Markov Decision Processes with Sure Parity and Multiple Reachability Objectives

TL;DR

This paper studies strategies in Markov decision processes that ensure a parity condition is always met while achieving multiple reachability objectives with certain probabilities, providing complexity results and algorithms.

Contribution

It introduces reductions of the problem to parity games and analyzes the complexity of various variants involving sure and threshold reachability objectives.

Findings

(a) and (c) reduce to solving parity games.

(b) can be solved in EXPTIME.

Strategies and algorithms are characterized for all cases.

Abstract

This paper considers the problem of finding strategies that satisfy a mixture of sure and threshold objectives in Markov decision processes. We focus on a single $\omega$-regular objective expressed as parity that must be surely met while satisfying $n$ reachability objectives towards sink states with some probability thresholds too. We consider three variants of the problem: (a) strict and (b) non-strict thresholds on all reachability objectives, and (c) maximizing the thresholds with respect to a lexicographic order. We show that (a) and (c) can be reduced to solving parity games, and (b) can be solved in $\sf{EXPTIME}$. Strategy complexities as well as algorithms are provided for all cases.