Energy and Mean-Payoff Parity Markov Decision Processes

TL;DR

This paper studies Markov Decision Processes with combined energy and mean-payoff parity objectives, revealing differences from two-player games and providing complexity results for decision problems.

Contribution

It demonstrates that energy and mean-payoff objectives differ in MDPs and improves the complexity bounds for almost-sure winning problems.

Findings

Deciding almost-sure winning in energy parity MDPs is in NP ∩ coNP.

Deciding almost-sure winning in mean-payoff parity MDPs is in P.

Energy and mean-payoff objectives differ in MDPs, unlike in two-player games.

Abstract

We consider Markov Decision Processes (MDPs) with mean-payoff parity and energy parity objectives. In system design, the parity objective is used to encode \omega-regular specifications, and the mean-payoff and energy objectives can be used to model quantitative resource constraints. The energy condition requires that the resource level never drops below 0, and the mean-payoff condition requires that the limit-average value of the resource consumption is within a threshold. While these two (energy and mean-payoff) classical conditions are equivalent for two-player games, we show that they differ for MDPs. We show that the problem of deciding whether a state is almost-sure winning (i.e., winning with probability 1) in energy parity MDPs is in NP \cap coNP, while for mean-payoff parity MDPs, the problem is solvable in polynomial time, improving a recent PSPACE bound.