Ziel der vorliegenden Arbeit ist die Entwicklung eines modular aufgebauten Erfüllbarkeitstesters für verschiedene unimodale und eine polymodale Beschreibungslogik, die zur Repräsentation von agentenabhängig modelliertem terminologischen Wissen genutzt werden kann. Es werden dazu Tableaukalküle eingeführt, und deren Implementationen vorgestellt. Die Tableaukalküle bearbeiten Beschreibungslogik- und Modallogikanteile getrennt, dieser Trennung folgt dann auch der Aufbau des Testers. Zur Bearbeitung der Beschreibungslogikanteile wird das System FaCT benutzt, welches als Modul in den Erfüllbarkeitstester integriert wird. Eine notwendige Vorverarbeitungsstufe wird von einem weiteren Modul realisiert. Für die Performance des gesamten modularen Systems spielt die interne Kommunikation eine entscheidene Rolle. Deshalb ist eine optimierte Variante der Kommunikation zwischen FaCT und dem Gesamtsystem entwickelt worden. Während des Tests verschiedener Formeln eines Benchmarks konnte das hier entwickelte modulare System im Vergleich mit anderen Systemen akzeptable Leistungen erbringen.

Constructing description logic ontologies is a difficult task that is normally conducted by experts. Recent results show that parts of ontologies can be constructed from description logic interpretations. However, these results assume the interpretations to be free of errors, which may not be the case for real-world data. To provide some mechanism to handle these errors, the notion of confidence from data mining is introduced into description logics, yielding confident general concept inclusions (confident GCIs) of finite interpretations. The main focus of this work is to prove the existence of finite bases of confident GCIs and to describe some of theses bases explicitly.

Description Logics (DLs) are a family of logic-based knowledge representation (KR) formalisms designed to represent and reason about static conceptual knowledge in a semantically well-understood way. On the other hand, standard action formalisms are KR formalisms based on classical logic designed to model and reason about dynamic systems. The largest part of the present work is dedicated to integrating DLs with action formalisms, with the main goal of obtaining decidable action formalisms with an expressiveness significantly beyond propositional. To this end, we offer DL-tailored solutions to the frame and ramification problem. One of the main technical results is that standard reasoning problems about actions (executability and projection), as well as the plan existence problem are decidable if one restricts the logic for describing action pre- and post-conditions and the state of the world to decidable Description Logics. A smaller part of the work is related to decidable extensions of Description Logics with concrete datatypes, most importantly with those allowing to refer to the notions of space and time.

Description Logics (DLs) form a family of knowledge representation formalisms which can be used to represent and reason with conceptual knowledge about a domain of interest. The knowledge represented by DLs is mainly static. In many applications, the domain knowledge is dynamic. This observation motivates the research on how to update the knowledge when changes in the application domain take place. This thesis is dedicated to the study of updating knowledge, more precisely, assertional knowledge represented in DLs. We explore whether the updated knowledge can be expressed in several standard DLs and, if so, whether it is computable and what is its size.

Description logics knowledge bases are a popular approach to represent terminological and assertional knowledge suitable for computers to work with. Despite that, the practicality of description logics is impaired by the difficulties one has to overcome to construct such knowledge bases. Previous work has addressed this issue by providing methods to learn valid terminological knowledge from data, making use of ideas from formal concept analysis. A basic assumption here is that the data is free of errors, an assumption that can in general not be made for practical applications. This thesis presents extensions of these results that allow to handle errors in the data. For this, knowledge that is "almost valid" in the data is retrieved, where the notion of "almost valid" is formalized using the notion of confidence from data mining. This thesis presents two algorithms which achieve this retrieval. The first algorithm just extracts all almost valid knowledge from the data, while the second algorithm utilizes expert interaction to distinguish errors from rare but valid counterexamples.

