Reasoning Algebraically with Description Logics

Semantic Web applications based on the Web Ontology Language (OWL) often require the use of numbers in class descriptions for expressing cardinality restrictions on properties or even classes. Some of these cardinalities are specified explicitly, but quite a few are entailed and need to be discovered by reasoning procedures. Due to the Description Logic (DL) foundation of OWL, those reasoning services are offered by DL reasoners. Existing DL reasoners employ reasoning procedures that are arithmetically uninformed and substitute arithmetic reasoning by "don't know" non-determinism in order to cover all possible cases. This lack of information about arithmetic problems dramatically degrades the performance of DL reasoners in many cases, especially with ontologies relying on the use of Nominals and Qualied Cardinality Restrictions. The contribution of this thesis is twofold: on the theoretical level, it presents algebra�ic reasoning with DL (ReAl DL) using a sound, complete, and terminating reasoning procedure for the DL SHOQ. ReAl DL combines tableau reasoning procedures with algebraic methods, namely Integer Programming, to ensure arithmetically better informed reasoning. SHOQ extends the standard DL ALC with transitive roles, role hierarchies, qualified cardinality restrictions (QCRs), and nominals, and forms an expressive subset of OWL. Although the proposed algebraic tableau is double exponential in the worst case, it deals with cardinalities with an additional level of information and properties that make the calculus amenable and well suited for optimizations. In order for ReAl DL to have a practical merit, suited optimizations are proposed towards achieving an efficient reasoning approach that addresses the sources of complexity related to nominals and QCRs. On the practical level, a running prototype reasoner (HARD) is implemented based on the proposed calculus and optimizations. HARD is used to evaluate the practical merit of ReAl DL, as well as the effectiveness of the proposed optimizations. Experimental results based on real world and synthetic ontologies show that ReAl DL outperforms existing reasoning approaches in handling the interactions between nominals and QCRs. ReAl DL also comes with some interesting features such as the ability to handle ontologies with cyclic descriptions without adopting special blocking strategies. ReAl DL can form a basis to provide more efficient reasoning support for ontologies using nominals or QCRs

Inductive Logic Programming in Databases: from Datalog to DL+log

In this paper we address an issue that has been brought to the attention of the database community with the advent of the Semantic Web, i.e. the issue of how ontologies (and semantics conveyed by them) can help solving typical database problems, through a better understanding of KR aspects related to databases. In particular, we investigate this issue from the ILP perspective by considering two database problems, (i) the definition of views and (ii) the definition of constraints, for a database whose schema is represented also by means of an ontology. Both can be reformulated as ILP problems and can benefit from the expressive and deductive power of the KR framework DL+log. We illustrate the application scenarios by means of examples. Keywords: Inductive Logic Programming, Relational Databases, Ontologies, Description Logics, Hybrid Knowledge Representation and Reasoning Systems. Note: To appear in Theory and Practice of Logic Programming (TPLP).Comment: 30 pages, 3 figures, 2 tables

Pseudo-contractions as Gentle Repairs

Updating a knowledge base to remove an unwanted consequence is a challenging task. Some of the original sentences must be either deleted or weakened in such a way that the sentence to be removed is no longer entailed by the resulting set. On the other hand, it is desirable that the existing knowledge be preserved as much as possible, minimising the loss of information. Several approaches to this problem can be found in the literature. In particular, when the knowledge is represented by an ontology, two different families of frameworks have been developed in the literature in the past decades with numerous ideas in common but with little interaction between the communities: applications of AGM-like Belief Change and justification-based Ontology Repair. In this paper, we investigate the relationship between pseudo-contraction operations and gentle repairs. Both aim to avoid the complete deletion of sentences when replacing them with weaker versions is enough to prevent the entailment of the unwanted formula. We show the correspondence between concepts on both sides and investigate under which conditions they are equivalent. Furthermore, we propose a unified notation for the two approaches, which might contribute to the integration of the two areas

Consequence-based Reasoning for Description Logics with Disjunction, Inverse Roles, Number Restrictions, and Nominals

We present a consequence-based calculus for concept subsumption and classification in the description logic ALCHOIQ, which extends ALC with role hierarchies, inverse roles, number restrictions, and nominals. By using standard transformations, our calculus extends to SROIQ, which covers all of OWL 2 DL except for datatypes. A key feature of our calculus is its pay-as-you-go behaviour: unlike existing algorithms, our calculus is worst-case optimal for all the well-known proper fragments of ALCHOIQ, albeit not for the full logic

Reasoning with Very Expressive Fuzzy Description Logics

It is widely recognized today that the management of imprecision and vagueness will yield more intelligent and realistic knowledge-based applications. Description Logics (DLs) are a family of knowledge representation languages that have gained considerable attention the last decade, mainly due to their decidability and the existence of empirically high performance of reasoning algorithms. In this paper, we extend the well known fuzzy ALC DL to the fuzzy SHIN DL, which extends the fuzzy ALC DL with transitive role axioms (S), inverse roles (I), role hierarchies (H) and number restrictions (N). We illustrate why transitive role axioms are difficult to handle in the presence of fuzzy interpretations and how to handle them properly. Then we extend these results by adding role hierarchies and finally number restrictions. The main contributions of the paper are the decidability proof of the fuzzy DL languages fuzzy-SI and fuzzy-SHIN, as well as decision procedures for the knowledge base satisfiability problem of the fuzzy-SI and fuzzy-SHIN

