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Amaury JOLY
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We consider a set of processes communicating asynchronously over reliable point-to-point channels. Each process maintains the following shared variables:
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\begin{itemize}
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\item \textbf{received}: the set of messages received (but not yet delivered).
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\item \textbf{delivered}: the set of messages that have been received, ordered, and delivered.
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\item \textbf{prop[$r$][$j$]}: the proposal set of process $j$ at round $r$. It contains the set of messages that process $j$ claims to have received but not yet delivered at round $r$, concatenated with its newly broadcast message.
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\item \textbf{proves}: the current content of the \texttt{DenyList} registry, accessible via the operation \texttt{READ()}. It returns a list of tuples $(j, \texttt{PROVE}(r))$, each indicating that process $j$ has issued a valid \texttt{PROVE} for round $r$.
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\item \textbf{winner$^r$}: the set of processes that have issued a valid \texttt{PROVE} operation for round $r$.
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\item \textbf{RB-cast}: a reliable broadcast primitive that satisfies the properties defined in Section~1.1.2.
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\item \textbf{APPEND$(r)$}, \textbf{PROVE$(r)$}: operations that respectively insert (APPEND) and attest (PROVE) the participation of a process in round $r$ in the DenyList registry.
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\item \textbf{READ()}: retrieves the current local view of valid operations (APPENDs and PROVEs) from the DenyList.
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\item \textbf{ordered$(S)$}: returns a deterministic total order over a set $S$ of messages (e.g., via hash or lexicographic order).
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\end{itemize}
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\resetalgline
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\begin{algorithm}
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\vspace{1em}
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\textbf{RB-received$(m, S, r_0, j_0)$}
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\begin{algorithmic}[1]
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\State \nextalgline $\textit{received} \gets \textit{received} \cup \{m\}$
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\State \nextalgline $\textit{prop}[r_0][j_0] \gets S$
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\end{algorithmic}
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\vspace{1em}
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\textbf{AB-broadcast$(m, j_0)$}
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\begin{algorithmic}[1]
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $r_0 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} + 1$
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\State \nextalgline $\texttt{RB-cast}(m, (\textit{received} \setminus \textit{delivered}) \cup \{m\}, r_0, j_0)$
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\State \nextalgline \texttt{PROVE}$(r_0)$
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\State \nextalgline \texttt{APPEND}$(r_0)$
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\Repeat
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $r_1 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} - 1$
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\State \nextalgline $\textit{winner}^{r_1} \gets \{j : (j, \texttt{PROVE}(r_1)) \in \textit{proves}\}$
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\State \nextalgline \textbf{wait} $\forall j \in \textit{winner}^{r_1},\ \textit{prop}[r_1][j] \neq \bot$
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\Until{\nextalgline $\exists r_2,\ \exists j_2 \in \textit{winner}^{r_2},\ m \in \textit{prop}[r_2][j_2]$} \nextalgline
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\end{algorithmic}
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\vspace{1em}
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\textbf{AB-listen}
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\begin{algorithmic}[1]
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\While{true}
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $r_1 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} - 1$
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\For{$r_2 \in [r_0, \dots, r_1]$} \nextalgline
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\State \nextalgline \texttt{APPEND}$(r_2)$
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\State \nextalgline $\textit{proves} \gets \texttt{READ}()$
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\State \nextalgline $\textit{winner}^{r_2} \gets \{j : (i, \texttt{PROVE}(r_2)) \in \textit{proves}\}$
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\State \nextalgline \textbf{wait} $\forall j \in \textit{winner}^{r_2},\ \textit{prop}[r_2][j] \neq \bot$
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\State \nextalgline $M^{r_2} \gets \bigcup_{j \in \textit{winner}^{r_2}} \textit{prop}[r_2][j]$
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\ForAll{$m \in \texttt{ordered}(M^{r_2})$} \nextalgline
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\State \nextalgline $\textit{delivered} \gets \textit{delivered} \cup \{m\}$
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\State \nextalgline \texttt{AB-deliver}$(m)$
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\EndFor
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\EndFor
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\EndWhile
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\end{algorithmic}
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\end{algorithm}
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\subsubsection{Model Properties}
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The system consists of \textit{n} asynchronous processes communicating via reliable point-to-point message passing. \\
