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Parity games can be used to solve satisfiability, verification and controller synthesis problems. As part of an effort to better understand their nature, or the nature of the problems they solve, preorders on parity games have been studied. Defining these relations, and in particular proving their transitivity, has proven quite difficult on occasion. We propose a uniform way of lifting certain preorders on Kripke structures to parity games and study the resulting preorders. We explore their relation with parity game preorders from the literature and we study new relations. Finally, we investigate whether these preorders can also be obtained via modal characterisations of the preorders.

It is well understood that solving parity games is equivalent, upi¾?to polynomial time, to model checking of the modal mu-calculus. It is a long-standing open problem whether solving parity games or model checking modal mu-calculus formulas can be done in polynomial time. Ai¾?recent approach to studying this problem has been the design of partial solvers, algorithms that run in polynomial time and that may only solve parts of a parity game. Although it was shown that such partial solvers can completely solve many practical benchmarks, the design of such partial solvers was somewhat ad hoc, limiting a deeper understanding of the potential of that approach. We here mean to provide such robust foundations for deeper analysis through a new form of game, alternating reachability under parity. We prove the determinacy of these games and use this determinacy to define, for each player, a monotone fixed point over an ordered domain of height linear in the size of the parity game such that all nodes in its greatest fixed point are won by said player in the parity game. We show, through theoretical and experimental work, that such greatest fixed points and their computation leads to partial solvers that run in polynomial time. These partial solvers are based on established principles of static analysis and are more effective than partial solvers studied in extant work.

We define governed stuttering bisimulation for parity games, weakening stuttering bisimulation by taking the ownership of vertices into account only when this might lead to observably different games. We show that governed stuttering bisimilarity is an equivalence for parity games and allows for a natural quotienting operation. Moreover, we prove that all pairs of vertices related by governed stuttering bisimilarity are won by the same player in the parity game. Thus, our equivalence can be used as a preprocessing step when solving parity games. Governed stuttering bisimilarity can be decided in $\mathcal{O}(n^2m)$ time for parity games with n vertices and m edges. Our experiments indicate that governed stuttering bisimilarity is mostly competitive with stuttering equivalence on parity games encoding typical verification problems.

Three-color parity games capture the disjunction of a Buchi and a co-Buchi condition. The most efficient known algorithm for these games is the progress measures algorithm by Jurdzinski. We present an acceleration technique that, while leaving the worst-case complexity unchanged, often leads to considerable speed-ups in games arising in practice. As an application, we consider games played in discrete real time, where players should be prevented from stopping time by always choosing moves with delay zero. The time progress condition can be encoded as a three-color parity game. Using the tool TICC as a platform, we compare the performance of a BDD-based symbolic implementation of the progress measure algorithm with acceleration, and of the symbolic implementation of the classical µ-calculus algorithm of Emerson and Jutla.