amaury_joly/bwconsistency
1
0
Fork 0
Code Issues 5 Pull Requests Packages Projects 3 Releases Wiki Activity

update

Browse Source
This commit is contained in:
Amaury JOLY
2026-01-07 17:40:57 +01:00
parent 926c3fdc51
commit e89e0e8d2a
3 changed files with 262 additions and 35 deletions
Download Patch File Download Diff File Expand all files Collapse all files

275
Recherche/BFT-ARBover/6_BFT_ARB/index.tex
View File

@@ -32,27 +32,22 @@ We denoted by $\PROVE^{(j)}(r)$ or $\APPEND^{(j)}(r)$ the operation $\PROVE(r)$
\subsubsection{t-BFT-DL}
We consider a t-Byzantine Fault Tolerant DenyList (t-$\BFTDL$) with the following properties.
There are 3 operations : $\PROVE(x), \APPEND(x), \READ(x)$ such that :
There are 3 operations : $\BFTPROVE(x), \BFTAPPEND(x), \BFTREAD(x)$ such that :
\paragraph{Termination.} Every operation $\APPEND(x)$, $\PROVE(x)$, and $\READ()$ invoked by a correct process always returns.
\paragraph{Termination.} Every operation $\BFTAPPEND(x)$, $\BFTPROVE(x)$, and $\BFTREAD()$ invoked by a correct process always returns.
\paragraph{APPEND Validity.} The invocation of $\APPEND(x)$ by a process $p$ is valid if:
\paragraph{APPEND Validity.} The invocation of $\BFTAPPEND(x)$ by a correct process $p_i$ is valid iff $i \in \Pi_M$. Otherwise the operation is invalid.
\paragraph{PROVE Validity.} If the invocation of a $op = \BFTPROVE(x)$ by a correct process $p$ is not valid, then:
\begin{itemize}
\item $p \in \Pi_M \subseteq \Pi$; \textbf{and}
\item $x \in S$, where $S$ is a predefined set.
\end{itemize}
Otherwise, the operation is invalid.
\paragraph{PROVE Validity.} If the invocation of a $op = \PROVE(x)$ by a correct process $p$ is not valid, then:
\begin{itemize}
\item $p \not\in \Pi_V \subseteq \Pi$; \textbf{or}
\item At least $t+1$ valid $\APPEND(x)$ appears before $op$ in $\Seq$.
\item $p \not\in \Pi_V$; \textbf{or}
\item At least $t+1$ valid $\BFTAPPEND(x)$ appears before $op$ in $\Seq$.
\end{itemize}
Otherwise, the operation is valid.
\paragraph{PROVE Anti-Flickering.} If the invocation of a operation $op = \PROVE(x)$ by a correct process $p \in \Pi_V$ is invalid, then any $\PROVE(x)$ operation that appears after $op$ in $\Seq$ is invalid.
\paragraph{PROVE Anti-Flickering.} If the invocation of a operation $op = \BFTPROVE(x)$ by a correct process $p \in \Pi_V$ is invalid, then any $\BFTPROVE(x)$ operation that appears after $op$ in $\Seq$ is invalid.
\paragraph{READ Validity.} The invocation of $op = \READ()$ by a process $p$ returns the list of valid invocations of $\PROVE$ that appears before $op$ in $\Seq$ along with the names of the processes that invoked each operation.
\paragraph{READ Validity.} The invocation of $op = \BFTREAD()$ by a process $p$ returns the list of valid invocations of $\BFTPROVE$ that appears before $op$ in $\Seq$ along with the names of the processes that invoked each operation.
\subsubsection{t-BFT-GE}
@@ -68,6 +63,188 @@ There are three operations: $\BFTVOTE(j, r)$, $\BFTCOMMIT(r)$, and $\BFTRESULT(r
\paragraph{Election.} If at least $f+1$ correct processes invoked a valid $\BFTVOTE(j, r)$ for the same process $j$ then $j$ will be enventually included in the set $W_r$ returned by $\BFTRESULT(r)$.
\subsection{DL $\Rightarrow$ t-BFT-DL}
\begin{lemma}
For any fixed value $t < |M|$, multiple DenyList Object can be used to implement a t-Byzantine Fault Tolerant DenyList Object.
\end{lemma}
\begin{proof}
Fix $t < |M|$. Let
\[
\mathcal{T} \triangleq \{\, T \subseteq M \mid |T| = t \,\}.
