vault backup: 2025-04-28 08:42:04

.obsidian/workspace.json

@@ -213,6 +213,7 @@
   },
   "active": "56150e6df7869900",
   "lastOpenFiles": [
+    "Pasted image 20250428083727.png",
     "Concurrent Systems/slides/class 13.pdf",
     "Concurrent Systems/notes/13 - Weak Bisimilarity.md",
     "Concurrent Systems/notes/12 - Calculus of communicating system.md",

Concurrent Systems/notes/13 - Weak Bisimilarity.md

@@ -24,4 +24,18 @@ $\approx$ is a
 1. equivalence
 2. congruence
 3. weak bisimulation
-4. $\sim a$
+4. $\sim \subset \approx$
+
+#### Examples of weakly bisimilar processes
+![](../../Pasted%20image%2020250428083727.png)
+
+**Theorem:** given any process P and any sum M, N, then:
+1. $P \approx \tau.{P}$
+2. $M+N+\tau.N \approx M + \tau.N$
+3. $M+\alpha.P+\alpha.(N+\tau.P) \approx M + \alpha.(N + \tau.P)$
+
+*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. $S=\\{ (M+\alpha.P+) \}$