typo et quelques lemmes

Recherche/BFT-ARBover/main.tex

@@ -63,10 +63,10 @@
 \newcommand{\BFTREAD}{\textsf{BFT\text{-}READ}}
 
 
-\newcommand{\ABbroadcast}{\textsf{AB-broadcast}}
-\newcommand{\ABdeliver}{\textsf{AB-deliver}}
-\newcommand{\RBcast}{\textsf{RB-cast}}
-\newcommand{\RBreceived}{\textsf{RB-received}}
+\newcommand{\ABbroadcast}{\textsf{ABroadcast}}
+\newcommand{\ABdeliver}{\textsf{ADeliver}}
+\newcommand{\RBcast}{\textsf{RBroadcast}}
+\newcommand{\RBreceived}{\textsf{RReceived}}
 \newcommand{\ordered}{\textsf{ordered}}
 \newcommand{\Winners}{\mathsf{Winners}}
 \newcommand{\Messages}{\mathsf{Messages}}
@@ -107,21 +107,23 @@
 
 \begin{document}
 
-\section{Model}
-We consider a static set of $n$ processes with known identities, communicating by reliable point-to-point channels, in a complete graph. Messages are uniquely identifiable. 
+\section{Model 1: Crash}
+We consider a static set $\Pi$ of $n$ processes with known identities, communicating by reliable point-to-point channels, in a complete graph. Messages are uniquely identifiable. At most $f$ processes can crash, with $n \geq f$.
 
-\paragraph{Synchrony.} The network is asynchronous. Processes may crash; at most $f$ crashes occur.
+\paragraph{Synchrony.} The network is asynchronous.
 
-\paragraph{Communication.} Processes can exchange through a Reliable Broadcast (\RB) primitive (defined below) which's invoked with the functions \RBcast$(m)$ and \RBreceived$(m)$. There exists a shared object called DenyList (\DL) (defined below) that is interfaced with the functions \APPEND$(x)$, \PROVE$(x)$ and \READ$()$. 
+\paragraph{Communication.} Processes can exchange through a Reliable Broadcast ($\RB$) primitive (defined below) which is invoked with the functions $\RBcast(m)$ and $m = \RBreceived()$. There exists a shared object called DenyList ($\DL$) (defined below) that is interfaced with a set $O$ of operations. There exist three types of these operations: $\APPEND(x)$, $\PROVE(x)$ and $\READ()$.   
 
-\paragraph{Notation.} Let $\Pi$ be the finite set of process identifiers and let $n \triangleq |\Pi|$. Two authorization subsets are $\Pi_M \subseteq \Pi$ (processes allowed to issue \APPEND) and $\Pi_V \subseteq \Pi$ (processes allowed to issue \PROVE). Indices $i,j \in \Pi$ refer to processes, and $p_i$ denotes the process with identifier $i$. Let $\mathcal{M}$ denote the universe of uniquely identifiable messages, with $m \in \mathcal{M}$. Let $\mathcal{R} \subseteq \mathbb{N}$ be the set of round identifiers; we write $r \in \mathcal{R}$ for a round. We use the precedence relation $\prec$ for the \DL{} linearization: $x \prec y$ means that operation $x$ appears strictly before $y$ in the linearized history of \DL. For any finite set $A \subseteq \mathcal{M}$, \ordered$(A)$ returns a deterministic total order over $A$ (e.g., lexicographic order on $(\textit{senderId},\textit{messageId})$ or on message hashes). For any round $r \in \mathcal{R}$, define $\Winners_r \triangleq \{\, j \in \Pi \mid (j,\PROVEtrace(r)) \prec \APPEND(r) \,\}$, i.e., the set of processes whose $\PROVE(r)$ appears before the first $\APPEND(r)$ in the \DL{} linearization.
-We denoted by $\PROVE^{(j)}(r)$ or $\APPEND^{(j)}(r)$ the operation $\PROVE(r)$ or $\APPEND(r)$ invoked by process $j$.
+\paragraph{Notation.} For any indice $x$ we defined by $\Pi_x$ a subset of $\Pi$. We consider two subsets $\Pi_M$ and $\Pi_V$ two authorization subsets. Indices $i \in \Pi$ refer to processes, and $p_i$ denotes the process with identifier $i$. Let $\mathcal{M}$ denote the universe of uniquely identifiable messages, with $m \in \mathcal{M}$. Let $\mathcal{R} \subseteq \mathbb{N}$ be the set of round identifiers; we write $r \in \mathcal{R}$ for a round. We use the precedence relation $\prec$ for the \DL{} linearization: $x \prec y$ means that operation $x$ appears strictly before $y$ in the linearized history of \DL. For any finite set $A \subseteq \mathcal{M}$, \ordered$(A)$ returns a deterministic total order over $A$ (e.g., lexicographic order on $(\textit{senderId},\textit{messageId})$ or on message hashes).
+For any operation $F \in O$,$F_i(...)$ denotes that the operation $F$ is invoked by process $p_i$.
+%For any round $r \in \mathcal{R}$, define $\Winners_r \triangleq \{\, j \in \Pi \mid (j,\PROVEtrace(r)) \prec \APPEND(r) \,\}$, i.e., the set of processes whose $\PROVE(r)$ appears before the first $\APPEND(r)$ in the \DL{} linearization.
+%We denoted by $\PROVE^{(j)}(r)$ or $\APPEND^{(j)}(r)$ the operation $\PROVE(r)$ or $\APPEND(r)$ invoked by process $j$.
 
 \section{Primitives}
 
 \input{2_Primitives/index.tex}
 
-\section{Target Abstraction: Atomic Reliable Broadcast (ARB)}
+\section{Atomic Reliable Broadcast (ARB)}
 
 \input{3_ARB_Def/index.tex}