What is Decidable about Partially Observable Markov Decision Processes with ω-Regular Objectives

We consider partially observable Markov decision processes (POMDPs) with ω-regular conditions specified as parity objectives. The class of ω-regular languages extends regular languages to infinite strings and provides a robust specification language to express all properties used in verification, and parity objectives are canonical forms to express ω-regular conditions. The qualitative analysis problem given a POMDP and a parity objective asks whether there is a strategy to ensure that the objective is satisfied with probability 1 (resp. positive probability). While the qualitative analysis problems are known to be undecidable even for very special cases of parity objectives, we establish decidability (with optimal complexity) of the qualitative analysis problems for POMDPs with all parity objectives under finitememory strategies. We establish asymptotically optimal (exponential) memory bounds and EXPTIMEcompleteness of the qualitative analysis problems under finite-memory strategies for POMDPs with parity objectives.