Deciding the synthesis problem for hybrid games through bisimulation

Hybrid games are games played on a finite graph endowed with real variables which may model behaviors of discrete controllers of continuous systems. The synthesis problem for hybrid games is decidable for classical objectives (like LTL formulas) when the games are initialized singular, meaning that the slopes of the continuous variables are piecewise constant and variables are reset whenever their slope changes. The known proof adapts the region construction from timed games. In this paper we show that initialized singular games can be reduced, via a sequence of alternating bisimulations, to timed games, generalizing the known reductions by bisimulation from initialized singular automata to timed automata. Alternating bisimulation is the generalization of bisimulation to games, accomodating a strategy translation lemma by which, when two games are bisimilar and carry the same observations, each strategy in one of the games can be translated to a strategy in the second game such that all the outcomes of the second strategy satisfies the same property that are satisfied by the first strategy. The advantage of the proposed approach is that one may then use realizability tools for timed games to synthesize a winning strategy for a given objective, and then use the strategy translation lemma to obtain a winning strategy in the hybrid game for the same objective.