  Bisimulation metrics provide a robust and accurate approach to study the
behavior of nondeterministic probabilistic processes. In this paper, we propose
a logical characterization of bisimulation metrics based on a simple
probabilistic variant of the Hennessy-Milner logic. Our approach is based on
the novel notions of mimicking formulae and distance between formulae. The
former are a weak version of the well known characteristic formulae and allow
us to characterize also (ready) probabilistic simulation and probabilistic
bisimilarity. The latter is a 1-bounded pseudometric on formulae that mirrors
the Hausdorff and Kantorovich lifting the defining bisimilarity pseudometric.
We show that the distance between two processes equals the distance between
their own mimicking formulae.
