Given two labelled Markov decision processes (MDPs), the trace-refinement problem asks whether for all strategies of the first MDP there exists a strategy of the second MDP such that the induced labelled Markov chains are trace-equivalent. We show that this problem is decidable in polynomial time if the second MDP is a Markov chain. The algorithm is based on new results on a particular notion of bisimulation between distributions over the states. However, we show that the general trace-refinement problem is undecidable, even if the first MDP is a Markov chain. Decidability of those problems was stated as open in 2008. We further study the decidability and complexity of the trace-refinement problem provided that the strategies are restricted to be memoryless.

@article{FijalkowKS16,
  title = {Trace Refinement in Labelled Markov Decision Processes},
  author = {Nathanaël Fijalkow and Stefan Kiefer and Mahsa Shirmohammadi},
  journal = {Logical Methods in Computer Science},
  volume = {Volume 16, Issue 2},
  year = {2020},
  doi = {10.23638/lmcs-16(2:10)2020}
}