Parity Games Research Articles

New deterministic algorithms for solving parity games

We study parity games in which one of the two players controls only a small number k of nodes and the other player controls the n−k other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time kO(k)⋅O(n3), and general parity games in time (p+k)O(k)⋅O(pnm), where p is the number of distinct priorities and m is the number of edges. For all games with k=o(n) this improves the previously fastest algorithm by Jurdziński, Paterson, and Zwick (SICOMP 2008).We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree.

Robust Time-Optimal Guidance in a Partially Uncertain Time-Varying Flow-Field

In this paper, we address the problem of guiding an aerial or aquatic vehicle to a fixed target point in a partially uncertain flow-field. We assume that the motion of the vehicle is described by a point-mass linear kinematic model. In addition, the velocity of the flow-field, which is taken to be time varying, can be decomposed into two components: one which is known a priori and another one which is uncertain and only a bound on its magnitude is known. We show that the guidance problem can be reformulated as an equivalent pursuit evasion game with time-varying affine dynamics. To solve the latter game, we propose an extension of a specialized solution approach, which transforms the pursuit–evasion game (whose terminal time is free) via a special state transformation into a family of games with fixed terminal time. In addition, we provide a simple method to visualize the level sets of the value function of the game, along with the corresponding reachable sets. Furthermore, we compare our conservative game-theoretic solution with a pure optimal control solution, for the special case in which the flow-field is perfectly known a priori.

Cooking Your Own Parity Game Preorders Through Matching Plays

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.

On the complexity of resource-bounded logics

We revisit decidability results for resource-bounded logics and use decision problems on vector addition systems with states (VASS) in order to establish complexity characterisations of (decidable) model checking problems. We show that the model checking problem for the logic RB±ATL is 2exptime-complete by using recent results on alternating VASS (and in exptime when the number of resources is bounded). Moreover, we establish that the model checking problem for RBTL is expspace-complete. The problem is decidable and of the same complexity for RBTL⁎, proving a new decidability result as a by-product of the approach. When the number of resources is bounded, the problem is in pspace. We also establish that the model checking problem for RB±ATL⁎, the extension of RB±ATL with arbitrary path formulae, is decidable by a reduction to parity games for single-sided VASS (a variant of alternating VASS). Furthermore, we are able to synthesise values for resource parameters. Hence, the paper establishes formal correspondences between model checking problems for resource-bounded logics advocated in the AI literature and decision problems on alternating VASS, paving the way for more applications and cross-fertilizations.

Solving parity games via priority promotion

We consider parity games, a special form of two-player infinite-duration games on numerically labeled graphs, whose winning condition requires that the maximal value of a label occurring infinitely often during a play be of some specific parity. The problem of identifying the corresponding winning regions has a rather intriguing status from a complexity theoretic viewpoint, since it belongs to the class $${\textsc {UPTime}} \cap {\textsc {CoUPTime}}$$ , and still open is the question whether it can be solved in polynomial time. Parity games also have great practical interest, as they arise in many fields of theoretical computer science, most notably logic, automata theory, and formal verification. In this paper, we propose a new algorithm for the solution of this decision problem, based on the idea of promoting vertexes to higher priorities during the search for winning regions. The proposed approach has nice computational properties, exhibiting the best space complexity among the currently known solutions. Experimental results on both random games and benchmark families show that the technique is also very effective in practice.

Memoryless determinacy of finite parity games: Another simple proof

Memoryless determinacy of (infinite) parity games is an important result with numerous applications. It was first independently established by Emerson and Jutla [1] and Mostowski [2] but their proofs involve elaborate developments. The elegant and simpler proof of Zielonka [3] still requires a nested induction on the finite number of priorities and on ordinals for sets of vertices. There are other proofs for finite games like the one of Björklund, Sandberg and Vorobyovin [4] that relies on relating infinite and finite duration games. We present here another simple proof that finite parity games are determined with memoryless strategies using induction on the number of relevant states. The closest proof that relies on induction over non absorbing states is the one of Grädel [5]. However instead of focusing on a single appropriate vertex for induction as we do here, he considers two reduced games per vertex, for all the vertices of the game. The idea of reasoning about a single state has been inspired to me by the analysis of finite stochastic priority games by Karelovic and Zielonka [6].

