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Marco Realacci 2025-03-25 15:42:22 +01:00
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Concurrent Systems/notes/8 - Enhancing Liveness Properties.md
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@ -36,8 +36,8 @@ stop_help(i) :=
*Proof:*
By contr., $\exists \tau$ s.t. $\exists$ many (> 0) op.'s invoked concurrently that never terminate.
Let Q be the set of proc.'s that performed these invocations.
- by enrichment, eventually `NEED_HELP[i]=T` ($\forall i\in Q$) forever
- since crashed are fail-stop, eventually `NEED_HELP[j]` is no longer modified ($\forall j \not \in Q$)
- by enrichment, eventually `NEED_HELP[i]=T` forever ($\forall i\in Q$)
- since crashes are fail-stop, eventually `NEED_HELP[j]` is no longer modified ($\forall j \not \in Q$)
- $\exists \tau' \geq \tau$ where all proc.'s in Q compute the same X
**Observation:** $Q \subseteq X$ (it is possible that $p_j$ sets `NEED_HELP[j]` and then fails)