Equivalence Relations in Fully and Partially Observable Markov Decision Processes

We explore equivalence relations between states in Markov Decision Processes and Partially Observable Markov Decision Processes. We focus on two different equivalence notions: bisimulation (Givan et al., 2003) and a notion of trace equivalence, under which states are considered equivalent if they generate the same conditional probability distributions over observation sequences (where the conditioning is on action sequences).  We show that the relationship between these two equivalence notions changes depending on the amount and nature of the partial observability. We also present an alternate characterization of bisimulation based on trajectory equivalence.