Index: papers/FDL2012/framework.tex
===================================================================
--- papers/FDL2012/framework.tex	(revision 52)
+++ papers/FDL2012/framework.tex	(revision 56)
@@ -5,5 +5,5 @@
 concrete model $M$, an abstraction of the concrete model $\widehat{M}$ is
 generated and tested in the model-checker. As the abstract model is an
-upper-approximation of the concrete model and we have restrained our
+over-approximation of the concrete model and we have restrained our
 verification to ACTL properties only, if $\Phi$ holds on the the abstract model then we are certain that it holds in the concrete model as well. 
 However, if $\Phi$ doesn't hold in the abstract model then we can't conclude anything regarding the concrete model until the counterexample, 
@@ -15,5 +15,5 @@
 %are CTL formulas with only universal path quantifiers: AX, AF, AG and AU.
 %\end{definition}
-\remark{Cecile}{remettre la dÃ©fintion du modÃšle concret}
+
 \begin{definition}
 Given $\widehat{M} = (\widehat{AP}, \widehat{S}, \widehat{S}_0, \widehat{L}, \widehat{R}, \widehat{F})$ an abstract model of a concrete model, $M$ and $\Phi$, a global property to be verified on $M$, the model-checking result can be interpreted as follows:
@@ -36,11 +36,54 @@
 \end{figure}
 
-As mention earlier, in our verification methodology, we have a concrete model which consists of several components and each component comes with its specification or more precisely, properties that hold in the component. Given a global property $\Phi$, the property to be verified by the composition of the concrete components model, an abstract model is generated by selecting some of the properties of the components which are relevant to $\varphi$. The generation of an abstract model in the form of AKS from CTL formulas, based on the works of Braunstein \cite{braunstein07ctl_abstraction}, has been successfully implemented by Bara \cite{bara08abs_composant}.
+As mention earlier, in our verification methodology, we have a concrete model which consists of several components and each component comes with its specification or more precisely, properties that hold in the component. Given a global property $\Phi$, the property to be verified by the composition of the concrete components model, an abstract model is generated by selecting some of the properties of the components which are relevant to $\varphi$. 
 
 In the case where model-checking failed, the counterexample given by the model-checker \cite{ucberkeley96vis}  has to be analysed. We use a SATSolver to check whether the counterexample is spurious or not. When a counterexample is proved to be spurious, we proceed to the refinement phase.
+\TODO{ref du papier sur le spurious + petite explication sur SAT et
+dÃ©roulement}
+\subsection{Concrete system definition}
 
-\subsection{Definition of the abstraction of a component and of the complete system}
+The concrete system is a synchronous compositon of components, each of which
+described as a Moore machine.
+\begin{definition}
+A \emph{Moore machine} $C$ is defined by a tuple $\langle I, O, R,$ $\delta, \lambda, \mathbf{R}_0 \rangle$, where,
+\begin{itemize}
+\item $I$ is a finite set of boolean inputs signals.
+\item $O$ is a finite set of boolean outputs signals.
+\item $R$ is a finite set of boolean sequential elements (registers).
+\item $\delta : 2^I \times 2^R \rightarrow 2^R$ is the transition function.
+\item $\lambda : 2^R \rightarrow 2^O$ is the output function.
+\item $\mathbf{R}_0 \subseteq 2^R$ is the set of initial states.
+\end{itemize}
+\end{definition}
 
