Games in system design and verification

Extended abstract. We consider graph games, where a token is moved along the edges of a directed graph in an infinite sequence of moves. The result of playing a graph game, or play, is an infinite path through the graph. Graph games can serve as models for discrete reactive systems: the vertices represent system states; the players represent system components; the moves represent system transitions; the plays represent system behaviors; and the objectives represent system specifications. Graph games come in different flavors depending, for example, on the number of players, the information that each player has available for choosing a move, the mechanism by which the successor vertex is determined from a given choice of moves, and the type of objective for each player. The variety of possible graph games permits the modeling of different interaction paradigms for component-based systems. Asynchronous interaction is naturally modeled using turn-based games: at each vertex, one of the players chooses a move, which determines the successor vertex [18]. Synchronous interaction, on the other hand, requires concurrent games: at each vertex, all players choose moves simultaneously and independently, and the combination of choices determines the successor vertex [7]. In deterministic games, a move or combination of moves determines a unique successor vertex; in stochastic games, a move or combination of moves determines a probability distribution over successor vertices [12]. Note that deterministic games can model nondeterministic choice through a choice of available moves; stochastic games model both nondeterministic and probabilistic choice. The objectives of the players in a graph game can be qualitative or quantitative. A qualitative objective is a set of winning plays. For modeling reactive systems, the interesting class of qualitative objectives are the ω-regular sets, which can express common safety and liveness properties [15]. A quantitative objective requires a labeled graph, where each vertex (or edge) is labeled with a rational payoff for each player. Then, the goal of a player is to maximize a reward, which is a function that maps each play to a real number, for instance, the supremum, limit sum, or limit average of all payoffs along the play. Payoffs can be used to represent costs, delays, or other resource values of reactive systems [4]. The solution problem for a graph game asks for the value of each player at each vertex. For turn-based deterministic games with qualitative objectives, the value of a player i at a vertex q is binary: the value Vi(q) is 1 if player i has a strategy to ensure that, when the game starts from vertex q, the outcome of the game is a play that is winning for player i; and else Vi(q) = 0. For other games, values are real numbers. For turn-based deterministic games with quantitative