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Each process has a unique, unforgeable identifier and knows the identifiers of all other processes. \\
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Up to $f<n$ processes may crash (fail-stop). \\
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The network is reliable: if a correct process sends a message to another correct process, it is eventually delivered. \\
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Messages are uniquely identifiable: two messages sent by distinct processes or at different rounds are distinguishable \\
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2 messages sent by the same processus in two differents rounds are differents \\
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\begin{property}[Message Uniqueness]
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If two messages are sent by different processes, or by the same process in different rounds, then the messages are distinct. \\
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Formally : \\
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\[
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\forall p_1, p_2,\ \forall r_1, r_2,\ \forall m_1, m_2,\
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\left(
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\begin{array}{l}
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\text{send}(p_1, r_1, m_1) \land \text{send}(p_2, r_2, m_2) \\
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\land\ (p_1 \ne p_2 \lor r_1 \ne r_2)
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\end{array}
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\right)
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\Rightarrow m_1 \ne m_2
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\]
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\end{property}
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\subsubsection{Reliable Broadcast Properties}
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\begin{property}{Integrity}
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Every message received was previously sent. \\
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Formally : \\
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$\forall p_i : \text{bc-recv}_i(m) \Rightarrow \exists p_j : \text{bc-send}_j(m)$
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\end{property}
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\begin{property}{No Duplicates}
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No message is received more than once at any single processor. \\
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Formally : \\
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$\forall m, \forall p_i: \text{bc-recv}_i(m) \text{ occurs at most once}$ \\
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\end{property}
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\begin{property}{Validity}
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All messages broadcast by a correct process are eventually received by all non faulty processors. \\
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Formally : \\
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$\forall m, \forall p_i: \text{correct}(p_i) \wedge \text{bc-send}_i(m) => \forall p_j : \text{correct}(p_j) \Rightarrow \text{bc-recv}_j(m)$
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\end{property}
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\subsubsection{AtomicBroadcast Properties}
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\begin{property}{AB Totally ordered}
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$\forall m_1, m_2, \forall p_i, p_j : \text{ab-recv}_{p_i}(m_1) < \text{ab-recv}_{p_i}(m_2) \Rightarrow \text{ab-recv}_{p_j}(m_1) < \text{ab-recv}_{p_j}(m_2)$
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\end{property}
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\subsubsection{DenyList Properties}
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Let $\Pi_M$ be the set of processes authorized to issue \texttt{APPEND} operations,
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and $\Pi_V$ the set of processes authorized to issue \texttt{PROVE} operations. \\
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Let $S$ be the set of valid values that may be appended. Let $\texttt{Seq}$ be
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the linearization of operations recorded in the DenyList.
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\begin{property}{APPEND Validity}
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An operation $\texttt{APPEND}(x)$ is valid iff :
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the issuing process $p \in \Pi_M$, and the value $x \in S$
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\end{property}
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\begin{property}{PROVE Validity}
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An operation $\texttt{PROVE}(x)$ is valid iff:
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the issuing process $p \in \Pi_V$, and there exists no $\texttt{APPEND}(x)$ that appears earlier in $\texttt{Seq}$.
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\end{property}
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\begin{property}{PROGRESS}
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If an APPEND(x) is invoked by a correct process, then all correct processes will eventually be unable to PROVE(x).
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\end{property}
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\begin{property}{READ Validity}
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READ() return a list of tuples who is a random permutation of all valids PROVE() associated to the identity of the emiter process.
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\end{property}
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\author{JOLY Amaury \\ \textbf{Encadrants :} GODARD Emmanuel, TRAVERS Corentin}
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\begin{theorem}[Integrity]
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Let $j$ be a process such that both $\text{AB-deliver}_j(m_0)$ and $\text{AB-deliver}_j(m_1)$ occur, with $m_0 = m_1$.
