  This paper studies a difference operator for stochastic systems whose
specifications are represented by Abstract Probabilistic Automata (APAs). In
the case refinement fails between two specifications, the target of this
operator is to produce a specification APA that represents all witness PAs of
this failure. Our contribution is an algorithm that allows to approximate the
difference of two APAs with arbitrary precision. Our technique relies on new
quantitative notions of distances between APAs used to assess convergence of
the approximations, as well as on an in-depth inspection of the refinement
relation for APAs. The procedure is effective and not more complex to implement
than refinement checking.