\]
For each $T \in \mathcal{T}$, we instantiate one DenyList object $DL_T$ whose authorization sets are
\[
\Pi_M(DL_T) \triangleq S_T \triangleq M \setminus T
\qquad\text{and}\qquad
\Pi_V(DL_T) \triangleq V.
\]
Let
\[
K \triangleq \{\, DL_T \mid T \in \mathcal{T} \,\},
\qquad\text{so that}\qquad
|K| = \binom{|M|}{t}.
\]
% The system consists of a set of $\Pi$ processes, we define a subset of processes $M \subseteq \Pi$ which are authorized to invoke the $\BFTAPPEND$ operation and a subset of processes $V \subseteq \Pi$ which are authorized to invoke the $\BFTPROVE$ operation.
% There exist $K$ a set of $\binom{|M|}{t+1}$ DenyList objects denoted by $DL_1, DL_2, \dots, DL_{|K|}$ such that each DenyList $DL_k$ is associated with a unique subset of processes $S_k \subseteq M$ of size $t+1$.
\begin{algorithmic}
\State $K \gets \{DL_1, DL_2, \dots, DL_{|K|}\}$ \Comment{Set of DenyList objects}
\vspace{1em}
\Function{BFTAPPEND}{x}
\If{$p_i \not\in \Pi_M$}
\State \Return \textbf{false}
\EndIf
\For{\textbf{each } $DL_k \in K$ \textbf{such that} $p_i \in S_k$}
\State $DL_k.\APPEND(x)$
\EndFor
\State \Return \textbf{true}
\EndFunction
\vspace{1em}
\Function{BFTPROVE}{x}
\For{\textbf{each } $DL_k \in K$}
\State $state \gets DL_k.\PROVE(x) || state$
\EndFor
\State \Return $state$
\EndFunction
\vspace{1em}
\Function{BFTREAD}{}
\State $results \gets \emptyset$
\For{\textbf{each } $DL_k \in K$}
\State $res \gets DL_k.\READ()$
\State $results \gets results \cup res$
\EndFor
\State \Return $results$
\EndFunction
\vspace{1em}
\end{algorithmic}
\textbf{APPEND Validity.} A process $p_i$ not in $\Pi_M$ who invokes $\BFTAPPEND(x)$ will never be able to submit a valid $\APPEND(x)$ to any DenyList $DL_k$ since each DenyList $DL_k$ is associated with a unique subset of processes $S_k \subseteq M$. Otherwise, if $p_i \in \Pi_M$, then any invocation of $\BFTAPPEND(x)$ are valid.
\textbf{PROVE Validity.}
Let set $op$ a particular invocation of $\BFTPROVE(x)$ by $p_i$.
\begin{itemize}
\item Case 1: If $p_i \not\in \Pi_V$, then $op$ is invalid since $p_i$ is not authorized to invoke $\PROVE$ operation in any DL.
\item Case 2: If at least $t+1$ valid $\BFTAPPEND(x)$ submitted by $t+1$ distincts processus appears before $op$ in $\Seq$, then for any set $S_k$ there is at least one process in $S_k$ who submitted a valid $\APPEND(x)$ before $op$. Therefore, $op$ will be invalid since all $DLs$ will be locked on the value $x$ by the PROVE Validity of DL Object.
\end{itemize}
Otherwise $op$ is always valid.
\textbf{PROVE Anti-Flickering.}
Let $op=\BFTPROVE(x)$ be an invocation by a correct process $p_i \in \Pi_V$ that is \emph{invalid} in $\Seq$.
By the \textbf{PROVE Validity} of t-$\BFTDL$, this implies that there exist at least $t+1$ \emph{valid}
invocations of $\BFTAPPEND(x)$ by $t+1$ distinct processes that appear before $op$ in $\Seq$.
Let $A \subseteq \Pi_M$ be the set of these (distinct) issuers, with $|A|\ge t+1$.
Consider any DenyList object $DL_k \in K$, associated with a subset $S_k \subseteq M$ of size $|S_k|=t+1$.
Since $K$ contains \emph{all} $(t+1)$-subsets of $M$ and $|A|\ge t+1$, we have
\[
S_k \cap A \neq \emptyset.