-The abstraction of a component is represented by an Abstract Kripke Structure (AKS for short), derived from a subset of the CTL properties the component satisfies. Roughly speaking, AKS($\varphi$), the AKS derived from a CTL property $\varphi$, simulates all execution trees whose initial state satisfies $\varphi$. In AKS($\varphi$), states are tagged with the truth values of $\varphi$'s atomic propositions, among four truth values : inconsistent, false, true and unknown (or undefined). States with inconsistent truth values are not represented since they refer to non possible assignments of the atomic propositions. A set of fairness constraints eliminates non-progress cycles.
+\emph{States} (or configurations) of the circuit correspond to boolean configurations of all the sequential elements. From now on, let $C = \langle I, O, R, \delta, \lambda, \mathbf{R}_0 \rangle$ be a sequential circuit.
+
+\begin{definition}
+A \emph{Concrete system} $M$ is obtained by synchronous composition of the
+component.\\
+$M = C_1 \parallel C_2 \parallel \ldots \parallel C_n$,where each $C_i$ is a
+Moore machine with a specification associated $\varphi_i = \{\varphi_i^1 \ldots
+\varphi_i^k\}$ Each $\varphi_i^j$ being a CTL$\setminus$X formula whose
+propositions $AP$ belongs to $\{I_i\cup O_i\cup R_i\}$ .
+\end{definition}
+
+\subsection{Abstraction definition}
+
+Our abstraction consists in reducing the size of the representation model by
+freeing some its variables. The point is to determine the good set of variable
+to be freed and when to free them. We take advantage of the CTL specification
+of each component: a CTL property may be seen as a partial view of the tree of
+behaviors of its variables. All the variables not specified by the property
+can be freed. We introduced the Abstract Kripke Structure (AKS for short) which exactly
+specifies when the variable of the prperty can be frreed.
+The abstraction of a component is represented by an AKS, 
+derived from a subset of the CTL properties the component satisfies.
+Roughly speaking, AKS($\varphi$), the AKS derived from a CTL property
+$\varphi$, simulates all execution trees whose initial state satisfies
+$\varphi$. In AKS($\varphi$), states are tagged with the truth values of
+$\varphi$'s atomic propositions, among four truth values : inconsistent,
+false, true and unknown (either true or false).
+States with inconsistent truth values are not represented since they refer to non possible
+assignments of the atomic propositions. A set of fairness constraints eliminates non-progress cycles.
 
  
@@ -71,4 +114,7 @@
 
 As the abstract model $\widehat{M}$ is generated from the conjunction of verified properties of the components in the concrete model $M$, it can be seen as the composition of the AKS of each property.
+The AKS composition has been defined in \cite{these_braunstein}; it extends
+the classical synchrounous composition of Moore machine to deal with
+four-valued variables.
 %\bigskip
 
@@ -79,5 +125,4 @@
 \item{$ \widehat{C}_j = AKS (\varphi_{C_j^1}) ~||~ AKS (\varphi_{C_j^2} ) ~||~...~||~ AKS (\varphi_{C_j^k}) ~||$\\ $ ...~||~ AKS (\varphi_{C_j^m}) $}
 \item{$ \widehat{M} = \widehat{C}_1 ~||~ \widehat{C}_2 ~||~ ... ~||~ \widehat{C}_j ~||~... ~||~ \widehat{C}_n $}
-\item{$ V_{\widehat{C}_j} \subseteq V_{C_j}$ (with $V_{\widehat{C}_j}$ and $V_{C_j}$ are variables of $\widehat{C}_j$ and $C_j$ respectively.) TODO : LES V ICI NE SONT-ELLES PAS L'UNION DES AP DES $\varphi_{C_j^k}$ ???????}
 \end{itemize}
 
@@ -90,4 +135,7 @@
 
 
+The generation of an abstract model in the form of AKS from CTL formulas,
+based on the works of Braunstein \cite{braunstein07ctl_abstraction}, 
+has been successfully implemented by Bara \cite{bara08abs_composant}.
 
 
@@ -95,15 +143,27 @@
 TODO : PEUT ETRE A VENTILER DANS DIFFERENTES PARTIES ??
 
-1. Ordering of AKS
+\begin{definition}
+A state $s$ is an abstract state if one its variable $p$ is {\it unknown}. It
+is concise representation of the set of more concrete states in which $p$
+is either true or false. 
+\end{definition}
 
-Def : Concrete and abstract variables in AKS
+\begin{definition}
+The {\emph concretization} of an abstract $s$ with respect to the variable $p$
+({\it unknown} in that state), assigns either true or false to $p$.
 
-Def : Concretization of an abstract variable
+The {\emph abstraction} of a state $s$ with respect to the variable $p$
+(either true or false in that state), assigns  {\it unknown} to $p$.
+\end{definition}
 
-Def (dual) : Abstraction of a concrete variable
+\begin{property}
+Let A1 and A2 two AKS such that A2 is obtained by concretizing one abstract variable of A1 (resp A1 is obtained by abstracting one variable in A2). Then A1 simulates A2.
+\end{property}
+\begin{proof}
+As the concretization of state reduces the set of concrete configuration the
+abstract state represents but does not affect the transition relation of the
+AKS. The unroll execution tree of A2 is a subtree of the one of A1. Then  A1 simulates A2.
+\end{proof}
 
-Prop : Let A1 and A2 two AKS such that A2 is obtained by concretizing one abstract variable of A1 (resp A1 is obtained by abstracting one variable in A2). Then A1 simulates A2.
-
-A possible refinement : concretization of selected abstract variables. How to choose variables and instants of concretization : introduce new CTL properties. The question is : how to select pertinent CTL properties ???
  
 2. Negation of states in an AKS
