Life is Random, Time is Not: Markov Decision Processes with Window Objectives
This work extends the window framework to stochastic models, enabling reasoning about time bounds in system specifications for probabilistic systems, which is incremental as it builds on prior game-based results.
The authors tackled the problem of extending the window mechanism, which strengthens classical game objectives with time bounds, to stochastic environments by considering Markov decision processes, and they solved the threshold probability problem for window objectives with a generic approach applicable to mean-payoff and parity objectives.
The window mechanism was introduced by Chatterjee et al. to strengthen classical game objectives with time bounds. It permits to synthesize system controllers that exhibit acceptable behaviors within a configurable time frame, all along their infinite execution, in contrast to the traditional objectives that only require correctness of behaviors in the limit. The window concept has proved its interest in a variety of two-player zero-sum games because it enables reasoning about such time bounds in system specifications, but also thanks to the increased tractability that it usually yields. In this work, we extend the window framework to stochastic environments by considering Markov decision processes. A fundamental problem in this context is the threshold probability problem: given an objective it aims to synthesize strategies that guarantee satisfying runs with a given probability. We solve it for the usual variants of window objectives, where either the time frame is set as a parameter, or we ask if such a time frame exists. We develop a generic approach for window-based objectives and instantiate it for the classical mean-payoff and parity objectives, already considered in games. Our work paves the way to a wide use of the window mechanism in stochastic models.