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

@@ -46,4 +46,10 @@ By definition of $\Omega_{X}, \exists \tau'' \geq t'$ s.t. all proc.'s in Q have
 - 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 
+It can be proved that there exists no wait-free implementation of $\Omega$ in an asynchronous system with atomic R/W registers and any number of crashes
+- crashes are indistinguishable from long delays
+- need of timing constraints
+
+1. $\exists$ time $\tau_{1}$, time interval $\nabla$ and correct process $p_{L}$ s.t. after $\tau_{1}$ every two consecutive writes to a specific SWMR atomic R/W by $p_{L}$ are at most $\nabla$ time units apart one from the other
+2. let t be an upper bound on the number of possible failing processes and f the real number of process failed (hence, $0\leq f\leq t\leq n-1$, with f unknown and t known in advance).
+   Then, there are at least $t-f$ correct processes different from $p_L$ with a timer s.t. $\exists$ time $\tau_{2} \forall$ time interval $\delta$, if their timer is set to $\delta$ after $\tau_{2}$, it expires at least after $\delta$