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1 & 0 % p6 \end{pmatrix}\\ M_0 &=& \begin{pmatrix} 0\\ 0\\ 0\\ 1\\ 2\\ 0\end{pmatrix} - \end{eqnarray*} + \end{eqnarray*} \end{correction} \question Donner la signification des six places du réseau $R$. \marks{2} @@ -177,20 +177,39 @@ \end{questions} + %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%% AUTOMATES %%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% - +\newpage \exercice{Automates} \begin{questions} \question \textbf{Équivalence pour des systèmes : } Rappeler, à l'aide d'un exemple, en quoi l'équivalence de traces n'est pas toujours suffisante pour comparer deux systèmes.\\ \emph{Réponse de 5 lignes maximum + un dessin éventuellement} -\question \textbf{Automates temporisés : } Considérons l'automate temporisé de la figure \ref{automateT} sur l'alphabet (ensemble de labels) - $\Sigma=\{a_1,a_2,a_3\}$, comportant une horloge $x$. Donner, sans justification, deux exemples de traces temporisées reconnues par cet - automate. (Donner des séquences sur $\Sigma \times \mathbb{R}_+$, c'est à dire des séquences de paires (\texttt{label}, - \texttt{durée d'attente})). +\marks{2} + +\begin{correction}{.8} + L'équivalence de traces ne prend pas en compte le branchement, c'est + à dire la prise en compte de différentes manières de poursuivre + l'exécution depuis un état donné. Dans l'exemple suivant, les deux + automates $\mathcal{A}_1$ et $\mathcal{A}_2$ sont équivalents au + sens de l'équivalence de traces, mais mis en présence de $\mathcal{B}$, + $\mathcal{A}_1||\mathcal{B}$ ne possède pas de deadlock, alors que + $\mathcal{A}_2||\mathcal{B}$ si. +\begin{center} +\includegraphics[width=6cm]{equiv_traces} +\end{center} +\end{correction} + +\question \textbf{Automates temporisés : } Considérons l'automate +temporisé de la figure \ref{automateT} sur l'alphabet (ensemble de +labels) $\Sigma=\{a_1,a_2,a_3\}$, comportant une horloge $x$. Donner, +sans justification, deux exemples de traces temporisées reconnues par +cet automate. (Donner des séquences sur $\Sigma \times \mathbb{R}_+$, +c'est à dire des séquences de paires (\texttt{label}, \texttt{durée + d'attente})). \marks{2} \begin{figure}[!h] \begin{center} \includegraphics[width=6cm]{automate_temp} @@ -198,70 +217,68 @@ \caption{Automate temporisé.} \label{automateT} \end{figure} -\question La sémantique d'un automate temporisé est donnée par un système de transitions temporisé, dont l'espace d'état est infini. - Rappeler les grandes idées de la construction d'un système de transitions fini (qui permet donc de vérifier certaines propriétés) + +\begin{correction}{.8} +Deux traces acceptées par l'automate:\\ +$(a_1,3);(a_3,1.2)$\\ +$(a_1,4.1);(a_2,3.3);(a_1,3.1);(a_3,1.5)$ +\end{correction} + + \question La sémantique d'un automate temporisé est donnée par + un système de transitions temporisé, dont l'espace d'état est + infini. Rappeler les grandes idées de la construction d'un + système de transitions fini (qui permet donc de vérifier + certaines propriétés) <<équivalent>> au système de transition temporisé infini. Dans quel sens est-il équivalent?