\]
Let $p_j \in S_k \cap A$. By construction of $\BFTAPPEND$, the invocation $\BFTAPPEND^{(j)}(x)$ triggers
the operation $DL_k.\APPEND(x)$ (because $p_j \in S_k$), and since $\BFTAPPEND^{(j)}(x) \prec op$ in $\Seq$,
this $DL_k.\APPEND(x)$ appears before the operation $DL_k.\PROVE(x)$ induced by $op$ in the linearization $\Seq_k$
of $DL_k$.
Therefore, in $\Seq_k$ there exists a valid $\APPEND(x)$ before the invocation $DL_k.\PROVE(x)$ induced by $op$.
By the \textbf{PROVE Validity} property of $\DL$, this implies that the invocation $DL_k.\PROVE(x)$ induced by $op$
is \emph{invalid}. Since this holds for every $k \in K$, all component $\PROVE$ operations of $op$ are invalid.
Now consider any invocation $op'=\BFTPROVE(x)$ that appears after $op$ in $\Seq$.
For each $k$, the invocation of $DL_k.\PROVE(x)$ induced by $op'$ appears after the one induced by $op$ in $\Seq_k$.
By the \textbf{PROVE Anti-Flickering} property of $\DL$, once a $\PROVE(x)$ is invalid in $\Seq_k$,
all subsequent $\PROVE(x)$ in $\Seq_k$ are invalid. Hence, for all $k$, the $\PROVE(x)$ induced by $op'$ is invalid,
which implies that $op'$ is invalid at the t-$\BFTDL$ level.
This proves the \textbf{PROVE Anti-Flickering} property for the construction.
\medskip
\textbf{READ Validity.}
Let $op=\BFTREAD()$ be an invocation by a correct process $p$.
The algorithm executes, for each $DL_k \in K$, a read $res_k \gets DL_k.\READ()$ and returns
\[
results \;=\; \bigcup_{k\in K} res_k.
\]
By \textbf{READ Validity} of each $\DL$, $res_k$ contains exactly the valid invocations of $\PROVE$
that appear before $DL_k.\READ()$ in $\Seq_k$, together with the identities of their issuers.
We show that $results$ is exactly the set of valid invocations of $\BFTPROVE$ that appear before $op$ in $\Seq$.
\smallskip
\emph{(i) Soundness (no false positives).}
Take any entry $(j,\PROVE(x)) \in results$. Then $(j,\PROVE(x)) \in res_k$ for some $k$,
hence the invocation $DL_k.\PROVE(x)$ by $p_j$ is valid and appears before $DL_k.\READ()$ in $\Seq_k$.
In particular, there is no valid $DL_k.\APPEND(x)$ before that $DL_k.\PROVE(x)$ in $\Seq_k$
(otherwise $DL_k.\PROVE(x)$ would be invalid by \textbf{PROVE Validity} of $\DL$).
Assume by contradiction that the corresponding invocation $\BFTPROVE^{(j)}(x)$ is invalid in $\Seq$.
Then, by \textbf{PROVE Validity} of t-$\BFTDL$, there exist at least $t+1$ valid invocations of $\BFTAPPEND(x)$
by $t+1$ distinct processes before $\BFTPROVE^{(j)}(x)$ in $\Seq$.
As in the argument used for \textbf{PROVE Anti-Flickering}, this implies that for every $DL_{k'} \in K$,
there exists a valid $DL_{k'}.\APPEND(x)$ before the $DL_{k'}.\PROVE(x)$ induced by $\BFTPROVE^{(j)}(x)$.
Therefore the induced $DL_k.\PROVE(x)$ would be invalid, contradicting $(j,\PROVE(x)) \in res_k$.
Hence $\BFTPROVE^{(j)}(x)$ is valid and appears before $op$ in $\Seq$.
\smallskip
\emph{(ii) Completeness (no omissions).}
Consider a valid invocation $\BFTPROVE^{(j)}(x)$ that appears before $op=\BFTREAD()$ in $\Seq$.
Let $A$ be the set of distinct processes that invoked a valid $\BFTAPPEND(x)$ before $\BFTPROVE^{(j)}(x)$ in $\Seq$.
Since $\BFTPROVE^{(j)}(x)$ is valid in t-$\BFTDL$, we have $|A| \le t$.
Because $K$ contains all subsets $S_k \subseteq M$ of size $t+1$ and $t<|M|$,
there exists some $S_k$ such that
\[
S_k \cap A = \emptyset.