Robust Exponential Worst Cases for Divide-et-Impera Algorithms for Parity Games

The McNaughton-Zielonka divide et impera algorithm is the simplest and most flexible approach available in the literature for determining the winner in a parity game. Despite its theoretical worst-case complexity and the negative reputation as a poorly effective algorithm in practice, it has been shown to rank among the best techniques for the solution of such games. Also, it proved to be resistant to a lower bound attack, even more than the strategy improvements approaches, and only recently a family of games on which the algorithm requires exponential time has been provided by Friedmann. An easy analysis of this family shows that a simple memoization technique can help the algorithm solve the family in polynomial time. The same result can also be achieved by exploiting an approach based on the dominion-decomposition techniques proposed in the literature. These observations raise the question whether a suitable combination of dynamic programming and game-decomposition techniques can improve on the exponential worst case of the original algorithm. In this paper we answer this question negatively, by providing a robustly exponential worst case, showing that no intertwining of the above mentioned techniques can help mitigating the exponential nature of the divide et impera approaches.

The Descriptive Complexity of Modal μ Model-checking Games

This paper revisits the well-established relationship between the modal mu calculus and parity games to show that it is even more robust than previously known. It addresses the question of whether the descriptive complexity of modal mu model-checking games, previously known to depend on the syntactic complexity of a formula, depends in fact on its semantic complexity. It shows that up to formulas of semantic co-B\"uchi complexity, the descriptive complexity of their model-checking games coincides exactly with their semantic complexity. Beyond co-B\"uchi, the descriptive complexity of the model-checking parity games of a formula is shown to be an upper bound on its semantic complexity; whether it is also a lower bound remains an open question.

An Existence Theorem of Nash Equilibrium in Coq and Isabelle

Nash equilibrium (NE) is a central concept in game theory. Here we prove formally a published theorem on existence of an NE in two proof assistants, Coq and Isabelle: starting from a game with finitely many outcomes, one may derive a game by rewriting each of these outcomes with either of two basic outcomes, namely that Player 1 wins or that Player 2 wins. If all ways of deriving such a win/lose game lead to a game where one player has a winning strategy, the original game also has a Nash equilibrium. This article makes three other contributions: first, while the original proof invoked linear extension of strict partial orders, here we avoid it by generalizing the relevant definition. Second, we notice that the theorem also implies the existence of a secure equilibrium, a stronger version of NE that was introduced for model checking. Third, we also notice that the constructive proof of the theorem computes secure equilibria for non-zero-sum priority games (generalizing parity games) in quasi-polynomial time.

Parity game reductions

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 vertices or beyond. Equivalence relations can be used to reduce the size of a parity game, thereby potentially alleviating part of the computational burden. We reconsider (governed) bisimulation and (governed) stuttering bisimulation, and we give detailed proofs that these relations are equivalences, have unique quotients and they approximate the winning regions of parity games. Furthermore, we present game-based characterisations of these relations. Using these characterisations our equivalences are compared to relations for parity games that can be found in the literature, such as direct simulation equivalence and delayed simulation equivalence. To complete the overview we develop coinductive characterisations of direct- and delayed simulation equivalence and we establish a lattice of equivalences for parity games.

Universally composable secure proxy re-signature scheme with effective calculation

In proxy re-signature scheme, the semi-trust proxy is given special information to convert user A’s signature into another user B’s signature on the same message m. This has numerous application, such as simplifying the certificate management by constructing a secure channel between two authentication centers, simplifying group signature management by making each signature transformed to the group’s signature, helping relieve some common key management headaches by resigning without the authentication center, and constructing digital right management (DRM) interoperable system by sharing the certification. In this paper, we proposed a novel proxy re-signature scheme with universally composable security. The proposed scheme is unidirectional, single-use, key optimal, non-transitive and non-temporary. Its security is based on Computational Diffie–Hellman assumption in the random oracle model. Furthermore, we present the universal composable based security definition of proxy re-signature, as well as the equivalent game based definition. If a scheme is proved to be safe in our game, it will have the universally composable property, which will simplify the future security analysis. Finally, we support the proposed scheme with detailed security and performance analysis.

