Index: papers/FDL2012/abstraction_refinement.tex =================================================================== --- papers/FDL2012/abstraction_refinement.tex (revision 75) +++ 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. Index: papers/FDL2012/framework.tex =================================================================== --- papers/FDL2012/framework.tex (revision 75) +++ papers/FDL2012/framework.tex (revision 76) @@ -1,7 +1,7 @@ The model-checking technique we propose is based on the Counterexample-guided Abstraction Refinement (CEGAR) methodology \cite{clarke00cegar}. The overall -descritpion of our methodology is shown in figure \ref{cegar}. +description of our methodology is shown in figure \ref{cegar}. We take into account the structure of the system as a set of synchronous components, -each of which has been previously verified and a set of CTL properties is attached to each component. This set refers to the specification of the component. We would like to verify whether a concrete model, $M$ presumedly huge sized composed of several components, satisfies a global property $\Phi$. Due to state space combinatorial explosion phenomenon that occurs when verifying huge and complex systems, an abstraction or approximation of the concrete model has to be done in order to be able to verify the system with model-checking techniques. Instead of building the product of the concrete components, we replace each concrete component by an abstraction of its behavior derived from a subset of the CTL properties it satisfies. Each abstract component represents an over-approximation of the set of behaviors of its related concrete component \cite{braunstein07ctl_abstraction}. +each of which has been previously verified and a set of CTL properties is attached to each component. This set refers to the specification of the component. We would like to verify whether a concrete model, $M$ presumedly huge sized and composed of several components, satisfies a global ACTL property $\Phi$. Due to state space combinatorial explosion phenomenon that occurs when verifying huge and complex systems, an abstraction or approximation of the concrete model has to be done in order to be able to verify the system with model-checking techniques. Instead of building the product of the concrete components, we replace each concrete component by an abstraction of its behavior derived from a subset of the CTL properties it satisfies. Each abstract component represents an over-approximation of the set of behaviors of its related concrete component \cite{braunstein07ctl_abstraction}. %\subsection{Overall Description of our methodology} @@ -9,13 +9,7 @@ concrete model $M$, an abstraction of the concrete model $\widehat{M}$ is generated and tested in the model-checker. As the abstract model is an -over-approximation of the concrete model and we have restrained our -verification to ACTL properties only. As shown in \cite{clarke94model} if $\Phi$ holds on the the abstract model then we are certain that it holds in the concrete model as well. -However, if $\Phi$ doesn't hold in the abstract model then we can't conclude anything regarding the concrete model until the counterexample, -$\sigma$ given by the model-checker has been analyzed. +over-approximation of the concrete model and the global property $\Phi$ is in the ACTL fragment, if $\Phi$ holds on the the abstract model then it holds in the concrete model as well \cite{clarke94model}. However, if $\Phi$ does not hold in the abstract model then one cannot conclude anything regarding the concrete model until the counterexample, $\sigma$, given by the model-checker has been analyzed. +In this last case, the test of spurious counter-example is translated into a SAT problem \TODO{citer ref SAT spurious}. When a counterexample is proven to be spurious, the refinement phase occurs, injecting more preciseness into the (abstract) model to be analyzed. -In the case where model-checking failed, the counterexample given by the -model-checker has to be analysed. We use a SAT solver to check whether the counterexample is spurious or not. When a counterexample is proved to be spurious, we proceed to the refinement phase. -\TODO{ref du papier sur le spurious + petite explication sur SAT et -déroulement} %\bigskip %\begin{definition} @@ -45,8 +39,8 @@ \subsection{Concrete