\]
For this $DL_k$, none of the $\BFTAPPEND(x)$ invocations issued by processes in $A$
triggers $DL_k.\APPEND(x)$ (since $\BFTAPPEND$ calls $DL_k.\APPEND(x)$ only for $k$ with issuer in $S_k$).
Thus, there is no valid $DL_k.\APPEND(x)$ before the invocation $DL_k.\PROVE(x)$ induced by $\BFTPROVE^{(j)}(x)$.
By \textbf{PROVE Validity} of $\DL$, the induced $DL_k.\PROVE(x)$ is valid.
Moreover, since $\BFTPROVE^{(j)}(x) \prec \BFTREAD()$ in $\Seq$, this induced $\PROVE(x)$ appears before
$DL_k.\READ()$ in $\Seq_k$, and therefore it is included in $res_k$.
Hence $(j,\PROVE(x)) \in results$.
Combining (i) and (ii), $results$ contains exactly the valid invocations of $\BFTPROVE$
that appear before $op$ in $\Seq$, together with the identities of their issuers.
This proves the \textbf{READ Validity} property.
\end{proof}
\subsection{t-BFT-DL $\Rightarrow$ t-BFT-GE}
\begin{lemma}
For any fixed value $r \in S$, multiple BFT-DenyList Object can be used to implement a BFT-Group Election Object.
\end{lemma}
\begin{proof}
\begin{algorithmic}
\State $Y[i]$ \Comment{Is a set of $n$ $\BFTDL$ with $\Pi_M = \Pi_V = \Pi$}
\vspace{1em}
\Function{BFTVOTE}{j, r}
\EndFunction
\vspace{1em}
\Function{BFTCOMMIT}{r}
\EndFunction
\vspace{1em}
\Function{BFTRESULT}{r}
\State $Z \gets \emptyset$
\For{\textbf{each } $j \in \Pi$}
\If{$|\{(\_, \PROVEtrace(\_, r)) \in Y[j].\BFTREAD(r)\}| \geq n-f$}
\State $P \gets \BFTREAD()$
\State \Return $\{j : (j, \PROVEtrace), \}$
\EndIf
\EndFor
\EndFunction
\vspace{1em}
\end{algorithmic}
\end{proof}
\subsection{Algorithm}
@@ -78,29 +255,73 @@ Each process $p_i$ maintains the following local variables:
\State $\received \gets \emptyset$
\State $\delivered \gets \emptyset$
\State $\prop[r][j] \gets \bot, \forall r, j$
\State $X_r \gets \bot, \forall r$
% \State $X_r \gets \bot, \forall r$
\State $W_r \gets \bot, \forall r$
\State $\resolved[r] \gets \bot, \forall r$
\end{algorithmic}
\renewcommand{\algletter}{A}
\begin{algorithm}[H]
\caption{ABbroadcast$(m)$}
\caption{ABroadcast$(m)$}
\begin{algorithmic}[1]
\State $r \gets \current$
\For{\textbf{each}\ $r \in \{\current, \current +1, \dots\}$}
\State $\RBcast(i, PROP, m, r)$
\State \textbf{wait} until $|W_r| \geq n - f$ where $W_r = \BFTRESULT[r]$
\State $\BFTCOMMIT(r)$
\State \textbf{wait} until $|\resolved[r]| \geq n - f$
\State $W \gets \BFTRESULT[r]$
\If{$i \in W_r \vee (\exists j, r': j \in W_r \wedge \prop[r'][j] \ni m)$}
\State \textbf{break}
\EndIf
\EndFor
\Function{ABroadcast}{$m$}
\State $r \gets \current$
\State $S \gets (\received \cup \{m\}$)
\For{\textbf{each}\ $r \in \{\current, \current +1, \dots\}$}
\State $\RBcast(i, PROP, S, r)$
\State \textbf{wait} until $|W_r| \geq n - f$ where $W_r = \BFTRESULT[r]$
\State $\BFTCOMMIT(r)$
\State \textbf{wait} until $|\resolved[r]| \geq n - f$
\State $W_r \gets \BFTRESULT[r]$