\\ - \emph{Réponse de 10 lignes maximum } + \emph{Réponse de 10 lignes maximum } \marks{3} + +\begin{correction}{.8} +L'ensemble des configurations d'un automate temporisé est infini à +cause du domaine des horloges ($\mathbb{R}_+$). L'idée est de +construire un ensemble fini de régions pour chaque horloge, telle que +quelle que soit la valeur d'une horloge dans une région, l'état du +système soit inchangé (les mêmes invariants et gardes sont +satisfaits). Une configuration du système abstrait peut alors être donnée par +un couple (localité, région) plutôt que (localité, valeur des +horloges). Ce système abstrait est en relation de bisimulation +temps-abstraite avec le système d'origine, ce qui garantit qu'il +satisfait les mêmes propriétés. +\end{correction} + \question Soient $T=(S,S_I,S_F,\rhd )$ et $T'=(S',S'_I,S'_F,\rhd')$ deux systèmes de transitions temporisés. Si $\mathcal{R}$ est une relation de bisimulation temps-abstraite qui respecte l'état - initial et l'état final, montrer que \linebreak Untime(L(T))=Untime(L(T')). + initial et l'état final, montrer que \linebreak + Untime(L(T))=Untime(L(T')). \marks{3} \end{questions} -\subsection*{Rappels de cours} -\begin{definition}[Système de transitions temporisé] -Un système de transition temporisé sur un alphabet $\Sigma$ est un quadruplet -$T=(S,S_I,S_F,\rhd)$, où -\begin{itemize} -\item $S$ est l'ensemble des états -\item $S_I \subseteq S$ (resp. $S_F\subseteq S$) est l'ensemble des états initiaux (resp. finaux) -\item $\rhd \subseteq S \times (\mathbb{R}_+ \times \Sigma) \times S$ - est la fonction de transition, $s_1 \overset{t,a}{\rhd}s_2$ exprime - qu'on peut passer de l'état $s_1$ à l'état $s_2$, en reconnaissant - le label $a\in \Sigma$, après avoir attendu $t\in \mathbb{R}^+$ - unités de temps. -\end{itemize} -\end{definition} - -\begin{definition}[Bisimulation temps-abstraite] -Soient $T=(S,S_I,S_F,\rhd)$ et $T'=(S',S_I',S_F',\rhd')$ deux systèmes -de transitions temporisés sur $\Sigma$. -$\mathcal{R} \subseteq S\times S'$ est une relation de bisimulation -temps-abstraite ssi elle est \textbf{symétrique} et : - -\begin{tabular}{ll} - si &$(s_1,s_1')\in \mathcal{R}$ et $s_1 \overset{t,a}{\rhd}s_2$ alors - $\exists s_2'\in S' \ \exists t'\in \mathbb{R}_+$ tels que\\ - &$(s_2,s_2')\in \mathcal{R}$ et $s_1'\overset{t',a}{\rhd}s_2'$\\ -\end{tabular} - -On dit que $\mathcal{R}$ \textbf{respecte les états initiaux} (resp. \textbf{finaux}) si \\ -$ -\begin{array}{llcl} -\forall s\in S & s\in S_I (\text{resp.