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Recherche/BFT-ARBover/5_ARB_with_RB_DL/index.tex

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+In this part we're using the consensus number of the GE Object to show that it's possible to implement F-ARB with RB and our defined object. Also we present an algorithm which achieve this in section 7. 
+
+\begin{theorem}[RB + Group Election implements F-ARB]\label{thm:ge-to-farb}
+  In an asynchronous message-passing system with crash failure. We can wait-free implement a FIFO-Atomic Reliable Broadcast from a Reliable Broadcast (RB) primitive and one Group Election object $\GE[r]$ per round $r \in \mathbb{N}$.
+\end{theorem}
+
+\begin{proof}
+  By Theorem~\ref{thm:ge-consensus}, the Group Election object type with verifier set $\Pi_V$ has consensus number $|\Pi_V|$. In particular, when $\Pi_V = \Pi$, using one instance $\GE[r]$ per round $r$ we can implement wait-free consensus among all processes in~$\Pi$.
+
+  It is well known that, in a crash-prone asynchronous message-passing system, consensus and atomic (total order) broadcast are equivalent (defago et al): given consensus, one can implement atomic broadcast by using an infinite sequence of consensus instances to decide the sequence of messages to deliver, and conversely atomic broadcast can be used to implement consensus by deciding a single value in the first
+  position of the total order.
+
+  In our setting, we already have a Reliable Broadcast (RB) primitive, which provides RB-Validity, RB-Agreement, and RB-Integrity for the dissemination of messages. Using the consensus power provided by the Group Election objects, we can therefore apply the standard reduction from consensus to atomic broadcast: each position (or \emph{slot}) in the global delivery sequence is chosen by a consensus instance, whose proposals are messages that have been RB-delivered but not yet ordered. This yields an atomic
+  reliable broadcast (ARB) primitive.
+
+  To obtain FIFO-Atomic Reliable Broadcast (F-ARB), it suffices to enforce per-sender FIFO order on top of ARB. This can be done in the usual way by tagging each message broadcast by a process $p_i$ with a local sequence number $s \in \mathbb{N}$, and by ensuring that only the message with the smallest pending sequence number for $p_i$ is ever proposed to a consensus instance. As a result, for every sender $p_i$, messages with tags $(p_i,s)$ and $(p_i,t)$ with $s < t$ are decided (and thus delivered) in this order at all processes.
+
+  Hence, RB plus Group Election objects is sufficient to implement FIFO-Atomic Reliable Broadcast.
+\end{proof}