We study countably infinite MDPs with parity objectives. Unlike in finite MDPs, optimal strategies need not exist, and may require infinite memory if they do. We provide a complete picture of the exact strategy complexity of ε-optimal strategies (and optimal strategies, where they exist) for all subclasses of parity objectives in the Mostowski hierarchy. Either MD-strategies, Markov strategies, or 1-bit Markov strategies are necessary and sufficient, depending on the number of colors, the branching degree of the MDP, and whether one considers ε-optimal or optimal strategies. In particular, 1-bit Markov strategies are necessary and sufficient for ε-optimal (resp. optimal) strategies for general parity objectives.
@InProceedings{kiefer_et_al:LIPIcs.CONCUR.2020.39, author = {Kiefer, Stefan and Mayr, Richard and Shirmohammadi, Mahsa and Totzke, Patrick}, title = {{Strategy Complexity of Parity Objectives in Countable MDPs}}, booktitle = {31st International Conference on Concurrency Theory (CONCUR 2020)}, pages = {39:1--39:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-160-3}, ISSN = {1868-8969}, year = {2020}, volume = {171}, editor = {Konnov, Igor and Kov\'{a}cs, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CONCUR.2020.39}, URN = {urn:nbn:de:0030-drops-128513}, doi = {10.4230/LIPIcs.CONCUR.2020.39}, annote = {Keywords: Markov decision processes, Parity objectives, Levy’s zero-one law} }
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