Universität Dortmund an der Fakultät für Informatik Matthias Thimm

Reasoning with inaccurate information is a major topic within the fields of artificial intelligence in general and knowledge representation and reasoning in particular. This thesis deals with information that can be incomplete, uncertain, and contradictory. We employ probabilistic conditional logic as a foundation for our investigation. This framework allows for the representation of uncertain pieces of information by using probabilistic conditionals, a specific approach to handle if-then-rules within a probabilistic framework. Uncertainty can be expressed by means of probabilities attached to those rules and incompleteness can be handled in this framework by reasoning based on the principle of maximum entropy. This principle is a powerful approach when only incomplete information is at hand as it allows for filling missing pieces of information in the most unbiased way. This principle is realized for reasoning by selecting the one unique probability function that both satisfies the knowledge base and has maximum entropy. In this thesis we focus on two major issues that arise when representing knowledge with probabilistic conditional logic. On the one hand, we look at the problem of contradictory information that, e. g., arises when multiple experts share their knowledge in order to come up with a common knowledge base consisting of probabilistic conditionals. As in classical logic this is a severe problem because inconsistency of a knowledge base forbids application of model-based inductive inference approaches such as reasoning based on the principle of maximum entropy. We investigate this issue by presenting ways to analyze inconsistencies and ways to restore consistency. More specifically, we introduce the concept of measuring inconsistencies in probabilistic conditional logic and present inconsistency measures that are apt for the application in a probabilistic setting. While analyzing inconsistencies is a first step to remove inconsistencies we also have a look at automatic approaches for restoring consistency. On the other hand, we investigate an extension of the syntactical and semantical notions of probabilistic conditional logic to the relational case. Until now, most approaches for reasoning in probabilistic conditional logic only consider propositional logic as the underlying foundation. When switching to the more expressive framework of first-order logic, traditional semantics seem to fail in the interpretation of relational probabilistic conditionals in a commonsensical manner. In particular, the problem of contradictory information is an issue in first-order extensions of probabilistic conditional logic as well and we present novel semantical approaches to probabilistic conditionals that circumvent non-satisfiability in an intuitive and rational manner. We also extend the approach of reasoning based on the principle of maximum entropy to the framework of relational probabilistic conditional logic and investigate its properties.