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Concurrent Systems/notes/8 - Enhancing Liveness Properties.md
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@ -57,7 +57,7 @@ It can be proved that there exists no wait-free implementation of $\Omega$ in an
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 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). 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 $\delta$. Then, there are at least $t-f$ correct processes different from $p_L$ with a timer s.t. $\exists$ time $\tau_{2}$ for each time interval $\delta$, if their timer is set to $\delta$ after $\tau_{2}$, it expires at least after $\delta$.
REMARK: $\tau_{1}, \tau_{2}, \nabla$ and $p_L$ are all unknown. REMARK: $\tau_{1}, \tau_{2}, \nabla$ and $p_L$ are all unknown.