A Testing Scenario for Probabilistic Processes

We introduce a notion of finite testing, based on statistical hypothesis tests, via a variant of the well-known trace machine. Under this scenario, two processes are deemed observationally equivalent if they cannot be distinguished by any finite test. We consider processes modeled as image finite probabilistic automata and prove that our notion of observational equivalence coincides with the trace distribution equivalence proposed by Segala. Along the way, we give an explicit characterization of the set of probabilistic executions of an arbitrary probabilistic automaton A and generalize the Approximation Induction Principle by defining an algebraic CPO structure on the set of trace distributions of A. We also prove limit and convex closure properties of trace distributions in an appropriate metric space