas). Der Lösungsraum kann riesig oder sogar unendlich sein. So können verschiedene Arten von Minimalitätsvorstellungen helfen, die Spreu vom Weizen zu trennen. Ich behaupte, dass die bestehenden unzureichend sind, und führe \emph{Verbindungsminimalität} ein. Dieses Kriterium bietet eine Interpretation von Ockhams Rasiermesser, bei der Hypothesen nur dann akzeptiert werden, wenn sie helfen, die Konsequenz zu erlangen, ohne willkürliche Axiome zu verwenden, die nichts mit dem vorliegenden Problem zu tun haben. Außerdem stelle ich eine Technik in Logik erster Ordnung zur Berechnung der verbindungsminimalen Hypothesen in zur Verfügung korrekte und vollständige Weise. Die Schlüsseltechnik beruht auf Primimplikanten. Während die Negation eines einzelnen Primimplikant bereits als Hypothese in Logik erster Ordnung dienen kann, würde eine Hypothese des Verbindungsminimums, die den syntaktischen Einschränkungen von \EL folgt (einer Menge einfacher Konzeptinklusionen), eine Kombination dieser beiden erfordern. Die Terminierung durch Begrenzung der Termtiefe in den Primimplikanten ist beweisbar, indem nur diejenigen betrachtet werden, die auch teilmengenminimal sind. Außerdem stelle ich eine Auswertung zu Ontologien aus der Medizin vor, Domäne durch die Implementierung eines Prototyps mit SPASS als Primimplikant-Generierungs-Engine

Efficient Maximum A-Posteriori Inference in Markov Logic and Application in Description Logics

Maximum a-posteriori (MAP) query in statistical relational models computes the most probable world given evidence and further knowledge about the domain. It is arguably one of the most important types of computational problems, since it is also used as a subroutine in weight learning algorithms. In this thesis, we discuss an improved inference algorithm and an application for MAP queries. We focus on Markov logic (ML) as statistical relational formalism. Markov logic combines Markov networks with first-order logic by attaching weights to first-order formulas. For inference, we improve existing work which translates MAP queries to integer linear programs (ILP). The motivation is that existing ILP solvers are very stable and fast and are able to precisely estimate the quality of an intermediate solution. In our work, we focus on improving the translation process such that we result in ILPs having fewer variables and fewer constraints. Our main contribution is the Cutting Plane Aggregation (CPA) approach which leverages symmetries in ML networks and parallelizes MAP inference. Additionally, we integrate the cutting plane inference (Riedel 2008) algorithm which significantly reduces the number of groundings by solving multiple smaller ILPs instead of one large ILP. We present the new Markov logic engine RockIt which outperforms state-of-the-art engines in standard Markov logic benchmarks. Afterwards, we apply the MAP query to description logics. Description logics (DL) are knowledge representation formalisms whose expressivity is higher than propositional logic but lower than first-order logic. The most popular DLs have been standardized in the ontology language OWL and are an elementary component in the Semantic Web. We combine Markov logic, which essentially follows the semantic of a log-linear model, with description logics to log-linear description logics. In log-linear description logic weights can be attached to any description logic axiom. Furthermore, we introduce a new query type which computes the most-probable 'coherent' world. Possible applications of log-linear description logics are mainly located in the area of ontology learning and data integration. With our novel log-linear description logic reasoner ELog, we experimentally show that more expressivity increases quality and that the solutions of optimal solving strategies have higher quality than the solutions of approximate solving strategies

Completeness and Consistency Analysis for Evolving Knowledge Bases

Assessing the quality of an evolving knowledge base is a challenging task as it often requires to identify correct quality assessment procedures. Since data is often derived from autonomous, and increasingly large data sources, it is impractical to manually curate the data, and challenging to continuously and automatically assess their quality. In this paper, we explore two main areas of quality assessment related to evolving knowledge bases: (i) identification of completeness issues using knowledge base evolution analysis, and (ii) identification of consistency issues based on integrity constraints, such as minimum and maximum cardinality, and range constraints. For completeness analysis, we use data profiling information from consecutive knowledge base releases to estimate completeness measures that allow predicting quality issues. Then, we perform consistency checks to validate the results of the completeness analysis using integrity constraints and learning models. The approach has been tested both quantitatively and qualitatively by using a subset of datasets from both DBpedia and 3cixty knowledge bases. The performance of the approach is evaluated using precision, recall, and F1 score. From completeness analysis, we observe a 94% precision for the English DBpedia KB and 95% precision for the 3cixty Nice KB. We also assessed the performance of our consistency analysis by using five learning models over three sub-tasks, namely minimum cardinality, maximum cardinality, and range constraint. We observed that the best performing model in our experimental setup is the Random Forest, reaching an F1 score greater than 90% for minimum and maximum cardinality and 84% for range constraints.Comment: Accepted for Journal of Web Semantic

Supporting Tools for Automated Generation and Visual Editing of Relational-to-Ontology Mappings

