Model Checking and Testing are two areas with a similar goal: to verify that a system satisfies a property. They start with different hypothesis on the systems and develop many techniques with different notions of approximation, as an exact verification may be computationally too hard. We present some of notions of approximation with their Logic and Statistics backgrounds, which

Preprint submitted to Elsevier January 27, 2010Abstract. We associate a statistical vector to a trace and a geometrical embedding to a Markov Decision Process, based on a distance on words, and study basic Membership and Equivalence problems. The Membership problem for a trace w and a Markov Decision Process S decides if there exists a strategy on S which generates with high probability traces close to w. We prove that Membership of a trace is testable and Equivalence of MDPs is polynomial time approximable. For Probabilistic Automata, Membership is not testable, and approximate Equivalence is undecidable. We give a class of properties, based on results concerning the structure of the tail σ-field of a finite Markov chain,