From 5e8a3d05f1f2e3ea533ef0e73272c55d28b0609a Mon Sep 17 00:00:00 2001 From: Marco Realacci Date: Mon, 5 May 2025 09:07:04 +0200 Subject: [PATCH] vault backup: 2025-05-05 09:07:04 --- .obsidian/workspace.json | 1 + .../notes/15 - A formal language for LTSs.md | 12 +++++++++++- Pasted image 20250505090603.png | Bin 0 -> 390190 bytes 3 files changed, 12 insertions(+), 1 deletion(-) create mode 100644 Pasted image 20250505090603.png diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json index 8169318..a02ef31 100644 --- a/.obsidian/workspace.json +++ b/.obsidian/workspace.json @@ -213,6 +213,7 @@ }, "active": "cdcc59f1bf6d4ae1", "lastOpenFiles": [ + "Pasted image 20250505090603.png", "Concurrent Systems/slides/class 15.pdf", "Concurrent Systems/notes/15 - A formal language for LTSs.md", "Pasted image 20250505085454.png", 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 047c942..bac9c0c 100644 --- a/Concurrent Systems/notes/15 - A formal language for LTSs.md +++ b/Concurrent Systems/notes/15 - A formal language for LTSs.md @@ -34,4 +34,14 @@ hence, P |= ☐aFF holds true only if P cannot perform any action a ![](../../Pasted%20image%2020250505085454.png) Let us now define the set of formulae satisfied by a process as $$L(P) = \{ \phi \in Form:P \models \phi \}$$ -To simplify the proof, let us modify the set of formulae \ No newline at end of file +To simplify the proof, let us modify the set of formulae by allowing conjunctions over a numerable set of formulae + +##### Theorem $$P \sim Q \space iif \space L(P)=L(Q)$$ +(=>) By induction on the syntax tree of the formula, let's show that $$φ∈L(P) \space iff \space φ∈L(Q)$$ +*Base case:* The only possible case is with φ = TT. +- by the satisfiability relation, $\phi$ belongs to the set of formulae of every process +- so also to L(P) and L(Q) + +*Inductive step:* Let's assume the thesis for every tree of height at most h. Let h+1 be the height of $\phi$. 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