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

## TL;DR

This paper extends the window mechanism for time-bounded objectives from deterministic games to stochastic Markov decision processes, providing solutions for probability thresholds and broadening applicability.

## Contribution

It introduces a generic approach to window-based objectives in MDPs and solves the threshold probability problem for classical objectives like mean-payoff and parity.

## Key findings

- Solved the threshold probability problem for window objectives in MDPs.
- Developed a generic framework applicable to various classical objectives.
- Enabled the use of window mechanisms in stochastic models.

## 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.