Changeset 81 for papers/FDL2012/ordering_filter_properties.tex
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papers/FDL2012/ordering_filter_properties.tex
r79 r81 8 8 %\bigskip 9 9 10 The ordering of the properties is based on the variable dependency graph 11 where the roots are primary variables. 12 The variables in the model are weighted according to their dependency level 13 \emph{vis-Ã -vis} primary variables and the properties is weighted according to the sum of the weights 14 of the variables present in it. We want to select the properties specifying 15 behaviors that may have an impact on the global property. We observed that 10 %The ordering of the properties is based on the variable dependency graph 11 %where the roots are primary variables. 12 %The variables in the model are weighted according to their dependency level 13 %\emph{vis-Ã -vis} primary variables and the properties is weighted according to the sum of the weights 14 %of the variables present in it. We want to select the properties specifying 15 %behaviors that may have an impact on the global property. 16 We observed that 16 17 the more closer a variable is from the primary 17 18 variable the more it affects the primary variable. Hence, a property … … 32 33 we consider the minimum depth. 33 34 \item Give a weight to each variable (see algorithm \ref{algo:weight}). 34 \item Compute the weight of properties for each component. 35 \item Compute the weight of properties for each component~: sum of the 36 property variables weight. 35 37 \end{enumerate} 36 38 … … 129 131 130 132 131 Each properties pertinence is evaluated by adding the weights of all the variables in it.133 %Each properties pertinence is evaluated by adding the weights of all the variables in it. 132 134 It is definitely not an exact pertinence calculation of properties but provides a good indicator 133 135 of their possible impact on the global property. … … 140 142 \subsection{Filtering properties} 141 143 The refinement step consists of adding new AKS of properties selected according to 142 their pertinence. This refinement respects items 1 and 2 of definition 143 \ref{def:goodrefinement}. The first item comes from AKS definition and the 144 composition property \ref{prop:concrete_compose}. 145 Adding a new AKS in the abstraction leads to an abstraction where more behaviors 146 are characterized. Hence there is more constrains behavior and more concretize 147 states. 148 149 144 their pertinence. 145 %This refinement respects items 1 and 2 of definition 146 %\ref{def:goodrefinement}. The first item comes from AKS definition and the 147 %composition property \ref{prop:concrete_compose}. 148 %Adding a new AKS in the abstraction leads to an abstraction where more behaviors 149 %are characterized. Hence there is more constrains behavior and more concretize 150 %states. 150 151 Unfortunately, this refinement does not ensure that the spurious counterexample 151 152 is evicted. … … 155 156 In order to reach this objective, a Abstract Kripke structure of the counterexample $\sigma$, $K(\sigma)$ 156 157 is generated. $K(\sigma)$ is a succession of states corresponding to the counterexample path which dissatisfies 157 the global property $\Phi$ as show in figure \ref{AKSNegCex}. In case where 158 the counter-example exibits a bounded path, we add a last 158 the global property $\Phi$ 159 %as show in figure \ref{AKSNegCex}. 