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Concurrent Systems/notes/13 - Weak Bisimilarity.md

@@ -36,6 +36,18 @@ $\approx$ is a
 
 *Proof:*
 take the symmetric closure of the following relations, that can be easily shown to be weak simulations:
-1. $S = \{ P,\tau.P \}\cup Id$
-2. $S=\{ M+N+\tau.N,M+\tau.N \}\cup Id$
-3. $S=\\{ (M+\alpha.P+) \}$
+1. $S = \{ (P,\tau.P )\}\cup Id$
+2. $S=\{ (M+N+\tau.N,M+\tau.N )\}\cup Id$
+3. $S=\{ ((M+\alpha.P+\alpha.(N+\tau.P), M+\alpha.(N+\tau.P)) \} \cup Id$
+
+#### Weak bisimilarity abstracts from any $\tau$
+![](../../Pasted%20image%2020250428084340.png)
+
+**There exists no weak bisimulation S that contains (P, Q).**
+*Proof:*
+By contr. suppose that a bisimulation exists
+Since Q −τ→ b.0, there must exist a P’ such that P ⇒ P and (P,b.0) ∈ S The only P that satisfies P ⇒ P’ is P itself
+hence it should be (P,b.0) ∈ S
+
+Contradiction: P can perform a whereas b.0 cannot !!
+Similarly, P/R and Q/R are NOT weakly bisimilar