We consider a set of processes communicating asynchronously over reliable point-to-point channels. Each process maintains the following shared variables:

\begin{itemize}
  \item \textbf{received}: the set of messages received (but not yet delivered).
  \item \textbf{delivered}: the set of messages that have been received, ordered, and delivered.
  \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.
  \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 $r$.
  \item \textbf{winner$^r$}: the set of processes that have issued a valid \texttt{PROVE} operation for round $r$.
  \item \textbf{RB-cast}: a reliable broadcast primitive that satisfies the properties defined in Section~1.1.2.
  \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.
  \item \textbf{READ()}: retrieves the current local view of valid operations (APPENDs and PROVEs) from the DenyList.
  \item \textbf{ordered$(S)$}: returns a deterministic total order over a set $S$ of messages (e.g., via hash or lexicographic order).
\end{itemize}

\resetalgline
\begin{algorithm}
  
  \vspace{1em}
  \textbf{RB-received$(m, S, r_0, j_0)$}
  \begin{algorithmic}[1]
    \State \nextalgline $\textit{received} \gets \textit{received} \cup \{m\}$
    \State \nextalgline $\textit{prop}[r_0][j_0] \gets S$
  \end{algorithmic}

  \vspace{1em}
  \textbf{AB-broadcast$(m, j_0)$}
  \begin{algorithmic}[1]
    \State \nextalgline $\textit{proves} \gets \texttt{READ}()$
    \State \nextalgline $r_0 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} + 1$
    \State \nextalgline $\texttt{RB-cast}(m, (\textit{received} \setminus \textit{delivered}) \cup \{m\}, r_0, j_0)$
    \State \nextalgline \texttt{PROVE}$(r_0)$
    \State \nextalgline \texttt{APPEND}$(r_0)$
    \Repeat
      \State \nextalgline $\textit{proves} \gets \texttt{READ}()$
      \State \nextalgline $r_1 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} - 1$
      \State \nextalgline $\textit{winner}^{r_1} \gets \{j : (j, \texttt{PROVE}(r_1)) \in \textit{proves}\}$
      \State \nextalgline \textbf{wait} $\forall j \in \textit{winner}^{r_1},\ \textit{prop}[r_1][j] \neq \bot$
    \Until{\nextalgline $\exists r_2,\ \exists j_2 \in \textit{winner}^{r_2},\ m \in \textit{prop}[r_2][j_2]$} \nextalgline
  \end{algorithmic}

  \vspace{1em}
  \textbf{AB-listen}
  \begin{algorithmic}[1]
    \While{true}
      \State \nextalgline $\textit{proves} \gets \texttt{READ}()$
      \State \nextalgline $r_1 \gets \max\{r : \exists j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} - 1$
      \For{$r_2 \in [r_0, \dots, r_1]$} \nextalgline
        \State \nextalgline \texttt{APPEND}$(r_2)$
        \State \nextalgline $\textit{proves} \gets \texttt{READ}()$
        \State \nextalgline $\textit{winner}^{r_2} \gets \{j : (i, \texttt{PROVE}(r_2)) \in \textit{proves}\}$
        \State \nextalgline \textbf{wait} $\forall j \in \textit{winner}^{r_2},\ \textit{prop}[r_2][j] \neq \bot$
        \State \nextalgline $M^{r_2} \gets \bigcup_{j \in \textit{winner}^{r_2}} \textit{prop}[r_2][j]$
        \ForAll{$m \in \texttt{ordered}(M^{r_2})$} \nextalgline
          \State \nextalgline $\textit{delivered} \gets \textit{delivered} \cup \{m\}$
          \State \nextalgline \texttt{AB-deliver}$(m)$
        \EndFor
      \EndFor
    \EndWhile
  \end{algorithmic}
\end{algorithm}