We take advantage of the specification of verified components to build more
accurate abstractions. The key, here, is how to select the part of the
specification relevent for the proof of the global property. We propose an
heuristic to order the properties of each component depending on the structure
of each component. 
%Before generating an abstract model to verify a global property $\phi$, the verified properties of all the components in the concrete model are ordered according to their pertinency in comparison to a global property $\phi$.
In order to do so, the variable dependency of the variables present in global property has to be analysed.
After this point, we refer to the variables present in the global property  as \emph{primary variables}.

%\bigskip

The ordering of the properties will be based on the variable dependency graph
where the roots are primary variables. 
The variables in the model are weighted according to their dependency level
\emph{vis-à-vis} primary variables and the properties will be weighted according to the sum of the weights
of the variables present in it. We want to select the properties that describe
behaviors that have an impact on the global one. Hence, the more closer a variable is from the primary
variable the more its behavior affects the primary variable. A property will
have higher priority the more of primary or close to primary variables it
contains.
Moreover, a global property often specifies the behavior at the interface of
components. Typicaly, a global property will ensure that a message sent is
always acknoledge or the good target get the message. This kind of behavior
relates the input-output behaviors of components. 
We have decided to allocate an extra weight for variables which are present at the interface of a component 
whereas variables which do not interfere in the obtention of a primary variable will be weighted 0. 
Here is how we proceed:
\begin{enumerate}
\item Build the dependency graph for all primary variables.
\item Compute the depth of all variables (DFS or BFS)
algorithm for all dependency graph.
Note that a variable may belong to more than one dependency graph, in that case
we consider the minimum depth.
\item Give a weight to each variables (see algorithm  \ref{algo:weight}).
\item Compute the weight of properties for each component.
\end{enumerate}

The algorithm \ref{algo:weight} gives weight according to the variable distance to the
primary variable with extra weight for interface variable and primary variable.

\begin{algorithm}[h]
\caption{Compute Weight}
\label{algo:weight}

\KwIn{ $\{V\}$, the set of all dependency graph variable}
\KwOut{$\{(v,w)| v \in V, w \in N\}$, The set of variables with their weight}

\Begin{
$p = $ max(depth(V))  \\
\For{$v\in V$}{
	d = depth(v) \;
	$w = 2^{p-d}*p$\;
	\If(v is primary variable){$d == 0$}
	{
		$w = 5 * w$\;
	}
	\If(v is an interface variable){$v\in{I\cup O}$}
	{
		$w = 3 * w $
	}
}
}
\end{algorithm}
%\begin{enumerate}
%
%\item {\emph{Establishment of primary variables' dependency and maximum graph depth}\\
%Each primary variable will be examined and their dependency graph is elobarated. The maximum graph depth among the primary variable dependency graphs will be identified and used to calibrate the weight of all the variables related to the global property.
%Given the primary variables of $\phi$, $V_{\phi} =  \langle v_{\phi_0}, v_{\phi_1}, ... , v_{\phi_k}, ... , v_{\phi_n} \rangle$ and $G{\_v_{\phi_k}}$ the dependency graph of primary variable $v_{\phi_k}$, we have the maximum graph depth $max_{d} = max(depth(Gv_{\phi_0}), depth(Gv_{\phi_1}), ... , depth(Gv_{\phi_k}), ... ,$\\$ depth(Gv_{\phi_n})) $.
%
%}
%
%\item {\emph{Weight allocation for each variables} \\
%Let's suppose $max_d$ is the maximum dependency graph depth calculated and $p$ is the unit weight. We allocate the variable weight as follows:
%\begin{itemize}
%\item{All the variables at degree $max_d$ of every dependency graph will be allocated the weight of $p$.}
% \\ \hspace*{20mm} $Wv_{max_d} = p$
%\item{All the variables at degree $max_d - 1$ of every dependency graph will be allocated the weight of $2Wv_{max_d}$.}
%\\ \hspace*{20mm} $Wv_{max_d - 1} = 2Wv_{max_d}$
%\item{...}
%\item{All the variables at degree $1$ of every dependency graph will be allocated the weight of $2Wv_{2}$.}
% \\ \hspace*{20mm} $Wv_{1} = 2Wv_{2}$
%\item{All the variables at degree $0$ (i.e. the primary variables) will be allocated the weight of $10Wv_{1}$.}
% \\ \hspace*{20mm} $Wv_{0} = 10Wv_{1}$
%\end{itemize}
%
%We can see here that the primary variables are given a considerable
%ponderation due to their pertinency \emph{vis-à-vis} global  property. Furthermore, we will allocate a supplementary weight of $3Wv_{1}$ to variables at the interface of a component as they are the variables which assure the connection between the components if there is at least one variable in the dependency graph established in the previous step in the property. All other non-related variables have a weight equals to $0$.
%}
%
%
%\item {\emph{Ordering of the properties} \\
%Properties will be ordered according to the sum of the weight of the variables in it. Therefore, given a property $\varphi_i$ which contains $n+1$ variables, $V_{\varphi_i} =  \langle v_{\varphi_{i0}}, v_{\varphi_{i1}}, ... , v_{\varphi_{ik}}, ... , v_{\varphi_{in}} \rangle$, the weight  of $\varphi_i$ , $W_{\varphi_i} = \sum_{k=0}^{n} Wv_{\varphi_{ik}}$ .
%After this stage, we will check all the properties with weight $>0$ and allocate a supplementary weight of $3Wv_{1}$ for every variable at the interface present in the propery. After this process, the final weight of a property is obtained and the properties will be ordered in a list with the weight  decreasing (the heaviest first). We will refer to the ordered list of properties related to the global property $\phi$ as $L_\phi$.
%
%
%}
%
%\end{enumerate}

