Analysing the complexity of games on graphs MSc thesis, Samson Tikitu de Jager =Abstract= This thesis does not have a thesis, in the sense of a central statement that is developed and argued for. Instead, it explores three different ways of describing complexity in simple games. The three approaches are drawn from set theory, from automata theory, and from theory of algorithms and computer science. These three areas, while obviously related, generally use different notation and different approaches even while ostensibly talking about the same things: `games'. The progression of ideas through this thesis could be seen as a movement from the mopst abstract of these representations to the concrete. The first complexity analysis is drawn from descriptive set theory and topology; a `game' here is an explicitly infinitary concept and one that (for sufficiently complicated cases) can be shown not to yield to constructive methods of proof. Chapter 1 presents the topological notion of complexity from descriptive set theory, and some abstract solution concepts utilising infinitary procedures on infinitary objects. In Chapter 2 we move to automata theory, restricting ourselves to a class of games that are played on finite structures (the games themselves remain infinite, but can be given finite representations). The focus here is on the finite computational resources needed to implement a strategy, which in the descriptive set theoretic context might easily be infinite. Chapter 3 takes an algorithmic approach to the process of *constructing* a strategy for an automaton game, and measures the complexity of the game by the time complexity of the algorithm to solve it. This is the most concrete of the three approaches, in fact a Java application implementing some of these algorithms accompanies the written text of this thesis [1]. Not only are the games restricted to finite representations (automata games, on finite structures) and the strategies implementable with finite computational resources, but the process of producing a strategy is also finitary and algorithmic. The final chapter contains information concerning the software, and a short user manual. There is also a short Appendix, applying the techniques of Chapters 1, 2 and 3 to more fine-grained analysis of one of the complexity classes considered in Chapter 1. --------------- [1] Download the software from http://www.illc.uva.nl/Publications/ResearchReports/MoL-2005-07.software.jar Bugfixes and updates will be posted to my homepage, at http://tikitu.dejager.net.nz/software/GameGraph/