A Parity Game Tale of Two Counters

Parity games are simple infinite games played on finite graphs with a winning condition that is expressive enough to capture nested least and greatest fixpoints. Through their tight relationship to the modal mu-calculus, they are used in practice for the model-checking and synthesis problems of the mu-calculus and related temporal logics like LTL and CTL. Solving parity games is a compelling complexity theoretic problem, as the problem lies in the intersection of UP and co-UP and is believed to admit a polynomial-time solution, motivating researchers to either find such a solution or to find superpolynomial lower bounds for existing algorithms to improve the understanding of parity games. We present a parameterized parity game called the Two Counters game, which provides an exponential lower bound for a wide range of attractor-based parity game solving algorithms. We are the first to provide an exponential lower bound to priority promotion with the delayed promotion policy, and the first to provide such a lower bound to tangle learning.Comment: In Proceedings GandALF 2019, arXiv:1909.0597

Symbolic Parity Game Solvers that Yield Winning Strategies

Parity games play an important role for LTL synthesis as evidenced by recent breakthroughs on LTL synthesis, which rely in part on parity game solving. Yet state space explosion remains a major issue if we want to scale to larger systems or specifications. In order to combat this problem, we need to investigate symbolic methods such as BDDs, which have been successful in the past to tackle exponentially large systems. It is therefore essential to have symbolic parity game solving algorithms, operating using BDDs, that are fast and that can produce the winning strategies used to synthesize the controller in LTL synthesis. Current symbolic parity game solving algorithms do not yield winning strategies. We now propose two symbolic algorithms that yield winning strategies, based on two recently proposed fixpoint algorithms. We implement the algorithms and empirically evaluate them using benchmarks obtained from SYNTCOMP 2020. Our conclusion is that the algorithms are competitive with or faster than an earlier symbolic implementation of Zielonka's recursive algorithm, while also providing the winning strategies.Comment: In Proceedings GandALF 2020, arXiv:2009.0936

Simple Fixpoint Iteration To Solve Parity Games

A naive way to solve the model-checking problem of the mu-calculus uses fixpoint iteration. Traditionally however mu-calculus model-checking is solved by a reduction in linear time to a parity game, which is then solved using one of the many algorithms for parity games. We now consider a method of solving parity games by means of a naive fixpoint iteration. Several fixpoint algorithms for parity games have been proposed in the literature. In this work, we introduce an algorithm that relies on the notion of a distraction. The idea is that this offers a novel perspective for understanding parity games. We then show that this algorithm is in fact identical to two earlier published fixpoint algorithms for parity games and thus that these earlier algorithms are the same. Furthermore, we modify our algorithm to only partially recompute deeper fixpoints after updating a higher set and show that this modification enables a simple method to obtain winning strategies. We show that the resulting algorithm is simple to implement and offers good performance on practical parity games. We empirically demonstrate this using games derived from model-checking, equivalence checking and reactive synthesis and show that our fixpoint algorithm is the fastest solution for model-checking games.Comment: In Proceedings GandALF 2019, arXiv:1909.0597

Improving parity games in practice

Parity games are infinite-round two-player games played on directed graphs whose nodes are labeled with priorities. The winner of a play is determined by the smallest priority (even or odd) that is encountered infinitely often along the play. In the last two decades, several algorithms for solving parity games have been proposed and implemented in PGSolver, a platform written in OCaml. PGSolver includes the Zielonka’s recursive algorithm (RE, for short) which is known to be the best performing one over random games. Notably, several attempts have been carried out with the aim of improving the performance of RE in PGSolver, but with small advances in practice. In this work, we deeply revisit the implementation of RE by dealing with the use of specific data structures and programming languages such as Scala, Java, C++, and Go. Our empirical evaluation shows that these choices are successful, gaining up to three orders of magnitude in running time over the classic version of the algorithm implemented in PGSolver

The Worst-Case Complexity of Symmetric Strategy Improvement

Symmetric strategy improvement is an algorithm introduced by Schewe et al. (ICALP 2015) that can be used to solve two-player games on directed graphs such as parity games and mean payoff games. In contrast to the usual well-known strategy improvement algorithm, it iterates over strategies of both players simultaneously. The symmetric version solves the known worst-case examples for strategy improvement quickly, however its worst-case complexity remained open. We present a class of worst-case examples for symmetric strategy improvement on which this symmetric version also takes exponentially many steps. Remarkably, our examples exhibit this behaviour for any choice of improvement rule, which is in contrast to classical strategy improvement where hard instances are usually hand-crafted for a specific improvement rule. We present a generalized version of symmetric strategy iteration depending less rigidly on the interplay of the strategies of both players. However, it turns out it has the same shortcomings

From Quasi-Dominions to Progress Measures

We revisit the approaches to the solution of parity games based on progress measures and show how the notion of quasi dominions can be integrated with those approaches. The idea is that, while progress measure based techniques typically focus on one of the two players, little information is gathered on the other player during the solution process. Adding quasi dominions provides additional information on this player that can be leveraged to greatly accelerate convergence to a progress measure. To accommodate quasi dominions, however, non trivial refinements of the approach are necessary. In particular, we need to introduce a novel notion of measure and a new method to prove correctness of the resulting solution technique