Parity games : descriptive complexity and algorithms for new solvers

Parity games are 2-person, 0-sum, graph-based, and determined games that form an important foundational concept in formal methods (see e.g., [Zie98]), and their exact computational complexity has been an open problem for over twenty years now. In this thesis, we study algorithms that solve parity games in that they determine which nodes are won by which player, and where such decisions are supported with winning strategies. We modify and so improve a known algorithm but also propose new algorithmic approaches to solving parity games and to understanding their descriptive complexity. For all of our contributions, we write our own custom frameworks, in the Scala programming language, to perform tailored experiments and empirical studies to demonstrate and support our theoretical findings. First, we improve on one of the solver algorithms, based on small progress measures [Jur00], by use of concurrency. We show that, for many parity games, it is possible to deliver extra performance using this technique in a multi-core environment. Second, we design algorithms to reduce the computational complexity of parity games, and create implementations to observe and evaluate the behaviours of these reductions in our experimental settings. The measure Rabin index, arising from the design of the said algorithm, is shown to be a new descriptive complexity for parity games. Finally, we define a new family of attractors and derive new parity game solvers from them. Although these new solvers are “partial”, in that they do not solve all parity games completely, our experiments show that they do solve a set of benchmark games (i.e., games with known structures) designed to stress test solvers from PGSolver toolkit [FL10] completely, and some of these partial solvers deliver favourable performance against a known high performance solver in many circumstances