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Parity games are abstract infinite-round games that take an important role in formal verification. In the basic setting, these games are two-player, turn-based, and played under perfect information on directed graphs, whose nodes are…

Computer Science and Game Theory · Computer Science 2019-10-31 Antonio Di Stasio , Aniello Murano , Giuseppe Perelli , Moshe Y. Vardi

Parity games are games that are played on directed graphs whose vertices are labeled by natural numbers, called priorities. The players push a token along the edges of the digraph. The winner is determined by the parity of the greatest…

Computer Science and Game Theory · Computer Science 2015-03-20 Christoph Dittmann , Stephan Kreutzer , Alexandru I. Tomescu

Symmetry plays a central role in accelerating symbolic computation involving polynomials. This chapter surveys recent developments and foundational methods that leverage the inherent symmetries of polynomial systems to reduce complexity,…

Algebraic Geometry · Mathematics 2025-08-01 Cordian Riener , Thi Xuan Vu

Calude et al. have recently shown that parity games can be solved in quasi-polynomial time, a landmark result that has led to a number of approaches with quasi-polynomial complexity. Jurdinski and Lasic have further improved the precise…

Data Structures and Algorithms · Computer Science 2022-11-18 Daniele Dell'Erba , Sven Schewe

We consider fixpoint algorithms for two-player games on graphs with $\omega$-regular winning conditions, where the environment is constrained by a strong transition fairness assumption. Strong transition fairness is a widely occurring…

Formal Languages and Automata Theory · Computer Science 2023-06-22 Tamajit Banerjee , Rupak Majumdar , Kaushik Mallik , Anne-Kathrin Schmuck , Sadegh Soudjani

Several distinct techniques have been proposed to design quasi-polynomial algorithms for solving parity games since the breakthrough result of Calude, Jain, Khoussainov, Li, and Stephan (2017): play summaries, progress measures and register…

Formal Languages and Automata Theory · Computer Science 2020-01-15 Wojciech Czerwiński , Laure Daviaud , Nathanaël Fijalkow , Marcin Jurdziński , Ranko Lazić , Paweł Parys

Parity games can be used to represent many different kinds of decision problems. In practice, tools that use parity games often rely on a specification in a higher-order logic from which the actual game can be obtained by means of an…

Logic in Computer Science · Computer Science 2022-01-25 Maurice Laveaux , Wieger Wesselink , Tim A. C. Willemse

This paper is a contribution to the study of parity games and the recent constructions of three quasipolynomial time algorithms for solving them. We revisit a result of Czerwi\'nski, Daviaud, Fijalkow, Jurdzi\'nski, Lazi\'c, and Parys…

Computer Science and Game Theory · Computer Science 2018-10-22 Thomas Colcombet , Nathanaël Fijalkow

Zielonka's classic recursive algorithm for solving parity games is perhaps the simplest among the many existing parity game algorithms. However, its complexity is exponential, while currently the state-of-the-art algorithms have…

Computer Science and Game Theory · Computer Science 2023-06-22 Karoliina Lehtinen , Paweł Parys , Sven Schewe , Dominik Wojtczak

Symbolic Computation algorithms and their implementation in computer algebra systems often contain choices which do not affect the correctness of the output but can significantly impact the resources required: such choices can benefit from…

Symbolic Computation · Computer Science 2024-09-12 Tereso del Río , Matthew England

Graph games provide the foundation for modeling and synthesis of reactive processes. Such games are played over graphs where the vertices are controlled by two adversarial players. We consider graph games where the objective of the first…

Computer Science and Game Theory · Computer Science 2017-06-21 Krishnendu Chatterjee , Monika Henzinger , Alexander Svozil

Recent advancements in algorithms for sequential decision-making under imperfect information have shown remarkable success in large games such as limit- and no-limit poker. These algorithms traditionally formalize the games using the…

Computer Science and Game Theory · Computer Science 2023-12-07 Vojtěch Kovařík , David Milec , Michal Šustr , Dominik Seitz , Viliam Lisý

This paper examines multiplayer symmetric constant-sum games with more than two players in a competitive setting, including examples like Mahjong, Poker, and various board and video games. In contrast to two-player zero-sum games,…

Machine Learning · Computer Science 2024-10-04 Jiawei Ge , Yuanhao Wang , Wenzhe Li , Chi Jin

Parity games have witnessed several new quasi-polynomial algorithms since the breakthrough result of Calude et al. (STOC 2017). The combinatorial object underlying these approaches is a universal tree, as identified by Czerwi\'nski et al.…

Data Structures and Algorithms · Computer Science 2025-06-25 Zhuan Khye Koh , Georg Loho

We consider the problem of computing the maximal probability of satisfying an omega-regular specification for stochastic nonlinear systems evolving in discrete time. The problem reduces, after automata-theoretic constructions, to finding…

Systems and Control · Electrical Eng. & Systems 2022-09-30 Rupak Majumdar , Kaushik Mallik , Anne-Kathrin Schmuck , Sadegh Soudjani

An attractor decomposition meta-algorithm for solving parity games is given that generalises the classic McNaughton-Zielonka algorithm and its recent quasi-polynomial variants due to Parys (2019), and to Lehtinen, Schewe, and Wojtczak…

Data Structures and Algorithms · Computer Science 2022-08-30 Marcin Jurdziński , Rémi Morvan , K. S. Thejaswini

Partial methods play an important role in formal methods and beyond. Recently such methods were developed for parity games, where polynomial-time partial solvers decide the winners of a subset of nodes. We investigate here how effective…

Logic in Computer Science · Computer Science 2016-09-15 Patrick Ah-Fat , Michael Huth

Energy games, which model quantitative consumption of a limited resource, e.g., time or energy, play a central role in quantitative models for reactive systems. Reactive synthesis constructs a controller which satisfies a given…

Logic in Computer Science · Computer Science 2016-11-24 Shahar Maoz , Or Pistiner , Jan Oliver Ringert

Parity games play a central role in model checking and satisfiability checking. Solving parity games is computationally expensive, among others due to the size of the games, which, for model checking problems, can easily contain $10^9$…

Logic in Computer Science · Computer Science 2016-03-22 S. Cranen , J. J. A. Keiren , T. A. C. Willemse

Solving parity games is a major building block for numerous applications in reactive program verification and synthesis. While they can be solved efficiently in practice, no known approach has a polynomial worst-case runtime complexity. We…

Computer Science and Game Theory · Computer Science 2023-07-28 Tobias Hecking , Swathy Muthukrishnan , Alexander Weinert