Markov Decision Processes with Sure Parity and Multiple Reachability Objectives

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 © maximizing the thresholds with respect to a lexicographic order. We show that (a) and © can be reduced to solving parity games, and (b) can be solved in $\mathsf{E}\mathsf{X}\mathsf{P}\mathsf{T}\mathsf{I}\mathsf{M}\mathsf{E}$. Strategy complexities as well as algorithms are provided for all cases.