Probabilistic synthesis against GR(1) winning condition

摘要

Reactive synthesis is a technique for automatic generation of a reactive system from a high level specification. The system is reactive in the sense that it reacts to the inputs from the environment. The specification is in general given as a linear temporal logic (LTL) formula. The behaviour of the system interacting with the environment can be represented as a game in which the system plays against the environment. Thus, a problem of reactive synthesis is commonly treated as solving such a game with the specification as the winning condition. Reactive synthesis has been thoroughly investigated for more two decades. A well-known challenge is to deal with the complex uncertainty of the environment. We understand that a major issue is due to the lack of a sufficient treatment of probabilistic properties in the traditional models. For example, a two-player game defined by a standard Kriple structure does not consider probabilistic transitions in reaction to the uncertain physical environment; and a Markov Decision Process (MDP) in general does not explicitly separate the system from its environment and it does not describe the interaction between system and the environment. In this paper, we propose a new and more general model which combines the two-player game and the MDP. Furthermore, we study probabilistic reactive synthesis for the games of General Reactivity of Rank 1 (i.e., GR(1)) defined in this model. More specifically, we present an algorithm, which for given model M, a location s and a GR(1) specification P, determines the strategy for each player how to maximize/minimize the probabilities of the satisfaction of P at location s. We use an example to describe the model of probabilistic games and demonstrate our algorithm.