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

@@ -75,4 +75,13 @@ Guessing the time duration for suspecting a process:
 #### From obstruction-freedom to wait-freedom
 **Eventually perfect:** failure detector ♢P provides each process $p_i$ a local variable $suspected_i$ such that
 1. *(Eventual completeness)* eventually, $suspended_{i}$ contains all the indexes of crashed processes, for all correct $p_i$
-2. (*Eventual accuracy*) 
+2. (*Eventual accuracy*) eventually, $suspected_{i}$ contains only indexes of crashed processes, for all correct $p_{i}$.
+
+**Definition:** A failure detector FD1 is **stronger** than a failure detector FD2 if there exists an algorithm that builds FD2 from instances of FD1 and atomic R/W registers.
+
+**Proposition:** ♢P is stronger than $\Omega_{X}$
+*Proof:*
+Forall i
+- i ∉ X $\to$ `ev_leader_i(X)` is any ID (and may change in time)
+- $i \in X \to$ `ev_leader_i(X)` $= min\left( suspected_{i} \cap X \right)$ where $Π$ denotes the set of all proc. IDs.
+