Using Model Theory to Find Decidable and Tractable Description Logics with Concrete Domains

Concrete domains have been introduced in the area of Description Logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. Unfortunately, in the presence of general concept inclusions (GCIs), which are supported by all modern DL systems, adding concrete domains may easily lead to undecidability. To regain decidability of the DL ALC in the presence of GCIs, quite strong restrictions, called ω-admissibility, were imposed on the concrete domain. On the one hand, we generalize the notion of ω-admissibility from concrete domains with only binary predicates to concrete domains with predicates of arbitrary arity. On the other hand, we relate ω-admissibility to well-known notions from model theory. In particular, we show that finitely bounded homogeneous structures yield ω-admissible concrete domains. This allows us to show ω-admissibility of concrete domains using existing results from model theory. When integrating concrete domains into lightweight DLs of the EL family, achieving decidability of reasoning is not enough. One wants the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. We investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical p-admissible concrete domain based on the rational numbers. Although ω-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs. DL systems that can handle concrete domains allow their users to employ a fixed set of predicates of one or more fixed concrete domains when modelling concepts. They do not provide their users with means for defining new predicates, let alone new concrete domains. The good news is that finitely bounded homogeneous structures offer precisely that. We show that integrating concrete domains based on finitely bounded homogeneous structures into ALC yields decidable DLs even if we allow predicates specified by first-order formulas. This class of structures also provides effective means for defining new ω-admissible concrete domains with at most binary predicates. The bad news is that defining ω-admissible concrete domains with predicates of higher arities is computationally hard. We obtain two new lower bounds for this meta-problem, but leave its decidability open. In contrast, we prove that there is no algorithm that would facilitate defining p-admissible concrete domains already for binary signatures.:1. Introduction . . . 1 2. Preliminaries . . . 5 3. Description Logics with Concrete Domains . . . 9 3.1. Basic definitions and undecidability results . . . 9 3.2. Decidable and tractable DLs with concrete domains . . . 16 4. A Model-Theoretic Analysis of ω-Admissibility . . . 23 4.1. Homomorphism ω-compactness via ω-categoricity . . . 23 4.2. Patchworks via homogeneity . . . 24 4.3. JDJEPD via decomposition into orbits . . . 27 4.4. Upper bounds via finite boundedness . . . 28 4.5. ω-admissible finitely bounded homogeneous structures . . . 32 4.6. ω-admissible homogeneous cores with a decidable CSP . . . 34 4.7. Coverage of the developed sufficient conditions . . . 36 4.8. Closure properties: homogeneity & finite boundedness . . . 39 5. A Model-Theoretic Analysis of p-Admissibility . . . 47 5.1. Convexity via square embeddings . . . 47 5.2. Convex ω-categorical structures . . . 50 5.3. Convex numerical structures . . . 52 5.4. Ages defined by forbidden substructures . . . 54 5.5. Ages defined by forbidden homomorphic images . . . 56 5.6. (Non-)closure properties of convexity . . . 59 6. Towards user-definable concrete domains . . . 61 6.1. A proof-theoretic perspective . . . 65 6.2. Universal Horn sentences and the JEP . . . 66 6.3. Universal sentences and the AP: the Horn case . . . 77 6.4. Universal sentences and the AP: the general case . . . 90 7. Conclusion . . . 99 7.1. Contributions and future outlook . . . 99 A. Concrete Domains without Equality . . . 103 Bibliography . . . 107 List of figures . . . 115 Alphabetical Index . . . 11

A Customized ILP-Based Solver for Description Logic Reasoners

Artificial intelligence based systems are known for conveying knowledge through machines. This knowledge is often represented using logic representation languages. One of the well-known families of such languages is called Description Logic (DL) which formally reasons and represents knowledge on the concepts, roles and individuals of an application domain. DL reasoners have been evolving and upgraded through the years, however when it comes to handling more complicated ontologies with big values occurring in number restrictions, the current reasoners mostly fail to perform efficiently. One of the techniques used in DL reasoners is the so-called atomic decomposition technique which combines arithmetic and logical reasoning. This thesis presents a customized CPLEX-based solver for enhancing DL reasoners through optimizing the atomic decomposition technique. Furthermore, we provide evidence on how this method can improve the reasoning performance by optimizing atomic decomposition. For such purpose, an empirical evaluation of our system for a set of synthesized benchmarks is demonstrated

Integrating Reasoning Services for Description Logics with Cardinality Constraints with Numerical Optimization Techniques

