From 9ed4f504a9b9dd9704b62305d743e177a9ae6f72 Mon Sep 17 00:00:00 2001 From: couchot Date: Tue, 14 Apr 2015 10:45:34 +0200 Subject: [PATCH] reprise chap 1 --- annexePreuveMixage.tex | 170 +++++++++++++++++++++-------------------- main.pdf | Bin 1826830 -> 1828033 bytes main.tex | 22 ++++-- mixage.tex | 134 +++++++++++++------------------- sdd.tex | 79 ++++++++++++------- 5 files changed, 206 insertions(+), 199 deletions(-) diff --git a/annexePreuveMixage.tex b/annexePreuveMixage.tex index 4757b0b..d4087d4 100644 --- a/annexePreuveMixage.tex +++ b/annexePreuveMixage.tex @@ -1,103 +1,105 @@ - -Let us introduce an ordering $\preceq$ between classes. Formally, \class{p} -$\preceq$ \class{q} if there exists a path of length $\alpha$ -($0<\alpha<|\mathcal{K}|$) from an element of \class{p} to an element of -\class{q}. One can remark that if \class{p}$\preceq$\class{q}, then it is not -possible to also have \class{q}$\preceq$\class{p}. - -% \begin{lemma} -% The relation $ \preceq$ is a partial order -% \begin{Proof} -% Reflexivity is established since for any $p_0 \in [p]$ there exists a path -% of length 0 from $p_0$ to $p_0$. -% For antisymmetry, suppose there exists two classes $[p]$ and -% $[q]$ such that $[p] \preceq [q]$ and $[q] \preceq [p]$. -% There exists then $p_0,p'_0 \in [p]$, $q_0, q'_0 \in [q]$ with paths -% from $p_0$ to $q_0$ and from $q'_0$ to $p_0$. -% Thus, elements $p_0$, $p'_0$, $q_0$ and $q'_0$ belong to the same -% strongly connected component and then -% $[p]$ = $[q]$. Transitivity is obviously established. -% \end{Proof} -% \end{lemma} +Introduisons tout d'abord une relation d'ordre +$\preceq$ entre les classes d'équivalences. +Formellement, \class{p} +$\preceq$ \class{q} +s'il existe un chemin de longueur $\alpha$ +($0<\alpha<|\mathcal{K}|$) entre un élément le la classe +\class{p} vers un élément de +\class{q}. +On remarque que si la \class{p}$\preceq$\class{q}, +il n'est alors pas possible que \class{q}$\preceq$\class{p}. \begin{lemma} - There exists a renaming process method that assigns new identifier numbers to - processes $i\in$ \class{p} and $j \in$ \class{q} s.t. $i \le j$ provided + Il existe un processus de renommage qui effecte un nouvel identifiant aux + élément $i\in$ \class{p} et $j \in$ \class{q} tel que + $i \le j$ si et seulement si \class{p} $\preceq$ \class{q}. \begin{Proof} - % We first define a renaming process method and later show that it fulfills - % the requirements. - First of all, let \class{p_1}, \ldots, \class{p_l} be classes of - $n_1$,\ldots, $n_l$ elements respectively that do not depend on other - classes. Elements of \class{p_1} are renamed by $1$, \ldots, $n_1$ and - elements of \class{p_i}, $2 \le i \le l$ are renamed by $1+ - \Sigma_{k=1}^{i-1} n_k$, \ldots, $\Sigma_{k=1}^{i} n_k$. We now consider - the classes \class{p_1}, \ldots, \class{p_{l'}} whose elements have been - renamed and let $m$ be the maximum index of elements of \class{p_1}, \ldots, - \class{p_{l'}}. Given another class \class{p} that exclusively depends on - some \class{p_i}, $1 \le i \le l'$ and that contains $k$ elements. Elements - of \class{p} are then renamed by $m+1$, \ldots, $m+k$. -% In the end for remaining classes, i.e. those which do not depend on