Benchmarks for Parity Games (extended version)

We propose a benchmark suite for parity games that includes all benchmarks that have been used in the literature, and make it available online. We give an overview of the parity games, including a description of how they have been generated. We also describe structural properties of parity games, and using these properties we show that our benchmarks are representative. With this work we provide a starting point for further experimentation with parity games.Comment: The corresponding tool and benchmarks are available from https://github.com/jkeiren/paritygame-generator. This is an extended version of the paper that has been accepted for FSEN 201

Solving parity games: Explicit vs symbolic

In this paper we provide a broad investigation of the symbolic approach for solving Parity Games. Specifically, we implement in a fresh tool, called, four symbolic algorithms to solve Parity Games and compare their performances to the corresponding explicit versions for different classes of games. By means of benchmarks, we show that for random games, even for constrained random games, explicit algorithms actually perform better than symbolic algorithms. The situation changes, however, for structured games, where symbolic algorithms seem to have the advantage. This suggests that when evaluating algorithms for parity-game solving, it would be useful to have real benchmarks and not only random benchmarks, as the common practice has been

Winning Cores in Parity Games

We introduce the novel notion of winning cores in parity games and develop a deterministic polynomial-time under-approximation algorithm for solving parity games based on winning core approximation. Underlying this algorithm are a number properties about winning cores which are interesting in their own right. In particular, we show that the winning core and the winning region for a player in a parity game are equivalently empty. Moreover, the winning core contains all fatal attractors but is not necessarily a dominion itself. Experimental results are very positive both with respect to quality of approximation and running time. It outperforms existing state-of-the-art algorithms significantly on most benchmarks

Fatal Attractors in Parity Games: Building Blocks for Partial Solvers

Attractors in parity games are a technical device for solving "alternating" reachability of given node sets. A well known solver of parity games - Zielonka's algorithm - uses such attractor computations recursively. We here propose new forms of attractors that are monotone in that they are aware of specific static patterns of colors encountered in reaching a given node set in alternating fashion. Then we demonstrate how these new forms of attractors can be embedded within greatest fixed-point computations to design solvers of parity games that run in polynomial time but are partial in that they may not decide the winning status of all nodes in the input game. Experimental results show that our partial solvers completely solve benchmarks that were constructed to challenge existing full solvers. Our partial solvers also have encouraging run times in practice. For one partial solver we prove that its run-time is at most cubic in the number of nodes in the parity game, that its output game is independent of the order in which monotone attractors are computed, and that it solves all Buechi games and weak games. We then define and study a transformation that converts partial solvers into more precise partial solvers, and we prove that this transformation is sound under very reasonable conditions on the input partial solvers. Noting that one of our partial solvers meets these conditions, we apply its transformation on 1.6 million randomly generated games and so experimentally validate that the transformation can be very effective in increasing the precision of partial solvers