La integració de dades amb formats heterogenis i de diversos dominis mitjançant tecnologies de la web semàntica permet solucionar la seva disparitat estructural i semàntica. L'accés a dades basat en ontologies (OBDA, en anglès) és una solució integral que es basa en l'ús d'ontologies com esquemes mediadors i el mapatge entre les dades i les ontologies per facilitar la consulta de les fonts de dades. No obstant això, una de les principals barreres que pot dificultar més l'adopció de OBDA és la manca d'eines per donar suport a la creació de mapatges entre dades i ontologies. L'objectiu d'aquesta investigació ha estat desenvolupar noves eines que permetin als experts sense coneixements d'ontologies la creació de mapatges entre dades i ontologies. Amb aquesta finalitat, s'han dut a terme dues línies de treball: la generació automàtica de mapatges entre dades relacionals i ontologies i l'edició dels mapatges a través de la seva representació visual. Les eines actualment disponibles per automatitzar la generació de mapatges estan lluny de proporcionar una solució completa, ja que es basen en els esquemes relacionals i amb prou feines tenen en compte els continguts de la font de dades relacional i les característiques de l'ontologia. No obstant això, les dades poden contenir relacions ocultes que poden ajudar a la generació de mapatges. Per superar aquesta limitació, hem desenvolupat AutoMap4OBDA, un sistema que genera automàticament mapatges R2RML a partir de l'anàlisi dels continguts de la font relacional i tenint en compte les característiques de l'ontologia. El sistema fa servir una tècnica d'aprenentatge d'ontologies per inferir jerarquies de classes, selecciona les mètriques de similitud de cadenes en base a les etiquetes de les ontologies i analitza les estructures de grafs per generar els mapatges a partir de l'estructura de l'ontologia. La representació visual per mitjà d'interfícies intuïtives pot ajudar els usuaris sense coneixements tècnics a establir mapatges entre una font relacional i una ontologia. No obstant això, les eines existents per a l'edició visual de mapatges mostren algunes limitacions. En particular, la representació visual de mapatges no contempla les estructures de la font relacional i de l'ontologia de forma conjunta. Per superar aquest inconvenient, hem desenvolupat Map-On, un entorn visual web per a l'edició manual de mapatges. AutoMap4OBDA ha demostrat que supera les prestacions de les solucions existents per a la generació de mapatges. Map-On s'ha aplicat en projectes d'investigació per verificar la seva eficàcia en la gestió de mapatges.La integración de datos con formatos heterogéneos y de diversos dominios mediante tecnologías de la Web Semántica permite solventar su disparidad estructural y semántica. El acceso a datos basado en ontologías (OBDA, en inglés) es una solución integral que se basa en el uso de ontologías como esquemas mediadores y mapeos entre los datos y las ontologías para facilitar la consulta de las fuentes de datos. Sin embargo, una de las principales barreras que puede dificultar más la adopción de OBDA es la falta de herramientas para apoyar la creación de mapeos entre datos y ontologías. El objetivo de esta investigación ha sido desarrollar nuevas herramientas que permitan a expertos sin conocimientos de ontologías la creación de mapeos entre datos y ontologías. Con este fin, se han llevado a cabo dos líneas de trabajo: la generación automática de mapeos entre datos relacionales y ontologías y la edición de los mapeos a través de su representación visual. Las herramientas actualmente disponibles para automatizar la generación de mapeos están lejos de proporcionar una solución completa, ya que se basan en los esquemas relacionales y apenas tienen en cuenta los contenidos de la fuente de datos relacional y las características de la ontología. Sin embargo, los datos pueden contener relaciones ocultas que pueden ayudar a la generación de mapeos. Para superar esta limitación, hemos desarrollado AutoMap4OBDA, un sistema que genera automáticamente mapeos R2RML a partir del análisis de los contenidos de la fuente relacional y teniendo en cuenta las características de la ontología. El sistema emplea una técnica de aprendizaje de ontologías para inferir jerarquías de clases, selecciona las métricas de similitud de cadenas en base a las etiquetas de las ontologías y analiza las estructuras de grafos para generar los mapeos a partir de la estructura de la ontología. La representación visual por medio de interfaces intuitivas puede ayudar a los usuarios sin conocimientos técnicos a establecer mapeos entre una fuente relacional y una ontología. Sin embargo, las herramientas existentes para la edición visual de mapeos muestran algunas limitaciones. En particular, la representación de mapeos no contempla las estructuras de la fuente relacional y de la ontología de forma conjunta. Para superar este inconveniente, hemos desarrollado Map-On, un entorno visual web para la edición manual de mapeos. AutoMap4OBDA ha demostrado que supera las prestaciones de las soluciones existentes para la generación de mapeos. Map-On se ha aplicado en proyectos de investigación para verificar su eficacia en la gestión de mapeos.Integration of data from heterogeneous formats and domains based on Semantic Web technologies enables us to solve their structural and semantic heterogeneity. Ontology-based data access (OBDA) is a comprehensive solution which relies on the use of ontologies as mediator schemas and relational-to-ontology mappings to facilitate data source querying. However, one of the greatest obstacles in the adoption of OBDA is the lack of tools to support the creation of mappings between physically stored data and ontologies. The objective of this research has been to develop new tools that allow non-ontology experts to create relational-to-ontology mappings. For this purpose, two lines of work have been carried out: the automated generation of relational-to-ontology mappings, and visual support for mapping editing. The tools currently available to automate the generation of mappings are far from providing a complete solution, since they rely on relational schemas and barely take into account the contents of the relational data source and features of the ontology. However, the data may contain hidden relationships that can help in the process of mapping generation. To overcome this limitation, we have developed AutoMap4OBDA, a system that automatically generates R2RML mappings from the analysis of the contents of the relational source and takes into account the characteristics of ontology. The system employs an ontology learning technique to infer class hierarchies, selects the string similarity metric based on the labels of ontologies, and analyses the graph structures to generate the mappings from the structure of the ontology. The visual representation through intuitive interfaces can help non-technical users to establish mappings between a relational source and an ontology. However, existing tools for visual editing of mappings show somewhat limitations. In particular, the visual representation of mapping does not embrace the structure of the relational source and the ontology at the same time. To overcome this problem, we have developed Map-On, a visual web environment for the manual editing of mappings. AutoMap4OBDA has been shown to outperform existing solutions in the generation of mappings. Map-On has been applied in research projects to verify its effectiveness in managing mappings