%\bigskip

%\emph{\underline{Example:}}  \\
%
%For example, if a global property $\phi$ consists of 3 variables: $ p, q, r $ where:
%\begin{itemize}
%\item{$p$ is dependent of $a$ and $b$}
%\item{$b$ is dependent of $c$}
%\item{$q$ is dependent of $x$}
%\item{$r$ is independent}
%\end{itemize}
%
%Example with unit weight= 50.
%The primary variables: $p$, $q$ and $r$ are weighted $100*10=1000$ each. \\
%The secondary level variables : $a$, $b$ and $x$ are weighted $50x2=100$ each. \\
%The tertiary level variable $c$ is weighted $50$. \\
%The weight of a non-related variable is $0$.
%
%
%\bigskip
%\begin{figure}[h!]
%   \centering
%%   \includegraphics[width=1.2\textwidth]{Dependency_graph_weight_PNG}
%%     \hspace*{-15mm}
%     \includegraphics{Dependency_graph_weight_PNG}
%   \caption{\label{DepGraphWeight} Example of weighting}
%\end{figure}

%Dans la figure~\ref{étiquette} page~\pageref{étiquette}, …



Each verified properties available pertinency will be evaluated by adding the weights of all the variables in it.
It is definitely not an exact pertinency calculation of properties but provides a good indicator 
of their possible impact on the global property.
After this pre-processing phase, we will have a list of properties $L_\phi$
ordered according to their pertinency in comparison to the global property.




\subsection{Filtering properties}
The refinement step consists of adding new AKS of properties selected according to
their pertinency. This refinement respects the points 1 and 2 of definition
\ref{def:goodrefinement}. The first item comes form AKS definition. 
Adding a new AKS in the abstraction leads to an abstraction where more behaviors
are characterized. Hence there is more constrains behavior and more concretize
states. 

\Remark{Cécile}{Mettre definition, property and proof ?????}

Unforutnatly, this refinement does not ensure that the spurious counter-example
is evicted. 
As we would like to ensure the elimination of the counterexample previously found,
we filter out properties that don't have an impact on the counterexample $\sigma_i$ thus won't eliminate it. 
In order to reach this obective, a AKS of the counterexample $\sigma_i$, $K(\sigma_i)$
is generated. $K(\sigma_i)$ is a succession of states corresponding to the counterexample path which dissatisfies 
the global property $\Phi$. It is an abstract kripke structure since not all
variables are concrete.
Each properties
All the properties available are then model-checked on $K(\sigma_i)$. If the
property holds then the property will not discriminate the counter-example.
Hence this property is not a good candidate for refinement.
Therefore all properties that are satisfied won't be chosen to be integrated in the next step of refinement. At this stage, we already have a list of potential properties that will definitely eliminate the current counterexample $\sigma_i$ and might converge the abstract model towards a model sufficient to verify the global property $\phi$.


