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We consider partially observable Markov decision processes (POMDPs) with{\omega}-regular conditions specified as parity objectives. The class of{\omega}-regular languages extends regular languages to infinite strings andprovides a robust specification language to express all properties used inverification, and parity objectives are canonical forms to express{\omega}-regular conditions. The qualitative analysis problem given a POMDP anda parity objective asks whether there is a strategy to ensure that theobjective is satis- fied with probability 1 (resp. positive probability). Whilethe qualitative analysis problems are known to be undecidable even for veryspecial cases of parity objectives, we establish decidability (with optimalcomplexity) of the qualitative analysis problems for POMDPs with all parityobjectives under finite- memory strategies. We establish optimal (exponential)memory bounds and EXPTIME-completeness of the qualitative analysis problemsunder finite-memory strategies for POMDPs with parity objectives.

What is Decidable about Partially Observable Markov Decision Processes with {\omega}-Regular Objectives