Recent research in the field of Description Logic (DL) investigated the complexity of the satisfiability problem for description logics that are obtained by enriching the well-known DL ALCQ with more complex set and cardinality constraints over role successors. The algorithms that have been proposed so far, despite providing worst-case optimal decision procedures for the concept satisfiability problem (both without and with a terminology) lack the efficiency needed to obtain usable implementations. In particular, the algorithm for the case without terminology is non-deterministic and the one for the case with a terminology is also best-case exponential. The goal of this thesis is to use well-established techniques from the field of numerical optimization, such as column generation, in order to obtain more practical algorithms. As a starting point, efficient approaches for dealing with counting quantifiers over unary predicates based on SAT-based column generation should be considered.:1. Introduction 2. Preliminaries 2.1. First-order logic 2.2. Linear Programming 2.3. The description logic ALCQ 2.4. Extending ALCQ with expressive role successor constraints 2.4.1. The logic QFBAPA 2.4.2 The description logic ALCSCC 3. The description logic ALCCQU 3.1. A normal form for ALCCQU 3.2. ALCQt as an equivalent formulation of ALCCQU 3.2.1. ALCQt is a sublogic of ALCCQU 3.2.2. ALCCQU is a sublogic of ALCQt 3.3. Model-theoretic characterization of ALCQt 3.3.1. ALCQt-bisimulation and invariance for ALCQt 3.3.2. Characterization of ALCQt concept descriptions 3.4. Expressive power 3.4.1. Relative expressivity of ALCQ and ALCCQU 3.4.2. Relative expressivity of ALCCQU and ALCSCC 3.5. ALCCQU as the first-order fragment of ALCSCC 4. Concept satisfiability in ALCCQU 4.1. The first-order fragment CQU 4.2. Column generation with SAT oracle 4.2.1. Column generation and CQU 4.2.2. From linear inequalities to propositional formulae 4.2.3. Column generation and ALCCQU 4.3. Branch-and-Price for ALCCQU concept satisfiability 4.4. Correctness of ALCCQU-BB 4.4.1. Complexity of ALCCQU-BB 5. Conclusion - Bibliograph

Dokumentverifikation mit Temporaler Beschreibungslogik

The thesis proposes a new formal framework for checking the content of web documents along individual reading paths. It is vital for the readability of web documents that their content is consistent and coherent along the possible browsing paths through the document. Manually ensuring the coherence of content along the possibly huge number of different browsing paths in a web document is time-consuming and error-prone. Existing methods for document validation and verification are not sufficiently expressive and efficient. The innovative core idea of this thesis is to combine the temporal logic CTL and description logic ALC for the representation of consistency criteria. The resulting new temporal description logics ALCCTL can - in contrast to existing specification formalisms - compactly represent coherence criteria on documents. Verification of web documents is modelled as a model checking problem of ALCCTL. The decidability and polynomial complexity of the ALCCTL model checking problem is proven and a sound, complete, and optimal model checking algorithm is presented. Case studies on real and realistic web documents demonstrate the performance and adequacy of the proposed methods. Existing methods such as symbolic model checking or XML-based document validation are outperformed in both expressiveness and speed.Die Dissertation stellt ein neues formales Framework für die automatische Prüfung inhaltlich-struktureller Konsistenzkriterien an Web-Dokumente vor. Viele Informationen werden heute in Form von Web-Dokumenten zugänglich gemacht. Komplexe Dokumente wie Lerndokumente oder technische Dokumentationen müssen dabei vielfältige Qualitätskriterien erfüllen. Der Informationsgehalt des Dokuments muss aktuell, vollständig und in sich stimmig sein. Die Präsentationsstruktur muss unterschiedlichen Zielgruppen mit unterschiedlichen Informationsbedürfnissen genügen. Die Sicherstellung grundlegender Konsistenzeigenschaften von Dokumenten ist angesichts der Vielzahl der Anforderungen und Nutzungskontexte eines elektronischen Dokuments nicht trivial. In dieser Arbeit werden aus der Hard-/Softwareverifikation bekannte Model-Checking-Verfahren mit Methoden zur Repräsentation von Ontologien kombiniert, um sowohl die Struktur des Dokuments als auch inhaltliche Zusammenhänge bei der Prüfung von Konsistenzkriterien berücksichtigen zu können. Als Spezifikationssprache für Konsistenzkriterien wird die neue temporale Beschreibungslogik ALCCTL vorgeschlagen. Grundlegende Eigenschaften wie Entscheidbarkeit, Ausdruckskraft und Komplexität werden untersucht. Die Adäquatheit und Praxistauglichkeit des Ansatzes werden in Fallstudien mit eLearning-Dokumenten evaluiert. Die Ergebnisse übertreffen bekannte Ansätze wie symbolisches Model-Checking oder Methoden zur Validierung von XML-Dokumenten in Performanz, Ausdruckskraft hinsichtlich der prüfbaren Kriterien und Flexibilität hinsichtlich des Dokumenttyps und -formats