Standard and Non-standard reasoning in Description Logics

The present work deals with Description Logics (DLs), a class of knowledge representation formalisms used to represent and reason about classes of individuals and relations between such classes in a formally well-defined way. We provide novel results in three main directions. (1) Tractable reasoning revisited: in the 1990s, DL research has largely answered the question for practically relevant yet tractable DL formalisms in the negative. Due to novel application domains, especially the Life Sciences, and a surprising tractability result by Baader, we have re-visited this question, this time looking in a new direction: general terminologies (TBoxes) and extensions thereof defined over the DL EL and extensions thereof. As main positive result, we devise EL++(D)-CBoxes as a tractable DL formalism with optimal expressivity in the sense that every additional standard DL constructor, every extension of the TBox formalism, or every more powerful concrete domain, makes reasoning intractable. (2) Non-standard inferences for knowledge maintenance: non-standard inferences, such as matching, can support domain experts in maintaining DL knowledge bases in a structured and well-defined way. In order to extend their availability and promote their use, the present work extends the state of the art of non-standard inferences both w.r.t. theory and implementation. Our main results are implementations and performance evaluations of known matching algorithms for the DLs ALE and ALN, optimal non-deterministic polynomial time algorithms for matching under acyclic side conditions in ALN and sublanguages, and optimal algorithms for matching w.r.t. cyclic (and hybrid) EL-TBoxes. (3) Non-standard inferences over general concept inclusion (GCI) axioms: the utility of GCIs in modern DL knowledge bases and the relevance of non-standard inferences to knowledge maintenance naturally motivate the question for tractable DL formalism in which both can be provided. As main result, we propose hybrid EL-TBoxes as a solution to this hitherto open question

On the Computation of Common Subsumers in Description Logics

Description logics (DL) knowledge bases are often build by users with expertise in the application domain, but little expertise in logic. To support this kind of users when building their knowledge bases a number of extension methods have been proposed to provide the user with concept descriptions as a starting point for new concept definitions. The inference service central to several of these approaches is the computation of (least) common subsumers of concept descriptions. In case disjunction of concepts can be expressed in the DL under consideration, the least common subsumer (lcs) is just the disjunction of the input concepts. Such a trivial lcs is of little use as a starting point for a new concept definition to be edited by the user. To address this problem we propose two approaches to obtain "meaningful" common subsumers in the presence of disjunction tailored to two different methods to extend DL knowledge bases. More precisely, we devise computation methods for the approximation-based approach and the customization of DL knowledge bases, extend these methods to DLs with number restrictions and discuss their efficient implementation

Hybrid Unification in the Description Logic EL

Unification in Description Logics (DLs) has been proposed as an inference service that can, for example, be used to detect redundancies in ontologies. For the DL EL, which is used to define several large biomedical ontologies, unification is NP-complete. However, the unification algorithms for EL developed until recently could not deal with ontologies containing general concept inclusions (GCIs). In a series of recent papers we have made some progress towards addressing this problem, but the ontologies the developed unification algorithms can deal with need to satisfy a certain cycle restriction. In the present paper, we follow a different approach. Instead of restricting the input ontologies, we generalize the notion of unifiers to so-called hybrid unifiers. Whereas classical unifiers can be viewed as acyclic TBoxes, hybrid unifiers are cyclic TBoxes, which are interpreted together with the ontology of the input using a hybrid semantics that combines fixpoint and descriptive semantics. We show that hybrid unification in EL is NP-complete and introduce a goal-oriented algorithm for computing hybrid unifiers

Polynomial-Time Reasoning Support for Design and Maintenance of Large-Scale Biomedical Ontologies

Description Logics (DLs) belong to a successful family of knowledge representation formalisms with two key assets: formally well-defined semantics which allows to represent knowledge in an unambiguous way and automated reasoning which allows to infer implicit knowledge from the one given explicitly. This thesis investigates various reasoning techniques for tractable DLs in the EL family which have been implemented in the CEL system. It suggests that the use of the lightweight DLs, in which reasoning is tractable, is beneficial for ontology design and maintenance both in terms of expressivity and scalability. The claim is supported by a case study on the renown medical ontology SNOMED CT and extensive empirical evaluation on several large-scale biomedical ontologies

