Life is Random, Time is Not: Markov Decision Processes with Window Objectives

Logical Methods in Computer Science, vol. Volume 16, Issue 4

Abstract

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.

Authors 4

1. Thomas Brihaye Aachen

   University of Mons · RWTH Aachen University

   Affiliation as printed

   RWTH Aachen , Germany

   UMONS -Université de Mons , Belgium

   RWTH Aachen, Germany

   UMONS -Université de Mons, Belgium

2. Florent Delgrange

   Vrije Universiteit Brussel

   Affiliation as printed

   Vrije Universiteit Brussel , Belgium

3. Youssouf Oualhadj

   University of Mons · Fund for Scientific Research · Université Paris-Est Créteil

   Affiliation as printed

   F.R.S.-FNRS , Belgium

   LACL -UPEC , France

   F.R.S.-FNRS, Belgium

   LACL -UPEC, France

   UMONS -Université de Mons, Belgium

4. Mickaël Randour

   Vrije Universiteit Brussel

   Affiliation as printed

   Vrije Universiteit Brussel , Belgium