vault backup: 2025-03-31 10:25:22

.obsidian/workspace.json

@@ -6,7 +6,6 @@
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@@ -14,31 +13,12 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "Concurrent Systems/notes/9 - Consensus.md",
+                "file": "Concurrent Systems/notes/2 - Fast mutex by Lamport.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "9 - Consensus"
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-                "file": "Concurrent Systems/slides/class 9.pdf"
-              },
-              "icon": "lucide-file-text",
-              "title": "class 9"
+              "title": "2 - Fast mutex by Lamport"
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@@ -213,8 +193,8 @@
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-    "Concurrent Systems/slides/class 9.pdf",
     "Concurrent Systems/notes/9 - Consensus.md",
+    "Concurrent Systems/slides/class 9.pdf",
     "Concurrent Systems/notes/8 - Enhancing Liveness Properties.md",
     "Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md",
     "Concurrent Systems/notes/4 - Semaphores.md",

Concurrent Systems/notes/9 - Consensus.md

@@ -228,4 +228,9 @@ bmv_propose(i,v) :=
 - Wait freedom: trivial
 - Integrity: trivial
 - Agreement: by agreement of the h binary consensus objects
-- Validity: for all k, we prove that, if dec is the decided value, then there exists a j such that $p_j$ is participating
+- Validity: for all k, we prove that, if dec is the decided value, then there exists a j such that $p_j$ is participating (`PROP[j] ≠ ⊥`) and `dec[1...k] = PROP[j][1...k]`
+	- by construction, P contains the k-th bits of the proposals whose first (k-1) bits coincide with the first k-1 bits decides so far:
+		- for every  b ∈ P, there exists a j such that `PROP[j] ≠ ⊥`,
+		- `PROP[j][1..k-1] = dec[1..k-1]` and `PROP[j][k] = b`
+	-  whatever b ∈ P is selected in the k-th binary consensus, there exists a j such that `PROP[j] ≠ ⊥` and `PROP[j][1..k] = dec[1..k]`
+	- by taking `k = h`, we can conclude.