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Mar 13, 2012, 4:53:14 PM (12 years ago)
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ema
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  • papers/FDL2012/abstraction_refinement.tex

    r58 r63  
    99\begin{definition} An efficient \emph{refinement} verified the following properties:
    1010\begin{enumerate}
    11 \item The new refinment is an over-approximation of the concrete model: $\widehat{M}_{i+1} \sqsubseteq \widehat{M}$.
    12 \item The new refinment is more concrete than the previous one:
     11\item The new refinement is an over-approximation of the concrete model: $\widehat{M}_{i+1} \sqsubseteq \widehat{M}$.
     12\item The new refinement is more concrete than the previous one:
    1313$\widehat{M}_{i} \sqsubseteq \widehat{M}_{i+1}$.
    1414\item The spurious counter-example in $\widehat{M}_i$ is  removed from
     
    2020convergence by the minimizing the number of iteration of the CEGAR loop.
    2121
    22 
    23 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 ???
    24 
    25 \TODO{discussion sur comment garantir les points 1/2/3 et le reste du bon
    26 rafinement}
    27 \subsection{Refinment by negation of the counterexample}
     22Refinements based on the concretization of selected abstract variables in $\widehat{M}_i$ ensure item 2. Concretization can be performed either in modifying the AKS of $\widehat{M}_i$, by changing some abstract value to concrete ones, but this approach is rude : in order to ensure item 1, concretization needs to be coherent with the sequences of values in the concrete system. The difficulty resides in defining the proper abstract variable to concretize, at which precise instant, and with which Boolean value.
     23Another way to concretize some variables at selected instants is to compose (by a synchronous product) the AKS  of $\widehat{M}_i$ with a new AKS, provided this latest represents over-approximations of the set of behaviors of $M$. By construction, this product satisfies items 1 and 2. We now have to compute an AKS eliminating the spurious counter-example, being easily computable and ensuring a quick convergence of the CEGAR loop.
     24
     25Several proposals can be made. The most straightforward consists in building the AKS representing all possible executions except the  spurious counter-example ; however the AKS representation may be huge and the process is not guaranteed to converge. A second possibility is to build an AKS with additional CTL properties of the components ; the AKS remains small but item 3 is not guaranteed, hence delaying the convergence. The final proposal combines both previous ones : first local CTL properties eliminating the spurious counter example are determined, and then the corresponding AKS is synchronized with the one of $\widehat{M}_i$.
     26
     27
     28\subsection{Refinement by negation of the counterexample}
    2829
    2930\TODO{Mettre la def avant ?}
     
    3839\widehat{L}_i, \widehat{R}_i, \widehat{F}_i \rangle$ and let the length of the
    3940counterexample, $|\sigma| = n$: $ \sigma = s_{0} \rightarrow s_{1} \ldots
    40 s_{n}$ with $(s_{k}, s_{k+1}) \in \widehat{R}_i$ $\forall k \in [0..n-1]$. 
     41s_{n}$ with $(s_{k}, s_{k+1}) \in \widehat{R}_i$ $\forall k \in [0..n-1]$.
    4142\begin{itemize}
    4243\item All its variables are concrete: $\forall s_i$ and $\forall p\in
    4344\widehat{AP}_i$, $p$ is either true or false
    44 (not {\it unknown}).
     45(not {\it unknown}), and $s_0 $ is an initial state of the concrete system: $s_0 \in \mathbf{R}_0$
    4546\item  $\sigma$ is a counter-example in  $\widehat{M}_i$: $s_0\not\models \Phi$.
    46 \item  $\sigma$ is not a path of the concrete system $M$: $\exists k$ such
    47 that $(s_{k}, s_{k+1}) \not\in R$.
     47\item  $\sigma$ is not a path of the concrete system $M$: $\exists k \in [1..n-1]$ such
     48that $\forall j < k, (s_j,s_{j+1}) \in R$ and $(s_{k}, s_{k+1}) \not\in R$.
    4849\end{itemize}
    4950\end{definition}
     
    5556b) The negation of an configuration may be represented by a set of abstract configurations
    5657
    57 c) building the AKS of a spurious counter-example may lead to a blow-up of the number of states of the AKS 
     58c) building the AKS of a spurious counter-example may lead to a blow-up of the number of states of the AKS
    5859
    5960
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