anything, -% elements are arbitrarily numbered. - This renaming process method has then been applied on $l'+1$ classes. It - ends since it decreases the number of elements to assign a number. Notice - that this process is not determinist. + Tout d'abord, soit \class{p_1}, \ldots, \class{p_l} des classes + contenant respectivement $n_1$,\ldots, $n_l$ éléments respectively + qui ne dépendent d'aucune autre classe. + Les éléments de \class{p_1} sont renommés par $1$, \ldots, $n_1$, + les elements de \class{p_i}, $2 \le i \le l$ sont renommés par + $1+ + \Sigma_{k=1}^{i-1} n_k$, \ldots, $\Sigma_{k=1}^{i} n_k$. + On considère maintenant les classes \class{p_1}, \ldots, \class{p_{l'}} + dont les éléments ont été renommés et soit + $m$ le plus grand indice des elements de \class{p_1}, \ldots, + \class{p_{l'}}. + Soit une autre classe \class{p} qui dépend exclusivement d'une classe + \class{p_i}, $1 \le i \le l'$ et qui contient $k$ elements. + Les éléments de \class{p} sont renommés par $m+1$, \ldots, $m+k$. + Ce processus a été appliqué sur $l'+1$ classes. Il se termine + puisqu'il diminue le nombre d'elements auquel il reste + à affecter un numéro. - We are then ready to prove that this renaming method verifies property - expressed in the lemma. The proof is done by induction on the length $l$ of - the longest dependency path between classes. + Il reste à montrer que cette méthode de renommage vérifie la propriété + énoncée dans le lemme. + Cette preuve se fait par induction sur la taille $l$ + du plus grand chemin de dépendance entre les classes. - First, if \class{p} $\preceq$ \class{q} and \class{q} immediately depends on - \class{p}, \textit{i.e.} the longest path from elements of \class{p} to - elements of \class{q} has length 1. Due to the renaming method, all element - numbers of \class{q} are greater than the ones of \class{p} and the result - is established. Let \class{p} and \class{q} s.t. the longest dependency - path from \class{p} to \class{q} has length $l+1$. Then there exists some - \class{q'} s.t. \class{q} immediately depends on \class{q'} and the longest - dependency path from \class{p} and \class{q'} has length $l$. We have then - \class{q'} $\preceq$ \class{q} and then for all $k$, $j$ s.t. $k \in$ - \class{q'} and $j \in$ \class{q}, $k \le j$. By induction hypothesis - \class{p} $\preceq$ \class{q'} and then for all $i$, $k$ s.t. $i \in$ - \class{p} and $k \in$ \class{q'}, $i \le k$ and the result is established. + Tout d'abord, si \class{p} $\preceq$ \class{q} et \class{q} + dépend immédiatement de + \class{p}, \textit{i.e.} + le chemin le plus long entre les éléments de \class{p} et les + elements de \class{q} est de longueur 1. + En raison de la méthode renommage, chaque numéro d'élément + \class{q} est plus grand que tous ceux de \class{p} et la preuve est + établie. + Soit \class{p} et \class{q} tels que le plus long chemin de dépendance + entre \class{p} et \class{q} a une longueur de $l+1$. + Il existe alors une classe + \class{q'} telle que \class{q} + dépend immédiatement de \class{q'} et le chemin de dépendance le + plus long entre \class{p} et \class{q'} a pour longueur $l$. + On a ainsi + \class{q'} $\preceq$ \class{q} + et pour tout $k$, $j$ tels que $k \in$ + \class{q'} et $j \in$ \class{q}, $k \le j$. + Par hypothèse d'induction, + \class{p} $\preceq$ \class{q'} et pour chaque $i$, $k$ tels que $i \in$ + \class{p} et $k \in$ \class{q'}, $i \le k$ + et le résultat est établi. \end{Proof} \end{lemma} -It can be noticed that the renaming process is inspired from \emph{graphs by - layer} of Golès and Salinas~\cite{GS08}. It ensures that identifier numbers of -a layer are greater than all the identifier numbers of any lower layers. +On peut remarquer que ce processus de renommage est inspiré des \emph{graphes + par couches } de Golès et Salinas~\cite{GS08}. -\begin{xpl} - We have \class{1} $=\{1,2\}$, \class{3} $=\{3\}$ and \class{4} $=\{4,5\}$. - Processes numbers are already compliant with the order $\preceq$. -\end{xpl} +% \begin{xpl} +% We have \class{1} $=\{1,2\}$, \class{3} $=\{3\}$ and \class{4} $=\{4,5\}$. +% Processes numbers are already compliant with the order $\preceq$. +% \end{xpl} \begin{Proof}[of Theorem~\ref{th:cvg}] - % Since $\preceq$ is a partial order, [[JFC citer un theoreme qui dit cela ou - % le prouver]] there exists a renaming process that assigns new identifier - % number to processes $i\in [p]$ and $j \in [q]$ s.t. $i \le j$ provided $[p] - % \preceq [q]$. - % % $i \in [p]$ and $j \in [q]$. - - The rest of the proof is done by induction on the class index. Let us - consider the first class \class{b_1} with $n_1$ elements \textit{i.e.} the - class with the smallest identifiers. + Le reste de la preuve est fait par induction sur le numéro de classe. + Considérons la première classe \class{b_1} de $n_1$ éléments + \textit{i.e.} la classe avec le plus petit identifiant. - By theorem hypotheses, %following the strategy $(S^t)$, - synchronous iterations converge to the fixed point in a finite number of - iterations. %pseudo periods. % [[JFC : borner m1/n1]]. - So, all \emph{source classes} (independant from any other class) will also - converge in mixed mode. Let us suppose now that mixed iterations with uniform - delays converge for classes \class{b_1}, \ldots, \class{b_k} in a time $t_k$. - By construction, \class{b_{k+1}} only depends on some \class{b_1}, \ldots, - \class{b_k} and \class{b_{k+1}}. There exists then a sufficiently large time - $t_0$ s.t. $D^{t_0}_{p_{k+1}p_j}$ is greater or equal to $t_k$ for any - $p_{k+1} \in$ \class{b_{k+1}} and $p_j \in$ \class{b_j}, $1 \le j \le k$. + D'après les hypothèses du théorème, %following the strategy $(S^t)$, + les itérations synchrones convergent vers un point fixe en un nombre + fini d'itérations. %pseudo periods. % [[JFC : borner m1/n1]]. + Ainsi toutes les \emph{classes sources} + (indépendantes de toutes les autres classes) vont aussi converger + dans le mode mixe. + On peut ainsi supposer que le mode d'itération mixe avec délais + uniformes fait converger les classes \class{b_1}, \ldots, \class{b_k} + en un temps $t_k$. + Par construction, la classe \class{b_{k+1}} dépend uniquement + de certaines classes de \class{b_1}, \ldots, + \class{b_k} et éventuellement d'elle-même. + Il existe un nombre d'iteration suffisamment grand + $t_0$ tel que $D^{t_0}_{p_{k+1}p_j}$ est suppérieur ou égal à $t_k$ + pour chaque + $p_{k+1} \in$ \class{b_{k+1}} et $p_j \in$ \class{b_j}, $1 \le j \le k$. - We are then left with synchronous iterations of elements of \class{b_{k+1}} - starting with configurations where all the elements of \class{b_j}, $1 \le j - \le k$, have constant values. By theorem hypotheses, it converges. + Il nous reste donc des itérations synchronous entre les + elements of \class{b_{k+1}} en démarant dans des configurations + où tous les éléments de \class{b_j}, $1 \le j + \le k$, on des valeurs constantes. + D'après les hypothèses du théorème, cela converge. \end{Proof} diff --git a/main.pdf b/main.pdf index 5ee54b773f66abdba39983cc386dbb88660d4309..b01f58d828c72e1efe2a214393bc4e2eba5c88c5 100644 GIT binary patch delta 147079 zcmZshV{jlr)UIP&8{4+6jkU3{;l>+lV%xTD+qUg&w6S&fTGllbW7K$QaD4J{Xz+Ct?kt(bAmr zuXG_}7~>UNk3~n5IV84HL|Weao?bW`WD+bO$AIsAh%P}%5%Z)Tt9S&{%#Q%7M{WV+zcS6b2jWaPAm>fHYD&2y zj9DF#nB`})_X10Ul0DT&%W5JCW*HlH1QIJ92LfIe>%Ra=NnS}B3;~v*G#ihWz#)kt zg=wg3H9$662bBebz@z3t&QA4B`oJTW6D!z|1ZpbD7bX@DrWR7YPz)Cc-1FZT2azof zPwPFzaL2+*=<=O`!SgqdCuW9CmJKPWMH(8QyaoZKj1^C+h5e~aA&4d&t|mfm1>Xm& z8p{edS}G3HD`J&|2?O00l$&M(BULZucaTe>viQ>~&3d;`mz*YrN&~WAUEdMMaFCD3 z<2I0zEj?51iZ(rSevTHX*r;<8SImiy)}Oy8{%#<#qsbd5ALLwAt*O+>%#;On0WQ zYnV`RWtyT(f6 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#&L(hr#)obtC=-saX9vnVx=#O9;!#&CCWv LPA;Y(4)cEiBpTN+ diff --git a/main.tex b/main.tex index c16595e..ce52fff 100644 --- a/main.tex +++ b/main.tex @@ -183,19 +183,29 @@ de l'asynchronisme en terme de vitesse de convergence. \chapter{Preuves sur les SDD} -\section{Théorème~\ref{th:Adrien}}\label{anx:sccg} -\input{annexesccg} +\section{Convergence du mode mixe}\label{anx:mix} +\input{annexePreuveMixage} + + +\section{Correction et complétude de la + vérification de convergence par SPIN}\label{anx:promela} +\input{annexePromelaProof} + + + +\chapter{Preuves sur les systèmes chaotiques} + \section{Continuité de $G_f$ dans $(\mathcal{X},d)$}\label{anx:cont} \input{annexecontinuite.tex} -\section{Convergence du mode mixe}\label{anx:mix} -\input{annexePreuveMixage} + +\section{Théorème~\ref{th:Adrien}}\label{anx:sccg} +\input{annexesccg} + -\section{Correction et complétude de la vérification de convergence par SPIN}\label{anx:promela} -\input{annexePromelaProof} \backmatter diff --git a/mixage.tex b/mixage.tex index d5f1981..a71a36d 100644 --- a/mixage.tex +++ b/mixage.tex @@ -3,7 +3,7 @@ Pour être exécuté, le mode des itérations généralisées nécessite que chaque élément connaisse la valeur de chaque autre élément dont il dépend. Pratiquement, cela se réalise en diffusant les valeurs des éléments de -proche en proche à tous les composants. +proche en proche à tous les composants avant chaque itération. Dans le mode généralisé \emph{asynchrone}, le composant n'attend pas: il met à jour sa valeur avec les dernières valeurs dont il dispose, même si celles-ci @@ -54,7 +54,7 @@ souvent. Dans le mode asynchrone, a chaque itération $t$, chaque composant peut mettre à jour son état en fonction des dernières valeurs qu'il connaît des autre composants. -Obtenir où non les valeurs les plus à jours dépend du temps de calcul et +Obtenir ou non les valeurs les plus à jours dépend du temps de calcul et du temps d'acheminement de celles-ci. On parle de latence, de délai. Formalisons le