MDPs with energy-parity objectives

Energy-parity objectives combine ω-regular with quantitative objectives of reward MDPs. The controller needs to avoid to run out of energy while satisfying a parity objective. We refute the common belief that, if an energy-parity objective holds almost-surely, then this can be realised by some finite memory strategy. We provide a surprisingly simple counterexample that only uses coBuchi conditions. We introduce the new class of bounded (energy) storage objectives that, when combined with parity objectives, preserve the finite memory property. Based on these, we show that almostsure and limit-sure energy-parity objectives, as well as almostsure and limit-sure storage parity objectives, are in NP ∩ coNP and can be solved in pseudo-polynomial time for energy-parity MDPs.