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 -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 EXPTIME. Strategy complexities as well as algorithms are provided for all cases.