Changeset 101 for papers/FDL2012/abstraction_refinement.tex
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papers/FDL2012/abstraction_refinement.tex
r93 r101 26 26 $\widehat{M}_i$ ensure item 2. Concretization can be performed by 27 27 modifying the AKS of $\widehat{M}_i$ by changing some abstract value to 28 concrete ones. However, this approach is rude 28 concrete ones. However, this approach is rude: in order to ensure item 1, 29 29 the concretization needs to be consistent 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. 30 30 %Another way to concretize some variables at selected instants is to compose … … 32 32 33 33 We propose to compose the abstraction with another AKS to build a good refinement 34 according to definition \ref{def:goodrefinement}.35 We have several options. The most straightforward consists in building36 an AKS representing all possible executions except the spurious counterexample ; 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 counterexample are determined, and then the corresponding AKS is synchronized with the one of $\widehat{M}_i$.34 according to Definition \ref{def:goodrefinement}. 35 We have several options. The most straightforward method consists in building 36 an AKS representing all possible executions except the spurious counterexample; 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 counterexample are determined, and then the corresponding AKS is synchronized with the one of $\widehat{M}_i$. 37 37 38 38 … … 43 43 counterexample given by the model-checker, the variable configuration in each 44 44 state is Boolean. We name $\widehat{L_i}$ this new labeling. 45 The spurious counterexample $\sigma$ is defined such that 45 The spurious counterexample $\sigma$ is defined such that: 46 46 \begin{definition} 47 47 Let $\sigma$ be a \emph{spurious counterexample} in $\widehat{M}_i =\langle \widehat{AP}_i, \widehat{S}_i, \widehat{S}_{0i}, … … 70 70 counterexample negation} $AKS(\overline{\sigma}) = \langle \widehat{AP}_{\overline{\sigma}}, \widehat{S}_{\overline{\sigma}}, \widehat{S}_{0{\overline{\sigma}}}, 71 71 \widehat{L}_{\overline{\sigma}}, \widehat{R}_{\overline{\sigma}}, 72 \widehat{F}_{\overline{\sigma}} \rangle$ is such that 72 \widehat{F}_{\overline{\sigma}} \rangle$ is such that: 73 73 \vspace*{-2mm} 74 74 \begin{itemize} … … 98 98 The labeling function of $s_i'$ represents (concrete) configuration of state $s_i$ and state $\bar{s_i}$ represents all 99 99 configurations {\it but} the one of $s_i$. This last set may not be representable by 100 the labeling function defined in def\ref{def-aks}. State labeling is treated100 the labeling function defined in Definition \ref{def-aks}. State labeling is treated 101 101 in the second step. $s_T$ is a state where all atomic propositions are {\it unknown}. 102 102 %The size of this structure is linear with the size of the counter-example. … … 139 139 model. The strengthening of the abstraction $\widehat{M}_i$ with the 140 140 addition of AKS of already verified local CTL properties eliminates sets of 141 behaviors and guarantees the over-approximation ( property141 behaviors and guarantees the over-approximation (Property 142 142 \ref{prop:concrete_compose}) but does not guarantee the elimination of the counterexample. We present in the following section a strategy to select sets of CTL properties eliminating the spurious counterexample. 143 143
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