1996 IEEE International Conference on Systems, Man and by IEEE

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Testing finitary probabilistic processes. In: CONCUR, pp. 274–288 (2009) 19. : Approximating labelled Markov processes. Inf. Comput. 184(1), 160–200 (2003) 20. : Weak bisimulation is sound and complete for PCTL*. Inf. Comput. 208(2), 203–219 (2010) 21. : Approximate analysis of probabilistic processes: Logic, simulation and games. In: QEST, pp. 264–273 (2008) 22. : On probabilistic automata in continuos time. In: LICS. IEEE, Los Alamitos (to appear, 2010) 23. : Probabilistic automata in continuous time.

Furthermore, ≈Δ has a selection of distinguishing properties. We refer the reader to [22] for details. We mention only briefly, that ≈Δ is a congruence with respect to parallel composition. The congruence property can be established for other standard process algebraic operators – with the usual root condition being needed to arrive at a congruence for non-deterministic choice. We finally want to correct our claim [22], that a reformulation of PA weak bisimilarity as a relation on distribution would not be compositional with respect to subdistribution composition, now turns out to be wrong.

The crucial point in this proof is the claim, that μ ≈◦ γ implies ∀C ∈ S/≈◦Δ : μ(C) = γ(C), since then the conditions of Definition 10 follow immediately. To see the claim, note that ≈◦ itself is a semi-weak bisimulation. Then, by repeated application of the left hand side of clause (i) in the definition of ≈◦ , we can split γ into a family of subdistribution {γE }E∈Supp(μ) , such that every E ∈ Supp(μ) is matched by one of these distributions, and μ(E)ΔE ≈◦ γE holds. In turn, we can split μ(E)ΔE into a family μE F F ∈Supp(γE ) accordingly, such that each state F ∈ Supp(γE ) is matched by the subdistribution μE F , satisfying ◦ E μE F ≈ γE (F )ΔF .

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