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Boolean games are a logical setting for representing static games in a succinct way, taking advantage of the expressive power and succinctness of propositional logic. A Boolean game consists of a set of players, each of them controlling a set of propositional variables and having a specific goal expressed by a propositional formula, or more generally a specification of the player's preference relation in some logical language for compact preference representation, such as prioritized goals. There is a lot of graphical structure hidden in a Boolean game: the satisfaction of each player's goal depends on players whose actions have an influence on her goals. Exploiting this dependency structure facilitates the computation of pure Nash equilibria, by partly decomposing a game into several sub-games that are only loosely related.