Reasoning about LTL Synthesis over finite and infinite games

In the last few years, research formal methods for the analysis and the verification of properties of systems has increased greatly. A meaningful contribution in this area has been given by algorithmic methods developed in the context of synthesis. The basic idea is simple and appealing: instead of developing a system and verifying that it satisfies its specification, we look for an automated procedure that, given the specification returns a system that is correct by construction. Synthesis of reactive systems is one of the most popular variants of this problem, in which we want to synthesize a system characterized by an ongoing interaction with the environment. In this setting, large effort has been devoted to analyze specifications given as formulas of linear temporal logic, i.e., LTL synthesis. Traditional approaches to LTL synthesis rely on transforming the LTL specification into parity deterministic automata, and then to parity games, for which a so-called winning region is computed. Computing such an automaton is, in the worst-case, double-exponential in the size of the LTL formula, and this becomes a computational bottleneck in using the synthesis process in practice. The first part of this thesis is devoted to improve the solution of parity games as they are used in solving LTL synthesis, trying to give efficient techniques, in terms of running time and space consumption, for solving parity games. We start with the study and the implementation of an automata-theoretic technique to solve parity games. More precisely, we consider an algorithm introduced by Kupferman and Vardi that solves a parity game by solving the emptiness problem of a corresponding alternating parity automaton. Our empirical evaluation demonstrates that this algorithm outperforms other algorithms when the game has a small number of priorities relative to the size of the game. In many concrete applications, we do indeed end up with parity games where the number of priorities is relatively small. This makes the new algorithm quite useful in practice. We then provide a broad investigation of the symbolic approach for solving parity games. Specifically, we implement in a fresh tool, called SPGSolver, four symbolic algorithms to solve parity games and compare their performances to the corresponding explicit versions for different classes of games. By means of benchmarks, we show that for random games, even for constrained random games, explicit algorithms actually perform better than symbolic algorithms. The situation changes, however, for structured games, where symbolic algorithms seem to have the advantage. This suggests that when evaluating algorithms for parity-game solving, it would be useful to have real benchmarks and not only random benchmarks, as the common practice has been. LTL synthesis has been largely investigated also in artificial intelligence, and specifically in automated planning. Indeed, LTL synthesis corresponds to fully observable nondeterministic planning in which the domain is given compactly and the goal is an LTL formula, that in turn is related to two-player games with LTL goals. Finding a strategy for these games means to synthesize a plan for the planning problem. The last part of this thesis is then dedicated to investigate LTL synthesis under this different view. In particular, we study a generalized form of planning under partial observability, in which we have multiple, possibly infinitely many, planning domains with the same actions and observations, and goals expressed over observations, which are possibly temporally extended. By building on work on two-player games with imperfect information in the Formal Methods literature, we devise a general technique, generalizing the belief-state construction, to remove partial observability. This reduces the planning problem to a game of perfect information with a tight correspondence between plans and strategies. Then we instantiate the technique and solve some generalized planning problems

A workbench for preprocessor design and evaluation: toward benchmarks for parity games

We describe a prototype workbench for the study of parity games and their solvers. This workbench is aimed at facilitating two activities: to aid in the design, validation, and evaluation of preprocessors for parity game solvers; and to aid in the generation of benchmark parity games that are meaningful for a wide range of solvers. Our workbench allows for easy composition of preprocessors, can populate databases with games and their meta-data, offers a query language for generating games of interest, and has already found potentially hard games

Static Analysis of Parity Games: Alternating Reachability Under Parity

It is well understood that solving parity games is equivalent, up 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. A 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

Sparse Positional Strategies for Safety Games

We consider the problem of obtaining sparse positional strategies for safety games. Such games are a commonly used model in many formal methods, as they make the interaction of a system with its environment explicit. Often, a winning strategy for one of the players is used as a certificate or as an artefact for further processing in the application. Small such certificates, i.e., strategies that can be written down very compactly, are typically preferred. For safety games, we only need to consider positional strategies. These map game positions of a player onto a move that is to be taken by the player whenever the play enters that position. For representing positional strategies compactly, a common goal is to minimize the number of positions for which a winning player's move needs to be defined such that the game is still won by the same player, without visiting a position with an undefined next move. We call winning strategies in which the next move is defined for few of the player's positions sparse. Unfortunately, even roughly approximating the density of the sparsest strategy for a safety game has been shown to be NP-hard. Thus, to obtain sparse strategies in practice, one either has to apply some heuristics, or use some exhaustive search technique, like ILP (integer linear programming) solving. In this paper, we perform a comparative study of currently available methods to obtain sparse winning strategies for the safety player in safety games. We consider techniques from common knowledge, such as using ILP or SAT (satisfiability) solving, and a novel technique based on iterative linear programming. The results of this paper tell us if current techniques are already scalable enough for practical use.Comment: In Proceedings SYNT 2012, arXiv:1207.055

Static analysis of parity games: alternating reachability under parity

It is well understood that solving parity games is equivalent, up 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. A 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