OBLIGATION BLACKWELL GAMES AND P-AUTOMATA

AbstractWe generalize winning conditions in two-player games by adding a structural acceptance condition called obligations. Obligations are orthogonal to the linear winning conditions that define whether a play is winning. Obligations are a declaration that player 0 can achieve a certain value from a configuration. If the obligation is met, the value of that configuration for player 0 is 1.We define the value in such games and show that obligation games are determined. For Markov chains with Borel objectives and obligations, and finite turn-based stochastic parity games with obligations we give an alternative and simpler characterization of the value function. Based on this simpler definition we show that the decision problem of winning finite turn-based stochastic parity games with obligations is in NP∩co-NP. We also show that obligation games provide a game framework for reasoning about p-automata.

Parallel Parity Games: a Multicore Attractor for the Zielonka Recursive Algorithm

Parity games are abstract infinite-duration two-player games, widely studied in computer science. Several solution algorithms have been proposed and also implemented in the community tool of choice called PGSolver, which has declared the Zielonka Recursive (ZR) algorithm the best performing on randomly generated games. With the aim of scaling and solving wider classes of parity games, several improvements and optimizations have been proposed over the existing algorithms. However, no one has yet explored the benefit of using the full computational power of which even common modern multicore processors are capable of. This is even more surprisingly by considering that most of the advanced algorithms in PGSolver are sequential.In this paper we introduce and implement, on a multicore architecture, a parallel version of the Attractor algorithm, that is the main kernel of the ZR algorithm. This choice follows our investigation that more of the 99% of the execution time of the ZR algorithm is spent in this module. We provide testing on graphs up to 20K nodes generated through PGSolver and we discuss performance analysis in terms of strong and weak scaling.

Solving parity games in big steps

This article proposes a new algorithm that improves the complexity bound for solving parity games. Our approach combines McNaughton's iterated fixed point algorithm with a preprocessing step, which is called prior to every recursive call. The preprocessing uses ranking functions similar to Jurdziński's, but with a restricted co-domain, to determine all winning regions smaller than a predefined parameter. The combination of the preprocessing step with the recursive call guarantees that McNaughton's algorithm proceeds in big steps, whose size is bounded from below by the chosen parameter. Higher parameters lead to smaller call trees, but they also result in an expensive preprocessing step. An optimal parameter balances the cost of the recursive call and the preprocessing step, resulting in an improvement of the known upper bound for solving parity games from O(m(2nc)12c) to approximately O(m(6e1.6‾nc2)13c).

Window Parity Games: An Alternative Approach Toward Parity Games with Time Bounds

Classical objectives in two-player zero-sum games played on graphs often deal with limit behaviors of infinite plays: e.g., mean-payoff and total-payoff in the quantitative setting, or parity in the qualitative one (a canonical way to encode omega-regular properties). Those objectives offer powerful abstraction mechanisms and often yield nice properties such as memoryless determinacy. However, their very nature provides no guarantee on time bounds within which something good can be witnessed. In this work, we consider two approaches toward inclusion of time bounds in parity games. The first one, parity-response games, is based on the notion of finitary parity games [CHH09] and parity games with costs [FZ14,WZ16]. The second one, window parity games, is inspired by window mean-payoff games [CDRR15]. We compare the two approaches and show that while they prove to be equivalent in some contexts, window parity games offer a more tractable alternative when the time bound is given as a parameter (P-c. vs. PSPACE-c.). In particular, it provides a conservative approximation of parity games computable in polynomial time. Furthermore, we extend both approaches to the multi-dimension setting. We give the full picture for both types of games with regard to complexity and memory bounds. [CHH09] K. Chatterjee, T.A. Henzinger, F. Horn (2009): Finitary winning in omega-regular games. ACM Trans. Comput. Log. 11(1). [FZ14] N. Fijalkow, M. Zimmermann (2014): Parity and Streett Games with Costs. LMCS 10(2). [WZ16] A. Weinert, M. Zimmermann (2016): Easy to Win, Hard to Master: Optimal Strategies in Parity Games with Costs. Proc. of CSL, LIPIcs, Schloss Dagstuhl - LZI. To appear. [CDRR15] K. Chatterjee, L. Doyen, M. Randour, J.-F. Raskin (2015): Looking at mean-payoff and total-payoff through windows. Information and Computation 242, pp. 25-52.

