Dennis Peuter

• The goal of this PhD thesis is to investigate possibilities of using symbol elimination for solving problems over complex theories and analyze the applicability of such uniform approaches in different areas of application, such as verification, knowledge representation and graph theory. In the thesis we propose an approach to symbol elimination in complex theories that follows the general idea of combining hierarchical reasoning with symbol elimination in standard theories. We analyze how this general approach can be specialized and used in different areas of application. In the verification of parametric systems it is important to prove that certain safety properties hold. This can be done by showing that a property is an inductive invariant of the system, i.e. it holds in the initial state of the system and is invariant under updates of the system. Sometimes this is not the case for the condition itself, but for a stronger condition it is. In this thesis we propose a method for goal-directed invariant strengthening. In knowledge representation we often have to deal with huge ontologies. Combining two ontologies usually leads to new consequences, some of which may be false or undesired. We are interested in finding explanations for such unwanted consequences. For this we propose a method for computing interpolants in the description logics EL and EL⁺, based on a translation to the theory of semilattices with monotone operators and a certain form of interpolation in this theory. In wireless network theory one often deals with classes of geometric graphs in which the existence or non-existence of an edge between two vertices in a graph relies on properties on their distances to other nodes. One possibility to prove properties of those graphs or to analyze relations between the graph classes is to prove or disprove that one graph class is contained in the other. In this thesis we propose a method for checking inclusions between geometric graph classes.