system definition} -As mention earlier, in our verification methodology, we have a concrete model which consists of several components and each component comes with its specification or more precisely, properties that hold in the component. Given a global property $\Phi$, the property to be verified by the composition of the concrete components model, an abstract model is generated by selecting some of the properties of the components which are relevant to $\varphi$. +As mention earlier, in our verification methodology, we have a concrete model which consists of several components and each component comes with its specification or more precisely, properties that hold in the component. Given a global property $\Phi$, the property to be verified by the composition of the concrete components model, an abstract model is generated by selecting some of the properties of the components which are relevant to $\Phi$. -The concrete system is a synchronous compositon of components, each of which +The concrete system is a synchronous composition of components, each of which described as a Moore machine. \begin{definition} @@ -70,5 +64,5 @@ Moore machine with a specification associated $\varphi_i = \{\varphi_i^1 \ldots \varphi_i^k\}$ Each $\varphi_i^j$ is a CTL$\setminus$X formula whose -propositions $AP$ belongs to $\{I_i\cup O_i\cup R_i\}$ . +propositions $AP$ belong to $\{I_i\cup O_i\cup R_i\}$ . \end{definition} @@ -76,10 +70,10 @@ Our abstraction consists in reducing the size of the representation model by -freeing some of its variables. The point is to determine the good set of variable +letting free some of its variables. The point is to determine the good set of variable to be freed and when to free them. We take advantage of the CTL specification of each component: a CTL property may be seen as a partial view of the tree of behaviors of its variables. All the variables not specified by the property can be freed. We introduced the Abstract Kripke Structure (AKS for short) which exactly -specifies when the variable of the prperty can be freed. +specifies when the variable of the property can be freed. The abstraction of a component is represented by an AKS, derived from a subset of the CTL properties the component satisfies. @@ -87,6 +81,6 @@ $\varphi$, simulates all execution trees whose initial state satisfies $\varphi$. In AKS($\varphi$), states are tagged with the truth values of -$\varphi$'s atomic propositions, among four truth values : inconsistent, -false, true and unknown (either true or false). +$\varphi$'s atomic propositions, among the four truth values of Belnap's logic \cite{belnap77}: inconsistent ($\bot$), +false ($\mathbf{f}$), true ($\mathbf{t}$) and unknown ($\top$)). States with inconsistent truth values are not represented since they refer to non possible assignments of the atomic propositions. A set of fairness constraints eliminates non-progress cycles. @@ -103,11 +97,5 @@ \item { $\widehat{S}$ : a finite set of states} \item { $\widehat{S}_0 \subseteq \widehat{S}$ : a set of initial states} -\item { $\widehat{L} : \widehat{S} \rightarrow 2^{Lit}$ : a labeling function which labels each state with the set of atomic propositions true in that state. Lit is a set of literals such that $Lit = AP \cup \{\bar{p} | p \in AP \}$. With this labeling definition, an atomic proposition in a state can have 4 different values as detailed below:} - \begin{itemize} - \item {$ p \notin \widehat{L}(s) \wedge \bar{p} \notin \widehat{L}(s) : p $\emph{ is \textbf{unknown} in} s } - \item {$ p \notin \widehat{L}(s) \wedge \bar{p} \in \widehat{L}(s) : p $\emph{ is \textbf{false} in} s} - \item {$ p \in \widehat{L}(s) \wedge \bar{p} \notin \widehat{L}(s) : p $\emph{ is \textbf{true} in} s} - \item {$ p \in \widehat{L}(s) \wedge \bar{p} \in \widehat{L}(s) : p $\emph{ is \textbf{inconsistent} in} s} - \end{itemize} +\item { $\widehat{L} : \widehat{S} \rightarrow \cal{B}^{\mid AP \mid}$ with $\cal{B} = \{\bot, \mathbf{f}, \mathbf{t},\top\}$: a labeling function which labels each state with configuration of current value of each atomic proposition.