Petri Games are Monotonic but Difficult to Decide?

In this paper, we study two-player games played on infinite but monotonic game structures. We concentrate on coverability games, a natural subclass of reachability games in the context of monotonic game structures. On the negative side, we show that surprisingly, and contrary to the one-player case, coverability is undecidable on two-player monotonic game structures. On the positive side, we identify an interesting subclass of two-player monotonic game structures, for which coverability is decidable and for which we can effectively construct winning strategies. Furthermore, we show how to define two-player game structures that belong to that subclass with Petri nets. The results of this paper are compared to recent results obtained independently by Abdulla, Bouajjani and d’Orso on similar game structures where they identify another subclass of monotonic game structures with decidable results.