The Descriptive Complexity of Parity Games

We study the logical definablity of the winning regions of parity games. For games with a bounded number of priorities, it is wellknown that the winning regions are definable in the modal μ-calculus. Here we investigate the case of an unbounded number of priorities, both for finite game graphs and for arbitrary ones. In the general case, winning regions are definable in guarded second-order logic (GSO), but not in least-fixed point logic (LFP). On finite game graphs, winning regions are LFP-definable if, and only if, they are computable in polynomial time, and this result extends to any class of finite games that is closed under taking bisimulation quotients.