} S_F)& \quad \text{si et - seulement si} \quad & \exists s'\in S_I' (\text{resp.} S_F') \text{ - tel que } s\mathcal{R}s'\\ -\forall s'\in S' & s'\in S'_I (\text{resp.} S'_F)& \quad \text{si et - seulement si} \quad & \exists s\in S_I (\text{resp.} S_F) \text{ - tel que } s'\mathcal{R}s\\ -\end{array} -$ -\end{definition} - -\begin{definition}[Langage reconnu] -Étant donné système de transition temporisé $T=(S,S_I,S_F,\rhd)$, le -langage $L(T)$ reconnu -par $T$ est l'ensemble des séquences sur $\Sigma\times \mathbb{R}_+$ $<(a_0,t_0), (a_1,t_1), \ldots, -(a_n,t_n)>$ telles qu'il existe une séquence d'états -$$ qui vérifie -\begin{itemize} -\item $s_0\in S_I$, $s_{n+1}\in S_F$ -\item $\forall i\in [0..n] \quad s_i \overset{a_i,t_i}{\rhd}s_{i+1}$ -\end{itemize} - - -Le langage $Untime(L(T))$ est l'ensemble des séquences sur $\Sigma$ -$$ telles qu'il existe une séquence sur $\mathbb{R}_+$ -$$ qui vérifie $<(a_0,t_0),\ldots ,(a_n,t_n)>\in L(T)$. -\end{definition} +\begin{correction}{.8} +Soit $(a_1,\ldots, a_n)\in Untime(L(T))$. +$\exists (t_1,\ldots , t_n), \exists (s_1,\ldots,s_{n+1})$ tels que +$s_1\overset{t_1,a_1}{\rhd}s_2 \overset{t_2,a_2}{\rhd} \ldots +\overset{t_n,a_n}{\rhd}s_{n+1}$ et $s_{n+1}\in S_F$. + +Montrons par récurrence sur $k$ que $\forall k\in [|1..n+1|] \exists +t'_k,s'_k,s'_{k+1}$ tels que $s_k\mathcal{R} s'_k$, +$s_{k+1}\mathcal{R} s'_{k+1}$, et $s'_k\overset{t'_k,a_k}{\rhd}s'_{k+1}$. + +$\bullet$ $\mathcal{R}$ étant une relation de bisimulation, l'hypothèse est +trivialement vérifiée au rang 1. + +$\bullet$ Supposons l'hypothèse de récurrence vraie au rang k. Alors, on a +$s_{k+1}\mathcal{R} s'_{k+1}$. Par ailleurs, +$s_{k+1}\overset{t_{k+1},a_{k+1}}{\rhd}s_{k+2}$. Puisque $\mathcal{R}$ est une +bisimulation temps-abstraite, $\exists t'_{k+1}, s'_{k+1}$ tels que +$s_{k+2}\mathcal{R}s'_{k+2}$ et $s'_{k+1}\overset{t'_{k+1},a_{k+1}}{\rhd}s'_{k+2}$. +Donc l'hypothèse de récurrence est vraie au rang k+1. + +On a donc prouvé qu'il existe $(s'_1, \ldots, s'_{n+1})$ et +$(t'_1,\ldots , t'_n)$ et tels que $s'_1\overset{t'_1,a_1}{\rhd}s'_2 +\overset{t'_2,a_2}{\rhd} \ldots \overset{t'_n,a_n}{\rhd}s'_{n+1}$. De +plus, puisque $\mathcal{R}$ respecte les états initiaux et finaux, +$s'_1\in S'_I$, $s'_{n+1}\in S'_F$. Donc $(a_1,\ldots, a_n)\in +Untime(L(T'))$. + +De manière symétrique, on peut prouver que $Untime(L(T')) \subseteq Untime(L(T))$. +\end{correction} \end{document} diff --git a/figures/automates/equiv_traces.pdf b/figures/automates/equiv_traces.