\If{$i \in W_r \vee (\exists j, r': j \in W_r \wedge \prop[r'][j] \ni m)$}
\State \textbf{break}
\EndIf
\EndFor
\EndFunction
\end{algorithmic}
\end{algorithm}
\renewcommand{\algletter}{B}
\begin{algorithm}[H]
\caption{ADeliver$(m)$}
\begin{algorithmic}[1]
\Function{ADeliver}{m}
\State $r \gets \current$
\If{$|\resolved[r]| < n - f$}
\State \Return $\bot$
\EndIf
\State $W_r \gets \BFTRESULT[r]$
\If{$\exists j \in W_r,\ \prop[r][j] = \bot$}
\State \Return $\bot$
\EndIf
\State $M_r \gets \bigcup_{j \in W_r} \prop[r][j]$
\State $m \gets \ordered(M_r \setminus \delivered)[0]$ \Comment{Set $m$ as the smaller message not already delivered}
\State $\delivered \leftarrow \delivered \cup \{m\}$
\If{$M_r \setminus \delivered = \emptyset$} \Comment{Check if all messages from round $r$ have been delivered}
\State $\current \leftarrow \current + 1$
\EndIf
\State \textbf{return} $m$
\EndFunction
\end{algorithmic}
\end{algorithm}
\renewcommand{\algletter}{C}
\begin{algorithm}[H]
\caption{RB handlers}
\begin{algorithmic}[1]
\Function{Rreceived}{j, PROP, S, r}
\State $\received \gets \received \cup \{S\}$
\State $\prop[r][j] \gets S$
\State $\BFTVOTE(j, r)$
\EndFunction
\vspace{1em}
\Function{Rreceived}{j, COMMIT, r}
\State $\received[r] \cup \{j\}$
\EndFunction
\end{algorithmic}
\end{algorithm}
% \subsection{Example execution}

BIN
Recherche/BFT-ARBover/main.pdf
View File

Binary file not shown.

22
Recherche/BFT-ARBover/main.tex
View File

@@ -50,6 +50,12 @@
\newcommand{\PROVE}{\textsf{PROVE}}
\newcommand{\PROVEtrace}{\textsf{prove}}
\newcommand{\READ}{\textsf{READ}}
\newcommand{\BFTAPPEND}{\textsf{BFT\text{-}APPEND}}
\newcommand{\BFTPROVE}{\textsf{BFT\text{-}PROVE}}
\newcommand{\BFTREAD}{\textsf{BFT\text{-}READ}}
\newcommand{\ABbroadcast}{\textsf{AB-broadcast}}
\newcommand{\ABdeliver}{\textsf{AB-deliver}}
\newcommand{\RBcast}{\textsf{RB-cast}}
@@ -59,9 +65,9 @@
\newcommand{\Messages}{\mathsf{Messages}}
\newcommand{\ABlisten}{\textsf{AB-listen}}
\newcommand{\CANDIDATE}{\mathsf{CANDIDATE}}
\newcommand{\CLOSE}{\mathsf{CLOSE}}
\newcommand{\READGE}{\mathsf{READGE}}
\newcommand{\CANDIDATE}{\textsf{VOTE}}
\newcommand{\CLOSE}{\textsf{COMMIT}}
\newcommand{\READGE}{\textsf{RESULT}}
\newcommand{\SHARE}{\mathsf{SHARE}}
\newcommand{\COMBINE}{\mathsf{COMBINE}}
@@ -78,12 +84,12 @@
\newcommand{\Seq}{\mathsf{Seq}}
\newcommand{\GE}{\mathsf{GE}}
\newcommand{\BFTDL}{\mathsf{BFT\text{-}DL}}
\newcommand{\BFTDL}{\textsf{BFT\text{-}DL}}
\newcommand{\BFTGE}{\mathsf{BFT\text{-}GE}}
\newcommand{\BFTVOTE}{\mathsf{BFT\text{-}VOTE}}
\newcommand{\BFTCOMMIT}{\mathsf{BFT\text{-}COMMIT}}
\newcommand{\BFTRESULT}{\mathsf{BFT\text{-}RESULT}}
\newcommand{\BFTGE}{\textsf{BFT\text{-}GE}}
\newcommand{\BFTVOTE}{\textsf{BFT\text{-}VOTE}}
\newcommand{\BFTCOMMIT}{\textsf{BFT\text{-}COMMIT}}
\newcommand{\BFTRESULT}{\textsf{BFT\text{-}RESULT}}
\crefname{theorem}{Theorem}{Theorems}