Index: papers/FDL2012/abstraction_refinement.tex
===================================================================
--- papers/FDL2012/abstraction_refinement.tex	(revision 70)
+++ papers/FDL2012/abstraction_refinement.tex	(revision 76)
@@ -19,5 +19,5 @@
 
 Moreover, the refinement steps should be easy to compute and ensure a fast
-convergence by the minimizing the number of iteration of the CEGAR loop.
+convergence by minimizing the number of iterations of the CEGAR loop.
 
 Refinements 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.
@@ -29,6 +29,6 @@
 \subsection{Refinement by negation of the counterexample}
 
-The counterexample at a refinement step $i$, $\sigma$ is a path in the
-abstract model $\widehat{M}_i$ which dissatisfy $\Phi$.  In the counterexample given by the model-checker, the variables value in each states are boolean.
+The counterexample at a refinement step $i$, $\sigma$, is a path in the
+abstract model $\widehat{M}_i$ which dissatisfies $\Phi$.  In the counterexample given by the model-checker, the variable configuration in each state is boolean.
 The spurious counterexample $\sigma$ is defined such that :
 \begin{definition}
@@ -66,10 +66,7 @@
 \emph{Example}. Assume $AP = \{v_0,v_1,v_2\}$ and $\sigma = s_0 \rightarrow s_1, \ldots$ and $\widehat{L}(s_0) = \{\mathbf{f},\mathbf{f},\mathbf{f}\}$ meaning the configuration associated with $s_0$ assigns false to each variable. The negation of this configuration represents a set of seven concrete configurations which are covered by three (abstract) configurations: $\{\{\mathbf{t},\top,\top\},\{\mathbf{f},\mathbf{t},\top\},\{\mathbf{f},\mathbf{f},\mathbf{t}\}\}$.
 
-To build the final AKS representing all sequences but spurious counter-example $\sigma$, one replaces in $AKS(\overline{\sigma})$ each state $s"_i$ by $n = \mid AP_{\overline\sigma} \mid$ states $s"_i^k$ with $k \in [0..n-1]$ and assigns to each of them a label of $n$ variables $\{v_0, \ldots, v_{n-1}\}$ defined such that : ${L}(s"_i^k) = \{\forall l \in [0..k], v_l = \neg  {L}_{i}(s'_i)[v_l], \forall l \in [k+1..n-1], v_l = \top\}$.
-\TODO{connexion des $s"_i^k$}
+To build the final AKS representing all sequences but spurious counter-example $\sigma$, one replaces in $AKS(\overline{\sigma})$ each state $s"_i$ by $n = \mid AP_{\overline\sigma} \mid$ states $s"_i^k$ with $k \in [0..n-1]$ and assigns to each of them a label of $n$ variables $\{v_0, \ldots, v_{n-1}\}$ defined such that : ${L}(s"_i^k) = \{\forall l \in [0..k], v_l = \neg  {L}_{i}(s'_i)[v_l], \forall l \in [k+1..n-1], v_l = \top\}$. each state $s"_i^k$ is connected to predecessor and successor states as state $s"_i$ was.
 
-\TODO{Revoir notation def 3 pour f,t,top et introduire notation ${L}_{i}(s'_i)[v_l]$}
-
-This final AKS presents a number of states which is linear with product of the length of the counter-example and the number of atomic propositions. However, removing, at each refinement step, the spurious counter-example {\em only} induces a low convergence. Moreover, in some cases, this strategy may not converge: suppose that all sequences of the form $a.b^*.c$ are spurious counter-examples. At a given refinement step $i$, a particular counter example $\sigma_i = s_0 \rightarrow s_1 \rightarrow \ldots s_n$ with $L(s_0) = a, \forall k \in [1, n-1], L(s_k) = b, L(s_n) = c$. Removing this counter-example does not prevent from a new spurious counter-example at step $i+1$ :  $\sigma_{i+1} = s_0 \rightarrow s_1 \rightarrow \ldots s_{n+1}$ with $L(s_0) = a, \forall k \in [1, n], L(s_k) = b, L(s_{n+1}) = c$. The strategy consisting of elimination spurious counter-example {\em one by one} diverges in this case. However, we cannot eliminate all the sequences of the form $a.b^*.c$ in a unique refinement step since we do not a priori know if at least one of these sequence is executable in the concrete model.
+This final AKS presents a number of states in $\cal{O}(\mid\sigma\mid\times\mid AP\mid)$. However, removing, at each refinement step, the spurious counter-example {\em only} induces a low convergence. Moreover, in some cases, this strategy may not converge: suppose that all sequences of the form $a.b^*.c$ are spurious counter-examples (here $a$, $b$ and $c$ represent concrete state configurations). At a given refinement step $i$, a particular counter example $\sigma_i = s_0 \rightarrow s_1 \rightarrow \ldots s_n$ with $L(s_0) = a, \forall k \in [1, n-1], L(s_k) = b, L(s_n) = c$. Removing this counter-example does not prevent from a new spurious counter-example at step $i+1$ :  $\sigma_{i+1} = s_0 \rightarrow s_1 \rightarrow \ldots s_{n+1}$ with $L(s_0) = a, \forall k \in [1, n], L(s_k) = b, L(s_{n+1}) = c$. The strategy consisting of elimination spurious counter-example {\em one by one} diverges in this case. However, we cannot eliminate all the sequences of the form $a.b^*.c$ in a unique refinement step since we do not a priori know if at least one of these sequence is executable in the concrete model.
 
 From these considerations, we are interested in removing {\em sets of behaviors encompassing the spurious counter-example} but still guaranteeing an over-approximation of the set of tree-organized behaviors of the concrete model. The strengthening of the abstraction $\widehat{M}_i$ with the adjunction of AKS of already verified local CTL properties eliminates sets of behaviors and guarantees the over-approximation but does not guarantee the elimination of the counter example. We present in the following section a strategy to select sets of CTL properties eliminating the spurious counter example.