Quantitative Methods for Similarity in Description Logics

Description Logics (DLs) are a family of logic-based knowledge representation languages used to describe the knowledge of an application domain and reason about it in formally well-defined way. They allow users to describe the important notions and classes of the knowledge domain as concepts, which formalize the necessary and sufficient conditions for individual objects to belong to that concept. A variety of different DLs exist, differing in the set of properties one can use to express concepts, the so-called concept constructors, as well as the set of axioms available to describe the relations between concepts or individuals. However, all classical DLs have in common that they can only express exact knowledge, and correspondingly only allow exact inferences. Either we can infer that some individual belongs to a concept, or we can't, there is no in-between. In practice though, knowledge is rarely exact. Many definitions have their exceptions or are vaguely formulated in the first place, and people might not only be interested in exact answers, but also in alternatives that are "close enough". This thesis is aimed at tackling how to express that something "close enough", and how to integrate this notion into the formalism of Description Logics. To this end, we will use the notion of similarity and dissimilarity measures as a way to quantify how close exactly two concepts are. We will look at how useful measures can be defined in the context of DLs, and how they can be incorporated into the formal framework in order to generalize it. In particular, we will look closer at two applications of thus measures to DLs: Relaxed instance queries will incorporate a similarity measure in order to not just give the exact answer to some query, but all answers that are reasonably similar. Prototypical definitions on the other hand use a measure of dissimilarity or distance between concepts in order to allow the definitions of and reasoning with concepts that capture not just those individuals that satisfy exactly the stated properties, but also those that are "close enough"

Learning Description Logic Knowledge Bases from Data Using Methods from Formal Concept Analysis

Description Logics (DLs) are a class of knowledge representation formalisms that can represent terminological and assertional knowledge using a well-defined semantics. Often, knowledge engineers are experts in their own fields, but not in logics, and require assistance in the process of ontology design. This thesis presents three methods that can extract terminological knowledge from existing data and thereby assist in the design process. They are based on similar formalisms from Formal Concept Analysis (FCA), in particular the Next-Closure Algorithm and Attribute-Exploration. The first of the three methods computes terminological knowledge from the data, without any expert interaction. The two other methods use expert interaction where a human expert can confirm each terminological axiom or refute it by providing a counterexample. These two methods differ only in the way counterexamples are provided

Model and Proof Theory of Constructive ALC, Constructive Description Logics

Description logics (DLs) represent a widely studied logical formalism with a significant impact in the field of knowledge representation and the Semantic Web. However, they are equipped with a classical descriptive semantics that is characterised by a platonic notion of truth, being insufficiently expressive to deal with evolving and incomplete information, as from data streams or ongoing processes. Such partially determined and incomplete knowledge can be expressed by relying on a constructive semantics. This thesis investigates the model and proof theory of a constructive variant of the basic description logic ALC, called cALC. The semantic dimension of constructive DLs is investigated by replacing the classical binary truth interpretation of ALC with a constructive notion of truth. This semantic characterisation is crucial to represent applications with partial information adequately, and to achieve both consistency under abstraction as well as robustness under refinement, and on the other hand is compatible with the Curry-Howard isomorphism in order to form the cornerstone for a DL-based type theory. The proof theory of cALC is investigated by giving a sound and complete Hilbert-style axiomatisation, a Gentzen-style sequent calculus and a labelled tableau calculus showing finite model property and decidability. Moreover, cALC can be strengthened towards normal intuitionistic modal logics and classical ALC in terms of sound and complete extensions and hereby forms a starting point for the systematic investigation of a constructive correspondence theory.Beschreibungslogiken (BLen) stellen einen vieluntersuchten logischen Formalismus dar, der den Bereich der Wissensrepräsentation und das Semantic Web signifikant geprägt hat. Allerdings basieren BLen meist auf einer klassischen deskriptiven Semantik, die gekennzeichnet ist durch einen idealisierten Wahrheitsbegriff nach Platons Ideenlehre, weshalb diese unzureichend ausdrucksstark sind, um in Entwicklung befindliches und unvollständiges Wissen zu repräsentieren, wie es beispielsweise durch Datenströme oder fortlaufende Prozesse generiert wird. Derartiges partiell festgelegtes und unvollständiges Wissen lässt sich auf der Basis einer konstruktiven Semantik ausdrücken. Diese Arbeit untersucht die Model- und Beweistheorie einer konstruktiven Variante der Basis-BL ALC, die im Folgenden als cALC bezeichnet wird. Die Semantik dieser konstruktiven Beschreibungslogik resultiert daraus, die traditionelle zweiwertige Interpretation logischer Aussagen des Systems ALC durch einen konstruktiven Wahrheitsbegriff zu ersetzen. Eine derartige Interpretation ist die Voraussetzung dafür, um einerseits Anwendungen mit partiellem Wissen angemessen zu repräsentieren, und sowohl die Konsistenz logischer Aussagen unter Abstraktion als auch ihre Robustheit unter Verfeinerung zu gewährleisten, und andererseits um den Grundstein für eine Beschreibungslogik-basierte Typentheorie gemäß dem Curry-Howard Isomorphismus zu legen. Die Ergebnisse der Untersuchung der Beweistheorie von cALC umfassen eine vollständige und korrekte Hilbert Axiomatisierung, einen Gentzen Sequenzenkalkül, und ein semantisches Tableaukalkül, sowie Beweise zur endlichen Modelleigenschaft und Entscheidbarkeit. Darüber hinaus kann cALC zu normaler intuitionistischer Modallogik und klassischem ALC durch vollständige und korrekte Erweiterungen ausgebaut werden, und bildet damit einen Startpunkt für die systematische Untersuchung einer konstruktiven Korrespondenztheorie

