From 2951dca7369d7566669f5ef629fd39edb1d8ebdb Mon Sep 17 00:00:00 2001
From: Marco Realacci <marco@marcorealacci.me>
Date: Mon, 5 May 2025 10:04:15 +0200
Subject: [PATCH] vault backup: 2025-05-05 10:04:15

---
 .obsidian/workspace.json                                    | 5 ++---
 Concurrent Systems/notes/15 - A formal language for LTSs.md | 6 ++++++
 2 files changed, 8 insertions(+), 3 deletions(-)

diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index ce7246a..28185d9 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -213,10 +213,10 @@
   },
   "active": "cdcc59f1bf6d4ae1",
   "lastOpenFiles": [
-    "Pasted image 20250505095743.png",
-    "Pasted image 20250505095249.png",
     "Concurrent Systems/slides/class 15.pdf",
     "Concurrent Systems/notes/15 - A formal language for LTSs.md",
+    "Pasted image 20250505095743.png",
+    "Pasted image 20250505095249.png",
     "Pasted image 20250505094841.png",
     "Pasted image 20250505090959.png",
     "Pasted image 20250505090603.png",
@@ -234,7 +234,6 @@
     "Concurrent Systems/notes/8 - Enhancing Liveness Properties.md",
     "Concurrent Systems/notes/11 - LTSs and Bisimulation.md",
     "Concurrent Systems/notes/13 - Weak Bisimilarity.md",
-    "Pasted image 20250430192526.png",
     "Concurrent Systems/slides/class 13.pdf",
     "Concurrent Systems/slides/class 14.pdf",
     "Concurrent Systems/slides/class 12.pdf",
diff --git a/Concurrent Systems/notes/15 - A formal language for LTSs.md b/Concurrent Systems/notes/15 - A formal language for LTSs.md
index 3a1380c..6d31ab0 100644
--- a/Concurrent Systems/notes/15 - A formal language for LTSs.md	
+++ b/Concurrent Systems/notes/15 - A formal language for LTSs.md	
@@ -90,3 +90,9 @@ indeed, P1 can simulate P2 and viceversa, but the simulations are not bisimulati
 
 ##### A negation and conjunction-free logic
 ![](../../Pasted%20image%2020250505095743.png)
+
+### A Logic for Weak Bisimilarity
+Very similar. $$\phi := TT | \lnot \phi | \phi \land \phi|\ll\gg a \phi \quad where \space a \in Action$$
+The only difference is the diamond operator, that here is turned into <<>>a.
+
+This is very similar to the difference between the definition of weak and strong simulation, where the former is obtained from the latter by replacing $\xrightarrow{a}$ with $\xRightarrow{\hat{a}}$. Satisfiability of this new operator is expectable: $$P \models \ll\gg a\phi \iff \exists P' : P \xRightarrow{P'} \land P' \models \phi$$