vault backup: 2025-03-25 08:58:01

Concurrent Systems/notes/8 - Enhancing Liveness Properties.md

@@ -38,4 +38,12 @@ Let Q be the set of proc.'s that performed these invocations.
 - since crashed 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:** 
+**Observation:** $Q \subseteq X$ (it is possible that $p_j$ sets `NEED_HELP[j]` and then fails)
+
+By definition of $\Omega_{X}, \exists \tau'' \geq t'$ s.t. all proc.'s in Q have the same `ev_leader(X)`
+- the leader belongs to Q, since it cannot be failed
+- this is the only process allowed to proceed
+- because run in isolation, it eventually terminates (because of obstruction freedom)
+
+#### On implementing $\Omega$
+It can be proved that there exists no wait-free implementation of $\Omega$ in an asynchronous