  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.