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\begin{align*}
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&\text{AB-deliver}_j(m_0) \wedge \text{AB-deliver}_j(m_1) & \text{(line 24)} \\
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\Rightarrow\; & m_0, m_1 \in \textit{delivered}_j & \text{(line 23)} \\
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\Rightarrow\; &\exists r_0, r_1 : m_0 \in M^{r_0} \wedge m_1 \in M^{r_1} & \text{(line 22)} \\
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\Rightarrow\; &\exists j_0, j_1 : m_0 \in \textit{prop}[r_0][j_0] \wedge m_1 \in \textit{prop}[r_1][j_1] \\
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&\hspace{2.5em} \wedge\ j_0 \in \textit{winner}^{r_0},\ j_1 \in \textit{winner}^{r_1} & \text{(line 21)}
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\end{align*}
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We now distinguish two cases:
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\vspace{0.5em}
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\noindent\textbf{Case 1:} $r_0 = r_1$:
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\begin{itemize}
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\item If $j_0 \neq j_1$: this contradicts message uniqueness, since two different processes would include the same message in round $r_0$.
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\item If $j_0 = j_1$:
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\begin{align*}
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\Rightarrow & |{(j_0, \texttt{PROVE}(r_0)) \in proves}| \geq 2 & \text{(line 19)}\\
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\Rightarrow &\texttt{PROVE}_{j_0}(r_0) \text{ occurs 2 times} & \text{(line 6)}\\
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\Rightarrow &\texttt{AB-Broadcast}_{j_0}(m_0) \text{ were invoked two times} \\
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\Rightarrow &(max\{r: \exists j, (j, \texttt{PROVE}(r)) \in proves\} + 1) & \text{(line 4)}\\
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&\text{ returned the same value in two differents invokations of \texttt{AB-Broadcast}} \\
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&\textbf{But } \texttt{PROVE}(r_0) \Rightarrow \texttt{max}\{r: \exists j, (j, \texttt{PROVE}(r)) \in proves\} + 1 > r_0 \\
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&\text{It's impossible for a single process to submit two messages in the same round} \hspace{1em} \\
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\end{align*}
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\end{itemize}
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% \vspace{0.5em}
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|
||||||
\noindent\textbf{Case 2:} $r_0 \ne r_1$:
|
|
||||||
\begin{itemize}
|
|
||||||
\item If $j_0 \neq j_1$: again, message uniqueness prohibits two different processes from broadcasting the same message in different rounds.
|
|
||||||
\item If $j_0 = j_1$: message uniqueness also prohibits the same process from broadcasting the same message in two different rounds.
|
|
||||||
\end{itemize}
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|
||||||
|
|
||||||