} \item { $\widehat{R} \subseteq \widehat{S} \times \widehat{S}$ : a transition relation where $ \forall s \in \widehat{S}, \exists s' \in \widehat{S}$ such that $(s,s') \in \widehat{R}$ } \item { $\widehat{F}$ : a set of fairness constraints (generalized B\"uchi @@ -117,14 +105,15 @@ %\bigskip +We denote by $\widehat{L}(s)$ the configuration of atomic propositions in state $s$ and by $\widehat{L}(s)[p]$ the projection of configuration $\widehat{L}(s)$ according to atomic proposition $p$. As the abstract model $\widehat{M}$ is generated from the conjunction of verified properties of the components in the concrete model $M$, it can be seen as the composition of the AKS of each property. The AKS composition has been defined in \cite{braunstein_phd07}; it extends -the classical synchrounous composition of Moore machine to deal with +the classical synchronous composition of Moore machine to deal with four-valued variables. %\bigskip \begin{definition} An \emph{Abstract model} $\widehat{M}$is obtained by -synchronous composition of components abstractions. -Let $C_j$ be a component of the concrete model $M$ and $\varphi_{j}^k$ is a CTL formula describing a satisfied property of component $C_j$. Let $AKS (\varphi_{C_j^k})$ the AKS generated from $\varphi_j^k$. We have $\forall j \in [1,n]$ and $\forall k \in [1,m]$: +synchronous composition of components abstractions. Let $n$ be the number of components in the model and $m$ be the number of selected verified properties of a component; +let $C_j$ be a component of the concrete model $M$ and $\varphi_{j}^k$ is a CTL formula describing a satisfied property of component $C_j$. Let $AKS (\varphi_{C_j^k})$ the AKS generated from $\varphi_j^k$. We have $\forall j \in [1,n]$ and $\forall k \in [1,m]$: \begin{itemize} @@ -133,22 +122,15 @@ \end{itemize} -\hspace*{3mm}with :\\ -\hspace*{5mm}- $ n \in \mathbb{N} $ : the number of components in the model \\ -\hspace*{5mm}- $ m \in \mathbb{N} $ : the number of selected verified properties of a component - \end{definition} -%\bigskip -The generation of an abstract model in the form of AKS from CTL formulas, -based on the works of Braunstein \cite{braunstein07ctl_abstraction}, -has been successfully implemented by Bara \cite{bara08abs_composant}. +The generation of an abstract model in the form of AKS from CTL formulas is described in \cite{braunstein07ctl_abstraction} and has been implemented (\cite{bara08abs_composant}). %\subsection{Characterization of AKS} -In an abstract kripke structure a state where a variable $p$ is {\it unknown} +In an abstract Kripke structure a state where a variable $p$ is {\it unknown} can simulate all states in which $p$ is either true or false. It -is concise representation of the set of more concrete states in which $p$ +is a concise representation of the set of more concrete states in which $p$ is either true or false. A state $s$ is said to be an \emph{abstract state} if one its variable $p$ is {\it unknown}. @@ -167,10 +149,10 @@ \begin{property} -Let A1 and A2 two abstractions such that A2 is obtained by concretizing one abstract variable of A1 (resp A1 is obtained by abstracting one variable in A2). Then A1 simulates A2. +Let $A_i$ and $A_j$ two abstractions such that $A_j$ is obtained by concretizing one abstract variable of $A_i$ (resp $A_i$ is obtained by abstracting one variable in $A_j$). Then $A_i$ simulates $A_j$, denoted by $A_j \sqsubseteq A_i$. \end{property} \begin{proof} As the concretization of state reduces the set of concrete configuration the abstract state represents but does not affect the transition relation of the -AKS. The unroll execution tree of A2 is a subtree of the one of A1. Then A1 simulates A2. +AKS. The unroll execution tree of $A_j$ is a subtree of the one of $A_i$. Then $A_i$ simulates $A_j$. \end{proof} Index: papers/FDL2012/myBib.bib =================================================================== --- papers/FDL2012/myBib.bib (revision 75) +++ papers/FDL2012/myBib.bib (revision 76) @@ -123,5 +123,5 @@ title = "{Efficient Generation of Counterexamples and Witnesses in Symbolic Model Checking}", journal = {32nd ACM/IEEE Design Automation Conference}, - year = 1995 + year = 1995 } @@ -132,4 +132,12 @@ pages = {80-89}, year = 2007 +} + +@article{ belnap77, + author = "N. Belnap", + title = "{A useful four-valued logic}", + journal = {Modern Uses of Multiple-Valued Logic}, + pages = {8-37}, + year = 1977 } @@ -342,5 +350,5 @@ title = "Using {BibTeX}. {D}ocumentation for general {B}ib{\TeX} users", year = 1988, - month = jan + month = jan } @@ -355,3 +363,2 @@ year = 2009 } -