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Additional info for CONCUR 2010 - Concurrency Theory: 21th International Conference, CONCUR 2010, Paris, France, August 31-September 3, 2010. Proceedings

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Fk } and Fi E for each Fi , and furthermore, ∀C ∈ S/ : γ s (C) = (μ−E)(C). Hence Condiα tion (i) is satisfied. Whenever E −→ μ , following Condition (ii), then for each Fi we α immediately deduce from E Fi that Fi =⇒C γFi and ∀C ∈ S/ : μ (C) = γFi (C). k γFi . It is then straightforward to show that in total γ g =⇒C ρ and that ∀C ∈ S/ : μ(E) · μ (C) = ρ(C). By the choice of R this immediately implies (μ(E)μ , ρ) ∈ R, which suffices to establish Condition (ii). ✷ Just like , most existing weak relations for systems with probabilistic or stochastic timed transitions can be recast as relations on distributions, which can be formulated as slight adaptations of ≈◦ and ◦ , respectively.

Concurrent processes may perform random experiments inside a transition. This is represented by transitions a μ, where s is a state, a is an action label, and μ is a probability distribuof the form s tion over states. Labelled transition systems are instances of this model family, obtained by restricting to Dirac distributions (assigning full probability to single states). Thus, foundational concepts and results of standard concurrency theory are retained in their full beauty, and extend smoothly to the model of probabilistic automata.

For σ, σ ∈ N>0 we write σ ≤ σ if there exists a (possibly empty) φ ∗ such that σφ = σ . A partial function T : N>0 → L, which satisfies ∗ – if for σ, σ ∈ N>0 : σ ≤ σ and σ ∈ dom(T ) then σ ∈ dom(T ) – if σi ∈ dom(T ) for i > 1, then also σ(i − 1) ∈ dom(T ) – ε ∈ dom(T ) is called an (infinite) L-labelled tree. Let σ ∈ dom(T ): σ is called a leaf of T if there is no σ ∈ dom(T ) such that σ < σ . The empty word ε is called the root of T . We denote the set of all leaves of T by LeafT and the set of all inner nodes by InnerT .

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