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

\begin{itemize}
  \item \textbf{\textit{received}}: the set of messages that have been received via the reliable broadcast primitive but not yet ordered.
  \item \textbf{\textit{delivered}}: the set of messages that have been ordered.
  \item \textbf{\textit{prop}[$r$][$j$]}: the proposal set announced by process $j$ at round $r$. It contains a set of messages that process $j$ claims to have received but not yet delivered.
  \item \textbf{\textit{winner}$^r$}: the set of processes that have issued a valid \texttt{PROVE} for round $r$, as observed through the registry.
  \item \textbf{\texttt{RB-cast}$(\texttt{PROP}, S, r, j)$}: a reliable broadcast invocation that disseminates the proposal $S$ from process $j$ for round $r$.
  \item \textbf{\texttt{RB-delivered}$(\texttt{PROP}, S, r, j)$}: the handler invoked upon reception of a \texttt{RB-cast}, which stores the received proposal $S$ into $\textit{prop}[r][j]$.
  \item \textbf{\texttt{READ}()} : returns the current view of all valid operations stored in the DenyList registry.
  \item \textbf{\texttt{ordered}$(S)$}: returns a deterministic total order over a set $S$ of messages.
\end{itemize}

\resetalgline
\begin{algorithm}
  \caption{Atomic Broadcast with DenyList}
  \begin{algorithmic}[1]
    \State $\textit{proves} \gets \emptyset$
    \State $\textit{received} \gets \emptyset$
    \State $\textit{delivered} \gets \emptyset$
    \State $r_1 \gets 0$

    \vspace{1em}
    % --- AB-Broadcast ---
    \State \nextalgline \textbf{AB-Broadcast}$_j(m)$
    \State \nextalgline \hspace{1em} $\texttt{RB-Broadcast}_j(m)$

    \vspace{1em}
    % --- RB-delivered ---
    \State \nextalgline \textbf{RB-delivered}$_j(m)$
    \State \nextalgline \hspace{1em} $\textit{received} \gets \textit{received} \cup \{m\}$
    \State \nextalgline \hspace{1em} \textbf{repeat until} $\textit{received} \setminus \textit{delivered} \neq \emptyset$
    \State \nextalgline \hspace{2em} $S \gets \textit{received} \setminus \textit{delivered}$
    \State \nextalgline \hspace{2em} $\textit{proves} \gets \texttt{READ}()$
    \State \nextalgline \hspace{2em} $r_2 \gets \max\{r : j,\ (j, \texttt{PROVE}(r)) \in \textit{proves}\} + 1$
    \State \nextalgline \hspace{2em} $\texttt{RB-cast}(\texttt{PROP}, S, r_2, j)$
    \State \nextalgline \hspace{2em} $\texttt{PROVE}(r_2)$

    \vspace{0.5em}
    \State \nextalgline \hspace{2em} \textbf{for } $r \in [r_1 + 1, \dots, r_2]$ \textbf{do}
    \State \nextalgline \hspace{3em} $\texttt{APPEND}(r)$
    \State \nextalgline \hspace{3em} $\textit{proves} \gets \texttt{READ}()$
    \State \nextalgline \hspace{3em} $\textit{winner}^r \gets \{j : (j, \texttt{PROVE}(r)) \in \textit{proves}\}$
    \State \nextalgline \hspace{3em} \textbf{wait } $\forall j \in \textit{winner}^r,\ \textit{prop}[r][j] \neq \bot$
    \State \nextalgline \hspace{3em} $T \gets \bigcup_{j \in \textit{winner}^r} \textit{prop}[r][j] \setminus \textit{delivered}$

    \vspace{0.5em}
    \State \nextalgline \hspace{3em} \textbf{for each } $m \in \texttt{ordered}(T)$
    \State \nextalgline \hspace{4em} $\textit{delivered} \gets \textit{delivered} \cup \{m\}$
    \State \nextalgline \hspace{4em} $\texttt{AB-deliver}_j(m)$
    \State \nextalgline \hspace{2em} $r_1 \gets r_2$

    \vspace{1em}
    % --- RB-deliver(Prop) handler ---
    \State \nextalgline \textbf{RB-delivered}$_j(\texttt{PROP}, S, r_1, j_1)$
    \State \nextalgline \hspace{1em} $\textit{prop}[r_1][j_1] \gets S$

  \end{algorithmic}
\end{algorithm}