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Concurrent Systems/notes/9 - Consensus.md

@@ -199,4 +199,26 @@ why wait forever? Because we want to prove that it is wait free, but not because
 - every $p_i$ that participates to the consensus reads the other proposals after that it has written `PROP[i]`
 	- all participants starts their for loops after $p_x$ has written its proposal
 - every $p_i$ that participates to the consensus finds `PROP[x] != ⊥` in their for loop
-- `BC[x]` o
+- `BC[x]` only receives proposals equal to 1
+- because of validity of binary consensus, `BC[x]` returns 1
+- every $p_i$ that participates to the consensus receives 1 at most in its x-th iteration of the for
+- let y (<= x) be the first index such that `BC[y]` returns 1
+	- `BC[z] = 0` for all $z < y$
+- since all participating process invoke the binary consensus in the same order, they all decide the value proposed by $p_y$ and terminate.
+
+
+##### Multivalued consensus (with bounded values)
+Let k be the number of possible proposals and $h = \lceil \log k \rceil$ be the number of bits needed to binary represent them (this value is known to all processes).
+IDEA: decide bit by bit the final outcome
+
+```
+PROP[1..n][1..h] array of n h-bits proposals, all init at ⊥ BC[1..h] array of h binary consensus objects
+
+bmv_propose(i,v) :=
+	PROP[i] <- binary_repr_h(v)
+	for k=1 to h do
+		P <- {PROP[j][k] | PROP[j]≠⊥ ∧ PROP[j][1..k-1]=res[1..k-1]}
+		let b be an element of P
+		res[k] <- BC[k].propose(b)
+	return value(res)
+```