Inevitably, it becomes more and more important to verify that the systems surrounding us have certain properties. This is indeed unavoidable for safety-critical systems such as power plants and intensive-care units. We refer to the term system in a broad sense: it may be man-made (e.g. a computer system) or natural (e.g. a patient in an intensive-care unit). Whereas in Model Checking it is assumed that one has complete knowledge about the functioning of the system, we consider an open-world scenario and assume that we can only observe the behaviour of the actual running system by sensors. Such an abstract sensor could sense e.g. the blood pressure of a patient or the air traffic observed by radar. Then the observed data are preprocessed appropriately and stored in a fact base. Based on the data available in the fact base, situation-awareness tools are supposed to help the user to detect certain situations that require intervention by an expert. Such situations could be that the heart-rate of a patient is rather high while the blood pressure is low, or that a collision of two aeroplanes is about to happen. Moreover, the information in the fact base can be used by monitors to verify that the system has certain properties. It is not realistic, however, to assume that the sensors always yield a complete description of the current state of the observed system. Thus, it makes sense to assume that information that is not present in the fact base is unknown rather than false. Moreover, very often one has some knowledge about the functioning of the system. This background knowledge can be used to draw conclusions about the possible future behaviour of the system. Employing description logics (DLs) is one way to deal with these requirements. In this thesis, we tackle the sketched problem in three different contexts: (i) runtime verification using a temporalised DL, (ii) temporalised query entailment, and (iii) verification in DL-based action formalisms.

Description logics (DLs) are used to represent knowledge of an application domain and provide standard reasoning services to infer consequences of this knowledge. However, classical DLs are not suited to represent vagueness in the description of the knowledge. We consider a combination of DLs and Fuzzy Logics to address this task. In particular, we consider the t-norm-based semantics for fuzzy DLs introduced by Hájek in 2005. Since then, many tableau algorithms have been developed for reasoning in fuzzy DLs. Another popular approach is to reduce fuzzy ontologies to classical ones and use existing highly optimized classical reasoners to deal with them. However, a systematic study of the computational complexity of the different reasoning problems is so far missing from the literature on fuzzy DLs. Recently, some of the developed tableau algorithms have been shown to be incorrect in the presence of general concept inclusion axioms (GCIs). In some fuzzy DLs, reasoning with GCIs has even turned out to be undecidable. This work provides a rigorous analysis of the boundary between decidable and undecidable reasoning problems in t-norm-based fuzzy DLs, in particular for GCIs. Existing undecidability proofs are extended to cover large classes of fuzzy DLs, and decidability is shown for most of the remaining logics considered here. Additionally, the computational complexity of reasoning in fuzzy DLs with semantics based on finite lattices is analyzed. For most decidability results, tight complexity bounds can be derived.

Inevitably, it becomes more and more important to verify that the systems surrounding us have certain properties. This is indeed unavoidable for safety-critical systems such as power plants and intensive-care units. We refer to the term system in a broad sense: it may be man-made (e.g. a computer system) or natural (e.g. a patient in an intensive-care unit). Whereas in Model Checking it is assumed that one has complete knowledge about the functioning of the system, we consider an open-world scenario and assume that we can only observe the behaviour of the actual running system by sensors. Such an abstract sensor could sense e.g. the blood pressure of a patient or the air traffic observed by radar. Then the observed data are preprocessed appropriately and stored in a fact base. Based on the data available in the fact base, situation-awareness tools are supposed to help the user to detect certain situations that require intervention by an expert. Such situations could be that the heart-rate of a patient is rather high while the blood pressure is low, or that a collision of two aeroplanes is about to happen. Moreover, the information in the fact base can be used by monitors to verify that the system has certain properties. It is not realistic, however, to assume that the sensors always yield a complete description of the current state of the observed system. Thus, it makes sense to assume that information that is not present in the fact base is unknown rather than false. Moreover, very often one has some knowledge about the functioning of the system. This background knowledge can be used to draw conclusions about the possible future behaviour of the system. Employing description logics (DLs) is one way to deal with these requirements. In this thesis, we tackle the sketched problem in three different contexts: (i) runtime verification using a temporalised DL, (ii) temporalised query entailment, and (iii) verification in DL-based action formalisms.

Description Logics (DLs) form a family of knowledge representation formalisms which can be used to represent and reason with conceptual knowledge about a domain of interest. The knowledge represented by DLs is mainly static. In many applications, the domain knowledge is dynamic. This observation motivates the research on how to update the knowledge when changes in the application domain take place. This thesis is dedicated to the study of updating knowledge, more precisely, assertional knowledge represented in DLs. We explore whether the updated knowledge can be expressed in several standard DLs and, if so, whether it is computable and what is its size.

Constructing description logic ontologies is a difficult task that is normally conducted by experts. Recent results show that parts of ontologies can be constructed from description logic interpretations. However, these results assume the interpretations to be free of errors, which may not be the case for real-world data. To provide some mechanism to handle these errors, the notion of confidence from data mining is introduced into description logics, yielding confident general concept inclusions (confident GCIs) of finite interpretations. The main focus of this work is to prove the existence of finite bases of confident GCIs and to describe some of theses bases explicitly.