Regular Games with Imperfect Information Are Not That Regular

We consider two-player games with imperfect information and the synthesis of a randomized strategy for one player that ensures the objective is satisfied almost-surely (i.e., with probability 1), regardless of the strategy of the other player. Imperfect information is modeled by an indistinguishability relation describing the pairs of histories that the first player cannot distinguish, a generalization of the traditional model with partial observations. The game is regular if it admits a regular function whose kernel commutes with the indistinguishability relation. The synthesis of pure strategies that ensure all possible outcomes satisfy the objective is possible in regular games, by a generic reduction that holds for all objectives. While the solution for pure strategies extends to randomized strategies in the traditional model with partial observations (which is always regular), we show that a similar reduction does not exist in the more general model. Despite that, we show that in regular games with Buechi objectives the synthesis problem is decidable for randomized strategies that ensure the outcome satisfies the objective almost-surely.