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Marco Realacci 2025-04-30 19:18:22 +02:00
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.obsidian/workspace.json vendored
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@ -194,6 +194,7 @@
"lastOpenFiles": [ "lastOpenFiles": [
"Concurrent Systems/notes/13 - Weak Bisimilarity.md", "Concurrent Systems/notes/13 - Weak Bisimilarity.md",
"Concurrent Systems/notes/14 Checking bisimilarity, an inference system.md", "Concurrent Systems/notes/14 Checking bisimilarity, an inference system.md",
"Concurrent Systems/notes/12b - CCS cose varie.md",
"Concurrent Systems/notes/12 - Calculus of communicating system.md", "Concurrent Systems/notes/12 - Calculus of communicating system.md",
"Concurrent Systems/notes/11 - LTSs and Bisimulation.md", "Concurrent Systems/notes/11 - LTSs and Bisimulation.md",
"Concurrent Systems/notes/10 - Implementing Consensus.md", "Concurrent Systems/notes/10 - Implementing Consensus.md",
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"Concurrent Systems/slides/class 13.pdf", "Concurrent Systems/slides/class 13.pdf",
"Concurrent Systems/notes/12b - CCS cose varie.md",
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"Concurrent Systems/slides/class 14.pdf", "Concurrent Systems/slides/class 14.pdf",
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Concurrent Systems/notes/14 Checking bisimilarity, an inference system.md
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@ -29,12 +29,15 @@ basically we can let the left or the right process evolve, leaving the other unc
#### Soundness theorem: $\vdash P=Q \implies P \sim Q$ #### Soundness theorem: $\vdash P=Q \implies P \sim Q$
*Proof:* *Proof:*
If $\vdash LHS=RHS$e need to consider the relation $\{ LHS,RHS \}$ If $\vdash LHS=RHS$, we need to consider the relation $\{ LHS,RHS \} \cup Id$ and prove it's a bisimulation (spoiler: it is).
![](images/Pasted%20image%2020250429083535.png) Since bisimilarity is an equivalence and a congruence, the inference rules holds.
>[!def] Standard form
$P$ is in standard form if and only if $P \triangleq \sum_{i}\alpha_{i}P_{i}$ and $\forall_{i}P_{i}$ is in standard form. $P$ is in standard form if and only if $P \triangleq \sum_{i}\alpha_{i}P_{i}$ and $\forall_{i}P_{i}$ is in standard form.
**Lemma:** $\forall P \exists P'$* in standard form such that $\vdash P = P'$ **Lemma:** $\forall P \space \exists \space P'$ in standard form such that $\vdash P = P'$
*Proof:* by induction on the structure of P. *Proof:* by induction on the structure of P.
**Base case:** $P \triangleq 0$. It suffices to consider $P' \triangleq 0$ and conclude reflexivity. **Base case:** $P \triangleq 0$. It suffices to consider $P' \triangleq 0$ and conclude reflexivity.