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Abstract
We study countably infinite MDPs with parity objectives, and special cases with a bounded number of colors in the Mostowski hierarchy (including reachability, safety, Buchi and coBuchi).
In finite MDPs there always exist optimal memoryless deterministic (MD) strategies for parity objectives, but this does not generally hold for countably infinite MDPs. In particular, optimal strategies need not exist. For countable infinite MDPs, we provide a complete picture of the memory requirements of optimal (resp., ϵoptimal) strategies for all objectives in the Mostowski hierarchy. In particular, there is a strong dichotomy between two different types of objectives. For the first type, optimal strategies, if they exist, can be chosen MD, while for the second type optimal strategies require infinite memory. (I.e., for all objectives in the Mostowski hierarchy, if finitememory randomized strategies suffice then also MD strategies suffice.) Similarly, some objectives admit ϵoptimal MDstrategies, while for others ϵoptimal strategies require infinite memory. Such a dichotomy also holds for the subclass of countably infinite MDPs that are finitely branching, though more objectives admit MDstrategies here.
In finite MDPs there always exist optimal memoryless deterministic (MD) strategies for parity objectives, but this does not generally hold for countably infinite MDPs. In particular, optimal strategies need not exist. For countable infinite MDPs, we provide a complete picture of the memory requirements of optimal (resp., ϵoptimal) strategies for all objectives in the Mostowski hierarchy. In particular, there is a strong dichotomy between two different types of objectives. For the first type, optimal strategies, if they exist, can be chosen MD, while for the second type optimal strategies require infinite memory. (I.e., for all objectives in the Mostowski hierarchy, if finitememory randomized strategies suffice then also MD strategies suffice.) Similarly, some objectives admit ϵoptimal MDstrategies, while for others ϵoptimal strategies require infinite memory. Such a dichotomy also holds for the subclass of countably infinite MDPs that are finitely branching, though more objectives admit MDstrategies here.
Original language  English 

Title of host publication  2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) 
Publisher  Institute of Electrical and Electronics Engineers (IEEE) 
Number of pages  21 
ISBN (Electronic)  9781509030187 
DOIs  
Publication status  Published  18 Aug 2017 
Event  2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science  Reykjavik, Reykjavik, Iceland Duration: 20 Jun 2017 → 23 Jun 2017 http://lics.siglog.org/lics17/ 
Conference
Conference  2017 32nd Annual ACM/IEEE Symposium on Logic in Computer Science 

Abbreviated title  LICS 2017 
Country/Territory  Iceland 
City  Reykjavik 
Period  20/06/17 → 23/06/17 
Internet address 
Keywords
 countable MDPs, parity objectives, strategies, memory requirement
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Richard Mayr
 School of Informatics  Reader
 Laboratory for Foundations of Computer Science
 Foundations of Computation
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