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. Example applications are the synthesis of finite-state systems from specifications in temporal logic and alternating-time temporal logic (ATL) model checking. 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. From a sparse winning positional strategy for the safety player in a synthesis game, we can compute a small implementation satisfying the specification used for building the game, and for ATL model checking, sparse strategies are easier to comprehend and thus help in analysing the cause of a model checking result. 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. Approaches considered include the techniques from common knowledge, such as using ILP or SAT (satisfiability) solving, and a novel technique based on iterative linear programming. The restriction to safety games is not only motivated by the fact that they are the simplest game model for continuous interaction between a system and its environment, and thus an evaluation of strategy extraction methods should start here, but also by the fact that they are sufficient for many applications, such as synthesis. The results of this paper shed light onto which directions of research in this area are the promising ones, and if current techniques are already scalable enough for practical use.