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7528a5ad5771365b743c3037d22d324e3ceecb04 GIT binary patch literal 8210 zcmb_h30PCt@=rh@jfho*E-G9%0xI0xtOQxgjzT~d1q6yAxnK$;AqlG}h$xUNf*arl zf;$yu5%FQsA{Fbt)CCtrYb&+d;)32_T>j^VMFih_{lCxn`Ek~nIWzM+GiUC!xo#fT zhz*N6?d0Qv3Mw0-L(-`4sP^`-4=RaKE`m5C#i0cd{wXC6i(jzRetn^ zm+dy;kJpSf?{S+`(Ha?D*LLHgQ;$RBt_JhVuX{Qko*7ph=X1Kwdah{+%=b49qyG4s zm3rTwH~pvI#IIss#lKzr%)8{>=E}03mJ7D0Qa7!iH1D>>_>l0DrkbUBf&9e*5W`Vef^S${(fL5dIhE;`JHvJ-n*JvjB~vkXPjEAAJAIny^c4omj2M| zLC^Mmw|{h7)1aYoWb|6V z-rsHr|6Ze$wMVC}%YMw$`uX;U?V3%!9fR!sIK>NM<6V5WHsxmdJrDJtV7hthRr@xh zNcSx;d7p7qOvlL!zf(^b1n5uM5kxN6*VB6k1z249Z0@RZ!I%rizm1!2wYAVS2ClgK zOZxn6CVE%TdKW}~eN_LVh5G!_f=I8P_1BCa)}D5}w1(t3)p6UOUyZoa{L)RQpnmQ( zLGdi1$?pea8GjqNwnc?MHr+h!xaoDfEot|!2&9$gv+mc`oZ4k3-K5~@?XbAnvy;8O zCEWj9heYjX^#m7OZ=h{) zE#84sTh?r(BnF>8TGzT$GJf*RD@&}uwttQ$e7WsZ{PyoHGB5o&-Q~&FktV0~PFDNw znY91qM7;+lPwi;5s`v#aw_jzuut(keQg3FC-M;+5A2*#ek>0On<`|iGa`hUV&XwXR zE!V9(Ue`s9Dma(?w98BU?%?#p2H$PFJ&C>hrgQx^(S)Q27w<{EW}9Dm+_PS+PNTBl z?F(rUK96`E;pau09K3D1Y|@_0gUs!^MHWBj@Bj6t>3-wa*S>XcmgWoSKhE+#tE*H` z-(OY!-Bic=juRP=9$YlN*`Q$Uw`q4bNyDAWv+y(DmRsad|F+x`uR4FG-mIqGj{Mu| z+FSLFe(O@IbzHl3k4@ZKPrDeh)qc6DfhgVIfU77qm(a`?N zjfI9=H(V2V%BJ2=KRL>GLHdSeYHnm%XfOT<6R=B>-`eq=PO z%7xq&5$+xPsM)7>!{-j|U!K16{My9%cQcQBr9+9;vC{1HTdTfE_pkA6vUn0vaXmYG zVR-YX30sQ8%0-XG{Iw;EKdXv}r~ND+z0~@8mGiamDJsr1-226(7yKsoexsCMm`(5g{_dWO#RW@)Z>RpNapNs!;L+OsetTvujhhn2M9vd%Pe4vIY^LZPVSw%AU{Hyw}!n`iCvtxfK*m?1%GP9>XX-;oOXaDNq zhe||!;Rm#ckVOP4;xEEsGpT)JtoPw8C{>2Sa|JOd%tu9Hfr~T+3ZsK*7Q%uM9xH+h zCgrF^2{DO)M}-4Wg)~tvL=_+u&Xr4rfv7SJ2Eh`7gU}QuSni&p^bAyja0nsV7)gaa zq+mP3)$Rw;5q1TAKv4k^Q8j=HhWf-JSoB6&Kn=(RX$4b2u(w#GfWoMR*a6fC2vA5Q zNDkVH3~cS&=_*hP;-oSCB>JT1qe_7Y%s>pr5Z;7%eGy)(l7VATVZXV22yt}m6H`B3hQ9Nv9%&hQX*U)s|=s+CIoKE&>Hgs1Ba( z@8`O}XND^uP|+&W4OIx`VwqAZhY+m}wP)5&0d0%r3Z?5JfgED-U>`wWn@xv<#UkY* zMHm9Q zP=r2AJgwe8E*^*dUOWzGKs+wb7UHnE|BHBpR)>n`@9gar$k!@wxO9Zq1bke&50r-t zNXMqrp}znBgM@^&hYINF;vM1|3LF;lAhsZj9awgGGlp9Ciee3&$vc3E3Q5D(tKf5|;;<0Js;(yijq> zqP{Lco|vOk;n@JIVxhAnCJrUKfl4%fKA=E_`C^3vIH>k;5R<_HM1W_5JI4~1N>Est zEVa_g2J&!}2_Zw#G)UjQnfl;F9q<|K=C1vjJ&VqAbR0H4A_rdH|)yLXuWj?n3JazqRC6J%Jca!~(~;n6Tv|LD6_Fpa4^ zpsq)C$8(sT(eW%WNa>&K_pk5-XVIoPTu(Di#otZsIYsi<8_$&9@t4jlUU%}) zKAew#4W+j??OP{wH#(AVvirrexpg*6-o1D>>k+lp0X`PfE_{3MdArBK^;O#|c{^$v zcUtY4)v)<~1AMF2o#qRJ?5*?2@q2IDyKIlvjlZov_jGehL<=iwadJo2&bs;Vfs*2m zD>vG&PU3dGZr&Qx$TNGKe2I(64WAXgn}{D-Rj)RfGAr^%R@uvuOX00A95SZg>iS9i zc*f!LvNH)aB^NsHZE;$W(|I<|>|1?((zl|rEyQi~VTeQC2afk({+Ra;zpMJ(*P2a- zh>M5IV0_$x!Ngz()^__H7;OGGE{q_&K_2T2yL{lST5u1#U5S3*?eNdNmCa)e@>a6D zPU9OgX@m}mOeP!j-pNFBs(L(GJwZo3Q5OsCsqKl@!whvX!x1~wN5|saF*sv1Zu2=w zmrVBX33HXoQsr8dTDn?+6oAKtoa0fsSSXM{e1UQi8ZS_SP8<{{6^c=1stx2E7Y79p zI}{Me4p4a#D(dCzDvg&V67OR^co)khBoax-*@Z-!OqxQ{1rm~UNJq#^jNa**rI129 z@zKZCNo8d9FL!m-ojA{=(5E<}Fr_G>FsT8f9JOOKmqeDs%KivcxUALmy! 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