Highly arc transitive digraphs

Unendliche, hochgradig bogentransitive Digraphen werden definiert und anhand von Beispielen vorgestellt. Die Erreichbarkeitsrelation und Eigenschaft–Z werden definiert und unter Verwendung von Knotengraden, Wachstum und anderen Eigenschaften, die von der Untersuchung von Nachkommen von Doppelstrahlen oder Automorphismengruppen herrühren, auf hochgradig bogentransitiven Digraphen untersucht. Seifters Theoreme über hochgradig bogentransitive Digraphen mit mehr als einem Ende, seine daherrührende Vermutung und deren sie widerlegende Gegenbeispiele werden vorgestellt. Eine Bedingung, unter der C–homogene Digraphen hochgradig bogentransitiv sind, wird angegeben und die Verbindung zwischen hochgradig bogentransitiven Digraphen und total unzusammenhängenden, topologischen Gruppen wird erwähnt. Einige Bemerkungen über die Vermutung von Cameron–Praeger–Wormald werden gemacht und eine verfeinerte Version vermutet. Die Eigenschaften der bekannten hochgradig bogentransitiven Digraphen werden gesammelt. Es wird festgestellt, dass einige, aber nicht alle unter ihnen Cayley–Graphen sind. Schließlich werden offen gebliebene Fragestellungen und Vermutungen zusammengefasst und neue hinzugefügt. Für die vorgestellten Lemmata, Propositionen und Theoreme sind entweder Beweise enthalten, oder Referenzen zu Beweisen werden angegeben.Infinite, highly arc transitive digraphs are defined and examples are given. The Reachability–Relation and Property-Z are defined and investigated on infinite, highly arc transitive digraphs using the valencies, spread and other properties arising from the investigation of the descendants of lines or the automorphism groups. Seifters theorems about highly arc transitive digraphs with more than one end, his conjecture on them and the counterexamples that disproved his conjecture, are given. A condition for C–homogeneous digraphs to be highly arc transitve is stated and the connection between highly arc transitive digraphs and totally disconnected, topological groups is mentioned. Some notes on the Cameron–Praeger–Wormald–Conjecture are made and a refined conjecture is stated. The properties of the known highly arc transitive digraphs are collected, some but not all of them are Cayley–graphs. Finally open questions and conjectures are stated and new ones are added. For the given lemmas, propositions and theorems either proofs or references to proofs are included

Model Checking Communicating Processes: Run Graphs, Graph Grammars, and MSO

The formal model of recursive communicating processes (RCPS) is important in practice but does not allows to derive decidability results for model checking questions easily. We focus a partial order representation of RCPS’s execution by graphs—so called run graphs, and suggest an underapproximative verification approach based on a bounded-treewidth requirement. This allows to directly derive positive decidability results for MSO model checking (seen as partial order logic on run graphs) based on a context-freeness argument for restricted classes run graph

Modal logics on rational Kripke structures

This dissertation is a contribution to the study of infinite graphs which can be presented in a finitary way. In particular, the class of rational graphs is studied. The vertices of a rational graph are labeled by a regular language in some finite alphabet and the set of edges of a rational graph is a rational relation on that language. While the first-order logics of these graphs are generally not decidable, the basic modal and tense logics are. A survey on the class of rational graphs is done, whereafter rational Kripke models are studied. These models have rational graphs as underlying frames and are equipped with rational valuations. A rational valuation assigns a regular language to each propositional variable. I investigate modal languages with decidable model checking on rational Kripke models. This leads me to consider regularity preserving relations to see if the class can be generalised even further. Then the concept of a graph being rationally presentable is examined - this is analogous to a graph being automatically presentable. Furthermore, some model theoretic properties of rational Kripke models are examined. In particular, bisimulation equivalences between rational Kripke models are studied. I study three subclasses of rational Kripke models. I give a summary of the results that have been obtained for these classes, look at examples (and non-examples in the case of automatic Kripke frames) and of particular interest is finding extensions of the basic tense logic with decidable model checking on these subclasses. An extension of rational Kripke models is considered next: omega-rational Kripke models. Some of their properties are examined, and again I am particularly interested in finding modal languages with decidable model checking on these classes. Finally I discuss some applications, for example bounded model checking on rational Kripke models, and mention possible directions for further research