A Delayed Promotion Policy for Parity Games

Parity games are two-player infinite-duration games on graphs that play a crucial role in various fields of theoretical computer science. Finding efficient algorithms to solve these games in practice is widely acknowledged as a core problem in formal verification, as it leads to efficient solutions of the model-checking and satisfiability problems of expressive temporal logics, e.g., the modal muCalculus. Their solution can be reduced to the problem of identifying sets of positions of the game, called dominions, in each of which a player can force a win by remaining in the set forever. Recently, a novel technique to compute dominions, called priority promotion, has been proposed, which is based on the notions of quasi dominion, a relaxed form of dominion, and dominion space. The underlying framework is general enough to accommodate different instantiations of the solution procedure, whose correctness is ensured by the nature of the space itself. In this paper we propose a new such instantiation, called delayed promotion, that tries to reduce the possible exponential behaviours exhibited by the original method in the worst case. The resulting procedure not only often outperforms the original priority promotion approach, but so far no exponential worst case is known.

Partial Solvers for Parity Games: Effective Polynomial-Time Composition

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 polynomial-time partial solvers can be by studying interactions of partial solvers based on generic composition patterns that preserve polynomial-time computability. We show that use of such composition patterns discovers new partial solvers - including those that merge node sets that have the same but unknown winner - by studying games that composed partial solvers can neither solve nor simplify. We experimentally validate that this data-driven approach to refinement leads to polynomial-time partial solvers that can solve all standard benchmarks of structured games. For one of these polynomial-time partial solvers not even a sole random game from a few billion random games of varying configuration was found that it won't solve completely.

An exponential lower bound for Cunningham’s rule

In this paper we give an exponential lower bound for Cunningham’s least recently considered (round-robin) rule as applied to parity games, Markov decision processes and linear programs. This improves a recent subexponential bound of Friedmann for this rule on these problems. The round-robin rule fixes a cyclical order of the variables and chooses the next pivot variable starting from the previously chosen variable and proceeding in the given circular order. It is perhaps the simplest example from the class of history-based pivot rules. Our results are based on a new lower bound construction for parity games. Due to the nature of the construction we are also able to obtain an exponential lower bound for the round-robin rule applied to acyclic unique sink orientations of hypercubes (AUSOs). Furthermore these AUSOs are realizable as polytopes. We believe these are the first such results for history based rules for AUSOs, realizable or not. The paper is self-contained and requires no previous knowledge of parity games.

Exponential examples of solving parity games

This paper is devoted to solving certain problems on the computational complexity of deciding the winner in cyclic games. The main result is the proof of the fact that the nondeterministic potential transformation algorithm designed for solving parity games is exponential in terms of computation time.

Linear game non-contextuality and Bell inequalities—a graph-theoretic approach

We study the classical and quantum values of a class of one- and two-party unique games, that generalizes the well-known XOR games to the case of non-binary outcomes. In the bipartite case the generalized XOR (XOR-d) games we study are a subclass of the well-known linear games. We introduce a ‘constraint graph’ associated to such a game, with the constraints defining the game represented by an edge-coloring of the graph. We use the graph-theoretic characterization to relate the task of finding equivalent games to the notion of signed graphs and switching equivalence from graph theory. We relate the problem of computing the classical value of single-party anti-correlation XOR games to finding the edge bipartization number of a graph, which is known to be MaxSNP hard, and connect the computation of the classical value of XOR-d games to the identification of specific cycles in the graph. We construct an orthogonality graph of the game from the constraint graph and study its Lovász theta number as a general upper bound on the quantum value even in the case of single-party contextual XOR-d games. XOR-d games possess appealing properties for use in device-independent applications such as randomness of the local correlated outcomes in the optimal quantum strategy. We study the possibility of obtaining quantum algebraic violation of these games, and show that no finite XOR-d game possesses the property of pseudo-telepathy leaving the frequently used chained Bell inequalities as the natural candidates for such applications. We also show this lack of pseudo-telepathy for multi-party XOR-type inequalities involving two-body correlation functions.