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This context can be regarded as an environment in which the behaviours of p and q differ. Following this programme, a suitable semantic equivalence on processes is defined in terms of two requirements. First of all the behaviour of processes is compared in the neutral environment. This entails isolating a class C of properties ϕ of processes that are deemed relevant in a given range of applications. One then requires for p ≡ q to hold that p and q have the same properties from this class: p ≡ q ⇒ ∀ϕ ∈ C.
The Coarsest Precongruences Respecting Safety and Liveness Properties 43 ⊥ Proof: Let ⊥ FDI be the preorder defined by: p FDI q iff the right-hand side of Theorem 7 holds. “⇐”: It suffices to establish that ⊥ FDI is a liveness respecting precongruence. ∗ ⊥ To show that ⊥ FDI respects liveness, let G ⊆ A , p FDI q, and suppose p |= liveness(G). I need to show that q |= liveness(G). So suppose σ ∈ CT (q). Then either σ ∈ diverg (g) ⊆ diverg⊥ (g) ⊆ diverg⊥ (p) or σ ∈ inf (q) ⊆ inf⊥ (q) ⊆ inf⊥ (p) or σ, A ∈ failures(q) ⊆ failures⊥ (q) ⊆ failures⊥ (q).
We assume that there are no free variables in a network (while there can be free channels). , each node identifier is unique and the corresponding transmission A Calculus for Power-Aware Multicast Communications in Ad Hoc Networks 23 Table 2. Structural Congruence n[[v = v]P, Q]μl,r ≡ n[P ]μl,r n[[v1 = v2 ]P, Q]μl,r ≡ n[Q]μl,r v1 = v2 (Struct Then) (Struct Else) v/˜ x}P ]μl,r if A(˜ x) = P ∧ |˜ x| = |˜ v | (Struct n[A v˜ ]μl,r ≡ n[{˜ M |N ≡ N |M (Struct (M |N )|M ≡ M |(N |M ) (Struct M |0 ≡ M (Struct (νc)0 ≡ 0 (Struct (νc)(νd)M ≡ (νd)(νc)M (Struct (νc)(M |N ) ≡ M |(νc)N If c ∈ / f c(M ) (Struct M ≡M (Struct N ≡ M if M ≡ N (Struct M ≡M if M ≡ M ∧ M ≡ M (Struct M |M ≡ N |M ∀M if M ≡ N (Struct (νc)M ≡ (νc)N ∀c if M ≡ N (Struct def Rec) Par Comm) Par Assoc) Zero Par) Zero Res) Res Res) Res Par) Refl) Symm) Trans) Cxt Par) Cxt Res) radius is compatible with the node’s power capacity.