vault backup: 2025-03-10 09:40:11

Concurrent Systems/notes/3.md

@@ -154,4 +154,13 @@ Without it, if two process $i$ and $j$ sets MY_TURN at the same time, then $i$ g
 
 *Proof:*
 - let t be the value of `MY_TURN[i]` after $p_{i}$ exits the doorway (after flag <- down)
-- when $p_j$ computes its ticket, it reads $t$ from `MY_TURN[i]`, it reads t from MY_TURNpi
+- when $p_j$ computes its ticket, it reads $t$ from `MY_TURN[i]`, it reads t from `MY_TURN[i]` (no write overlapping with this read)
+- Hence, `MY_TURN[j]` is at least $t+1$.
+
+**Lemma 2:** Let $p_i$ be in the CS and $p_j$ is in the doorway or in the bakery; then, $$⟨MY\_TURN[i] , i⟩ < ⟨MY\_TURN[j] , j⟩$$*Proof:*
+If $p_i$ is in the CS, it has terminated its first wait for $j$
+	let's consider the read of `FLAG[j]` done by $p_i$ that terminates such wait
+	
+W.r.t. the execution of $p_j$, it can be that
+- this read overlaps with `FLAG[j] <- up` by Lemma 1, `MY_TURN[i] < MY_TURN[j]` and ok
+- this read is contained within the computation of `MY_TURN[j]`, but it's not possible, since `MY_TURN` is computed with the