# MDPs with Energy-Parity Objectives

## TL;DR

This paper investigates energy-parity objectives in Markov decision processes, challenging previous assumptions, and introduces bounded storage objectives that maintain finite-memory strategies, providing complexity results for solving these problems.

## Contribution

It refutes the belief that almost-sure energy-parity objectives require infinite memory and introduces bounded storage objectives that preserve finite-memory strategies.

## Key findings

- Counterexample shows finite memory may not suffice for energy-parity objectives
- Bounded storage objectives combined with parity preserve finite-memory strategies
- Complexity of solving energy-parity MDPs is in NP ∩ coNP and solvable in pseudo-polynomial time

## Abstract

Energy-parity objectives combine $\omega$-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 coB\"uchi 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 almost-sure and limit-sure energy-parity objectives, as well as almost-sure and limit-sure storage parity objectives, are in $\mathit{NP}\cap \mathit{coNP}$ and can be solved in pseudo-polynomial time for energy-parity MDPs.