In all cases, we reach a contradiction. Therefore, it is impossible for a process to deliver the same message more than once. $\square$
|
|
||||||
\end{proof}
|
|
||||||
|
|
||||||
% \subsubsection{No Duplication}
|
|
||||||
|
|
||||||
% $M = (\bigcup_{i \rightarrow |P|} AB\_receieved_{i}(m)), \not\exists m_0 m_1 \in M \text{s.t. } m_1 = m_2$
|
|
||||||
% \\
|
|
||||||
% Proof \\
|
|
||||||
% \begin{align*}
|
|
||||||
% &\text{Soit } i, m_0, m_1 \text{ tels que } m_0 = m_1 \\
|
|
||||||
% &\exists r_0,\ m_0 \in M^{r_0} \\
|
|
||||||
% &\exists r_1,\ m_1 \in M^{r_1} \\
|
|
||||||
% &\text{if } r_0 = r_1 \\
|
|
||||||
% &\quad \exists j_0, j_1,\ \text{prop tq } \text{prop}[r_0][j_0] = m_0,\ \text{prop}[r_0][j_1] = m_1 \quad j_0, j_1 \in \text{winnner}^{r_0} \\
|
|
||||||
% &\quad \text{if} j_0 \neq j_1 \\
|
|
||||||
% &\quad\quad \text{On admet qu'il est impossible pour un processus de soumettre le même msg qu'un autre} \\
|
|
||||||
% &\quad \text{if } j_0 = j_1 \\
|
|
||||||
% &\quad\quad j_0 \text{ a émis son } \text{PROVE}(r_0) \text{ valide 2 fois}\\
|
|
||||||
% &\quad\quad \text{Impossible si } j_0 \text{ correct} \\
|
|
||||||
% &\text{else} \\
|
|
||||||
% &\quad \exists j_0, j_1,\ \text{prop tq } \text{prop}[r_0][j_0] = m_0,\ \text{prop}[r_0][j_1] = m_1 \quad j_0, j_1 \in \text{winnner}^{r_0} \\
|
|
||||||
% &\quad \text{if } j_0 \neq j_1 \\
|
|
||||||
% &\quad\quad \text{On admet qu'il est impossible pour un processus de soumettre le même msg qu'un autre} \\
|
|
||||||
% &\quad \text{if } j_0 = j_1 \\
|
|
||||||
% &j_0 \text{ à emis et validé 2 fois le même messages a des rounds différents.}\\
|
|
||||||
% &\text{On admet que deux message identiques soumis a des rounds différents ne peuvent être identiques}
|
|
||||||
% \end{align*}
|
|
||||||
|
|
||||||
\subsubsection{Broadcast Validity}
|
|
||||||
$\exists j_0, m_0 \quad AB\_broadcast_{j_0}(m_0) \Rightarrow \forall j_1 \quad AB\_received_{j_1}(m_0)$ \\
|
|
||||||
|
|
||||||
Proof:
|
|
||||||
\begin{align*}
|
|
||||||
&\exists j_0, m_0 \quad AB\_broadcast_{j_0}(m_0) \\
|
|
||||||
&\forall j_1, \exists r_1 \quad RB\_deliver_{j_1}^{r_1}(m_0) \\
|
|
||||||
&\exists receieved : m_0 \in receieved_{j_1} \\
|
|
||||||
&\exists r_0 : RB\_deliver(PROP, r_0, m_0) & LOOP\\
|
|
||||||
&\exists prop: \text{prop}[r_0][j_0] = m_0 \\
|
|
||||||
&\text{if } \not\exists (j_0, PROVE(r_0)) \in \text{proves} \\
|
|
||||||
&\quad r_0 += 1 \\
|
|
||||||
&\quad \text{jump to LOOP} \\
|
|
||||||
&\text{else} \\
|
|
||||||
&\quad \exists \text{winner}, \text{winner}^{r_0} \ni j_0 \\
|
|
||||||
&\quad \exists M^{r_0} \ni (\text{prop}[r_0][j_0] = m_0) \\
|
|
||||||
&\quad \forall j_1, \quad AB\_deliver_{j_1}(m_0)
|
|
||||||
\end{align*}
|
|
||||||
|
|
||||||
\subsubsection*{AB receive width}
|
|
||||||
\[
|
|
||||||
\exists j_0, m_0 \quad AB\_deliver_{j_0}(m_0) \Rightarrow \forall j_1\ AB\_deliver_{j_1}
|
|
||||||
\]
|
|
||||||
|
|
||||||
Proof:
|
|
||||||
\begin{align*}
|
|
||||||
&\forall j_0, m_0\ AB\_deliver_{j_0}(m_0) \Rightarrow \exists j_1 \text{ correct }, AB\_broadcast(m_0) \\
|
|
||||||
&\exists j_0, m_0 \quad AB\_broadcast_{j_0}(m_0) \Rightarrow \forall j_1,\ AB\_deliver_{j_1}(m_0)
|
|
||||||
\end{align*}
|
|
||||||
Loading…
Reference in New Issue
Block a user