\TODO{Revoir la definition et Proof}

The refinement process from $\widehat{M}_i$ to $\widehat{M}_{i+1}$ can be seperated into 2 steps:

\begin{enumerate}

\item {\emph{\underline{Step 1:}} \\

As we would like to ensure the elimination of the counterexample previously found, we filter out properties that don't have an impact on the counterexample $\sigma_i$ thus won't eliminate it. In order to reach this obective, a Kripke Structure of the counterexample $\sigma_i$, $K(\sigma_i)$ is generated. $K(\sigma_i)$ is a succession of states corresponding to the counterexample path which dissatisfies the global property $\phi$.

\bigskip

\begin{definition}
\textbf{\emph{The counterexample $\sigma_i$ Kripke Structure $K(\sigma_i)$ :}} \\
Let a counterexample of length $n$, $ \sigma_i = \langle s_{\bar{a}i,0}, s_{\bar{a}i,1},\\ s_{\bar{a}i,2}, ... , s_{\bar{a}i,k}, s_{\bar{a}i,k+1}, ... , s_{\bar{a}i,n}\rangle $ with $ \forall k \in [0,n-1]$, we have \\
$K(\sigma_i) = (AP_{\sigma_i}, S_{\sigma_i}, S_{0\sigma_i}, L_{\sigma_i}, R_{\sigma_i})$ a 5-tuple consisting of :

\begin{itemize}
\item { $AP_{\sigma_i}$ : a finite set of atomic propositions which corresponds to the variables in the abstract model $\widehat{V}_{i}$ }	
\item { $S_{\sigma_i} = \{s_{\bar{a}i,0}, s_{\bar{a}i,1}, s_{\bar{a}i,2}, ... , s_{\bar{a}i,k}, s_{\bar{a}i,k+1}, ... , s_{\bar{a}i,n}\}$}
\item { $S_{0\sigma_i} = \{s_{\bar{a}i,0}\}$}
\item { $L_{\sigma_i}$ : $S_{\sigma_i} \rightarrow 2^{AP_{\sigma_i}}$ : a labeling function which labels each state with the set of atomic propositions true in that state. }
\item { $R_{\sigma_i}$ = $ (s_{\bar{a}i,k}, s_{\bar{a}i,k+1})$ }
\end{itemize}
\end{definition}

%\bigskip
All the properties available are then model-checked on $K(\sigma_i)$.

If:
\begin{itemize}
\item {\textbf{$K(\sigma_i) \vDash \varphi  \Rightarrow \varphi $ will not eliminate $\sigma_i$}}
\item {\textbf{$K(\sigma_i) \nvDash \varphi  \Rightarrow \varphi $ will eliminate $\sigma_i$}}
\end{itemize}

%\bigskip


%\begin{figure}[h!]
%   \centering
%%   \includegraphics[width=1.2\textwidth]{K_sigma_i_S_PNG}
%%     \hspace*{-15mm}
%     \includegraphics{K_sigma_i_S_PNG}
%   \caption{\label{AKSNegCex} Kripke Structure of counterexample $\sigma_i$, $K(\sigma_i)$}
%\end{figure}

%Dans la figure~\ref{étiquette} page~\pageref{étiquette}, …

%\bigskip


\begin{figure}[h!]
   \centering

\begin{tikzpicture}[->,>=stealth',shorten >=1.5pt,auto,node distance=2cm,
                    thick]
  \tikzstyle{every state}=[fill=none,draw=blue,text=black, minimum size=1.5cm]

  \node[initial,state] (A)                            {$s_{\bar{a}i,0}$};
  \node[state]           (B) [below of=A]     {$s_{\bar{a}i,1}$};

  \node[state]           (C) [below of=B]        {$s_{\bar{a}i,k}$};

  \node[state]           (D) [below of=C]       {$s_{\bar{a}i,n-1}$};
  \node[state]           (E) [below of=D]       {$s_{\bar{a}i,n}$};

  \path (A) edge              node {} (B)
            (B) edge 	   node {} (C)
            (C) edge             node {} (D)
            (D) edge             node {} (E);

\end{tikzpicture}

   \caption{\label{AKSNegCex} Kripke Structure of counterexample $\sigma_i$, $K(\sigma_i)$}
\end{figure}


}
%\bigskip

\item {\emph{\underline{Step 2:}} \\

The property at the top of the list (not yet selected and excluding the properties which are satisfied by $K(\sigma_i)$) is selected to be integrated in the generation of $\widehat{M}_{i+1}$.
%\bigskip

}
\end{enumerate}

$\widehat{M}_{i+1}$ is model-checked and the refinement process is repeated until the model satisfies the global property or there is no property left to be integrated in next abstraction.