160 In case where the spurious counter-example exhibits a bounded path, we add a last 159 161 state $s_T$ where all variable are free({\it unknown}). The tree starting from this 160 162 state represents all the possible future of the counterexample. … … 183 185 \item $S_{\sigma} = \{s_{i}|s_i\in \sigma\}\cup\{s_T\}$ 184 186 \item $S_{0\sigma} = \{s_{0}\}$ 185 \item $L_{\sigma} = \ check{L}_i$187 \item $L_{\sigma} = \widehat{L}_i$ 186 188 \item $R_{\sigma} = \{(s_{k}, s_{k+1})|(s_{k}\rightarrow s_{k+1})\in 187 189 \sigma\}\cup\{(s_{n-1},s_T)\}$ … … 215 217 % 216 218 % 217 \begin{figure}[h!]218 \centering219 220 \begin{tikzpicture}[->,>=stealth',shorten >=1.5pt,auto,node distance=2cm,221 thick]222 \tikzstyle{every state}=[fill=none,draw=blue,text=black, minimum size=1.1cm]223 224 \node[initial,state] (A) {$s_{0}$};225 \node[state] (B) [below of=A] {$s_{1}$};226 \node[node distance=1.5cm] (C) [below of=B] {$\ldots$};227 \node[state,node distance=1.5cm] (D) [below of=C] {$s_{n-1}$};228 \node[state] (E) [below of=D] {$s_T$};229 230 \path (A) edge node {} (B)231 (B) edge node {} (C)232 (C) edge node {} (D)233 (D) edge node {} (E)234 (E) edge[loop right] node {} (E);235 236 \end{tikzpicture}237 238 \caption{\label{AKSNegCex} Kripke Structure of counterexample $\sigma$, $K(\sigma)$}239 \end{figure}219 %\begin{figure}[h!] 220 % \centering 221 % 222 %\begin{tikzpicture}[->,>=stealth',shorten >=1.5pt,auto,node distance=2cm, 223 % thick] 224 % \tikzstyle{every state}=[fill=none,draw=blue,text=black, minimum size=1.1cm] 225 % 226 % \node[initial,state] (A) {$s_{0}$}; 227 % \node[state] (B) [below of=A] {$s_{1}$}; 228 % \node[node distance=1.5cm] (C) [below of=B] {$\ldots$}; 229 % \node[state,node distance=1.5cm] (D) [below of=C] {$s_{n-1}$}; 230 % \node[state] (E) [below of=D] {$s_T$}; 231 % 232 % \path (A) edge node {} (B) 233 % (B) edge node {} (C) 234 % (C) edge node {} (D) 235 % (D) edge node {} (E) 236 % (E) edge[loop right] node {} (E); 237 % 238 %\end{tikzpicture} 239 % 240 % \caption{\label{AKSNegCex} Kripke Structure of counterexample $\sigma$, $K(\sigma)$} 241 %\end{figure} 240 242 241 243 All the properties available for refinement are then model-checked on $K(\sigma)$. If the 242 244 property holds then the property will not discriminate the counterexample. 243 245 Hence this property is not a good candidate for refinement. 244 Therefore all properties that are satisfied are chosen to be246 Therefore all properties that are not satisfied are chosen to be 245 247 integrated in the next step of refinement. At this stage, we already have a 246 248 list of potential properties that definitely eliminates the current counterexample $\sigma$ and might converge the abstract model towards a model sufficient to verify the global property $\Phi$. … … 262 264 $AKS(\varphi)$, thus $\sigma$ is not a possible path in $AKS(\varphi)$ 263 265 otherwise $AKS(\varphi)\not\models \varphi$ that is not feasible due to AKS 264 definition and the composition with $M_i$ with $AKS(\varphi)$ will el eiminate266 definition and the composition with $M_i$ with $AKS(\varphi)$ will eliminate 265 267 $\sigma$. 266 268 \end{enumerate} 267 269 \end{proof} 268 270 269 The property at the top of the list (not yet selected and excluding the properties 270 which are satisfied by $K(\sigma)$) is selected to be integrated in the generation of $\widehat{M}_{i+1}$. 271 We ensure that our refinement respects the definition \ref{def:goodrefinement}. 272 Moreover, the time needed to build an AKS is neglectible and building the 273 next abstraction is just a parallel composition with the previous one. Thus the refinement 274 we propose is not time consuming. 271 The propose approach ensure that the refinement excludes the counter-example 272 and respects the definition \ref{def:goodrefinement}. 273 We will show in our experiments that first the time needed to build an AKS is 274 negligible and secondly the refinement converges rapidly. 275 %The property at the top of the list (not yet selected and excluding the properties 276 %which are satisfied by $K(\sigma)$) is selected to be integrated in the generation of $\widehat{M}_{i+1}$. 277 %We ensure that our refinement respects the definition \ref{def:goodrefinement}. 278 %Moreover, the time needed to build an AKS is neglectible and building the 279 %next abstraction is just a parallel composition with the previous one. Thus the refinement 280 % we propose is not time consuming. 275 281 276 282
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