Verification of general Markov decision processes by approximate similarity relations and policy refinement

In this work we introduce new approximate similarity relations that are shown to be key for policy (or control) synthesis over general Markov decision processes. The models of interest are discrete-time Markov decision processes, endowed with uncountably infinite state spaces and metric output (or observation) spaces. The new relations, underpinned by the use of metrics, allow, in particular, for a useful trade-off between deviations over probability distributions on states, and distances between model outputs. We show that the new probabilistic similarity relations, inspired by a notion of simulation developed for finite-state models, can be effectively employed over general Markov decision processes for verification purposes, and specifically for control refinement from abstract models.