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ISBN-10: 0931011442

ISBN-13: 9780931011443

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Definition 3. n uj i∈I z@y uj ρ vj ρ Ji Pi | R → (z @ y) Pk ρ | R and closed under ≡, | and (z @ z ) . 38 C. Laneve and L. Padovani The last premise of the reduction rule requires that the interacting processes are underneath a sequence of news that is long enough. For example, the process P = x u | x(u @ v) Q is inactive. On the contrary (u @ v)P reduces to Q. We refer to the introduction for few sample processes in our calculus. The semantics is completed by the notion of barbed congruence [12].

The proof proceeds by case analysis of the different types of µ and then by induction on the derivation of the β-move. Proposition 1. Suppose Γ N |==⇒β Γ M . Then Γ N ≈Γ M. Proof. We prove the above statement by defining R = {Γ N, Γ M|Γ N |==⇒β Γ M } and showing that R is a bisimulation, which follows as a consequence of Lemma 1. Definition 7 (Bisimulation up-to β-moves). Bisimulation up-to β-moves, denoted as ≈β , is the largest typed relation between configurations such that Γ1 M1 ≈β Γ2 M2 and – Γ1 M1 ◦ ≈ Γ2 – Γ2 M 2 ◦ ≈ Γ1 µ µ ˆ −→ Γ1 M1 implies Γ2 M2 =⇒ Γ2 M2 such that Γ1 M1 Al ◦ ≈β M2 µ µ ˆ −→ Γ2 M2 implies Γ1 M1 =⇒ Γ1 M1 such that Γ2 M2 Al ◦ ≈β M1 where Al is the relation |==⇒β ◦ ≡.

Thus sPassive is not 1-dynamic fault tolerant; k1 formally one can show Γe FD (sPassive) ∼ = Γe sPassive. 3 . (V ) or bound output, can be an internal action, τ , a bound input, (˜ n:T ˜ (˜ n : T)l : a! V . These actions are defined by transition rules given in Table 5, inspired by [7, 6, 5]. In accordance with Definition 2 (observable barbs) and Definitions 1 (valid observers), (l-in) and (l-out) restrict external communication to public channels at public locations (Γ obs l, a). Furthermore, in (l-in) we require that the type of the values inputted, V , match the object type of channel a; since a is public and configurations are well-typed, this also implies that V are public values defined in Γ .

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Calculus on the HP-48G by Grapevine Publications

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