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.

翻译：我们考虑具有不完美信息的双人博弈，以及为其中一方玩家合成随机策略的问题，该策略能确保无论对手采用何种策略，目标都以概率1（即几乎必然）被满足。不完美信息通过一种不可区分关系来建模，该关系描述了先手玩家无法区分的博弈历史对，这是对传统部分可观测模型的推广。若博弈存在一个正则函数，其核与不可区分关系可交换，则称该博弈是正则的。对于正则博弈，通过一种适用于所有目标的通用规约方法，可以合成确保所有可能结果均满足目标的纯策略。虽然在传统部分可观测模型（总是正则的）中，纯策略的解可以推广到随机策略，但我们证明在更一般的模型中不存在类似的规约。尽管如此，我们证明了对于具有Buechi目标的正则博弈，为随机策略合成几乎必然满足目标的策略问题是可判定的。