Topics in Knowledge Bases: Epistemic Ontologies and Secrecy-preserving Reasoning

Applications of ontologies/knowledge bases (KBs) in many domains (healthcare, national security, intelligence) have become increasingly important. In this dissertation, we focus on developing techniques for answering queries posed to KBs under the open world assumption (OWA). In the first part of this dissertation, we study the problem of query answering in KBs that contain epistemic information, i.e., knowledge of different experts. We study ALCKm, which extends the description logic ALC by adding modal operators of the basic multi-modal logic Km. We develop a sound and complete tableau algorithm for answering ALCKm queries w.r.t. an ALCKm knowledge base with an acyclic TBox. We then consider answering ALCKm queries w.r.t. an ALCKm knowledge base in which the epistemic operators correspond to those of classical multi-modal logic S4m and provide a sound and complete tableau algorithm. Both algorithms can be implemented in PSpace. In the second part, we study problems that allow autonomous entities or organizations (collectively called querying agents) to be able to selectively share information. In this scenario, the KB must make sure its answers are informative but do not disclose sensitive information. Most of the work in this area has focused on access control mechanisms that prohibit access to sensitive information (secrets). However, such an approach can be too restrictive in that it prohibits the use of sensitive information in answering queries against knowledge bases even when it is possible to do so without compromising secrets. We investigate techniques for secrecy-preserving query answering (SPQA) against KBs under the OWA. We consider two scenarios of increasing difficulty: (a) a KB queried by a single agent; and (b) a KB queried by multiple agents where the secrecy policies can differ across the different agents and the agents can selectively communicate the answers that they receive from the KB with each other subject to the applicable answer sharing policies. We consider classes of KBs that are of interest from the standpoint of practical applications (e.g., description logics and Horn KBs). Given a KB and secrets that need to be protected against the querying agent(s), the SPQA problem aims at designing a secrecy-preserving reasoner that answers queries without compromising secrecy under OWA. Whenever truthfully answering a query risks compromising secrets, the reasoner is allowed to hide the answer to the query by feigning ignorance, i.e., answering the query as Unknown . Under the OWA, the querying agent is not able to infer whether an Unknown answer to a query is obtained because of the incomplete information in the KB or because secrecy protection mechanism is being applied. In each scenario, we provide a general framework for the problem. In the single-agent case, we apply the general framework to the description logic EL and provide algorithms for answering queries as informatively as possible without compromising secrecy. In the multiagent case, we extend the general framework for the single-agent case. To model the communication between querying agents, we use a communication graph, a directed acyclic graph (DAG) with self-loops, where each node represents an agent and each edge represents the possibility of information sharing in the direction of the edge. We discuss the relationship between secrecy-preserving reasoners and envelopes (used to protect secrets) and present a special case of the communication graph that helps construct tight envelopes in the sense that removing any information from them will leave some secrets vulnerable. To illustrate our general idea of constructing envelopes, Horn KBs are considered