mode les itérations asynchrone. @@ -65,7 +65,7 @@ dont chaque élément $D_{ij}^{t}$ représente la date (inférieure ou égale à disponible au composant $i$. On considère que le délai entre l'émission par $j$ et la réception par $i$, défini par $\delta_{ij}^t = t - D_{ij}^{t}$ est borné par une constante $\delta_0$ pour tous les $i$, $j$. -Le \emph{mode des itérations généralisées sans attente} +Le \emph{mode des itérations généralisées asynchrone} est défini pour chaque $i \in \{1,\ldots,n\}$ et chaque $t=0,1,2,...$ par: @@ -74,7 +74,7 @@ est défini pour chaque $i x^{t+1}_i= \left\{ \begin{array}{l} f_i( x_1^{D_{i1}^t},\ldots, x_{n}^{D_{i{n}}^t}) - \textrm{ if } \textit{bin}(s^t)[i] = 1\\ + \textrm{ si } \textit{bin}(s^t)[i] = 1\\ x^{t}_i \textrm{ sinon } \end{array} \right. @@ -99,20 +99,20 @@ $(D^{(t)})^{t \in \Nats}$, alors les itérations de $f$ sont \emph{universellement convergentes}. -\subsection{Exemple jouet} -On considère cinq éléments prenant à valeurs dans $\Bool$. +\begin{xpl} +On considère cinq éléments à valeurs dans $\Bool$. Une configuration dans $\Bool^5$ est représentée par un entier entre -0 et 31. La~\Fig{fig:mix:map} donne la fonction définissant la dynamique du -système. La~\Fig{fig:mix:xplgraph} donne le graphe d'interaction associé à cette fonction. +0 et 31. +La~\Fig{fig:mix:map} donne la fonction définissant la dynamique du +système et son graphe d'interaction. On note que le graphe d'interaction contient cinq cycles. Les résultats connus~\cite{Bah00} de conditions suffisantes établissant la convergence du système pour les itérations généralisées sont basés sur l'absence de cycles. Ils ne peuvent donc pas être appliqués ici. \begin{figure}[ht] -\begin{minipage}[b]{0.55\linewidth} - \centering - $ f(x)= \left \{ + \begin{center} + $$ f(x)= \left \{ \begin{array}{lll} f_1(x_1,x_2,x_3,x_4,x_5) & = & x_1.\overline{x_2} + \overline{x_1}.x_2 \\ f_2(x_1,x_2,x_3,x_4,x_5) & = & \overline{x_1 + x_2} \\ @@ -120,48 +120,47 @@ basés sur l'absence de cycles. Ils ne peuvent donc pas être appliqués ici. f_4(x_1,x_2,x_3,x_4,x_5) & = & x_5 \\ f_5(x_1,x_2,x_3,x_4,x_5) & = & \overline{x_3} + x_4 \end{array} - \right. $ -\caption{Fonction $f$ de l'exemple jouet.} -\label{fig:mix:map} -\end{minipage}\hfill -\begin{minipage}[b]{.40\linewidth} - \begin{center} - \includegraphics[scale=0.55]{xplgraphmix} + \right. + $$ + + \includegraphics[scale=0.55]{xplgraphmix} \end{center} - \caption{Graphe d'interaction associé à $f$.} - \label{fig:mix:xplgraph} -\end{minipage} + \caption{Définition de $f:\Bool^5 \rightarrow \Bool^5$ et son graphe d'interaction} + \label{fig:mix:map} \end{figure} \begin{figure} -\begin{minipage}{0.56\linewidth} - \includegraphics[scale=0.55]{para_iterate_dec} - \caption{Itérations parallèles de $f$.}\label{fig:mix:xplparaFig} -\end{minipage} -\hfill -\begin{minipage}{0.39\linewidth} - \includegraphics[scale=0.55]{chao_iterate_excerpt} - \caption{Extrait d'itérations chaotiques.} - \label{fig:mix:xplchaoFig} -\end{minipage} + \begin{center} + \subfigure[Itérations synchrones de $f$.]{ + \includegraphics[scale=0.50]{para_iterate_dec} + \label{fig:mix:xplparaFig} + } + \subfigure[Extrait des itérations unaires.]{ + \includegraphics[scale=0.49]{chao_iterate_excerpt} + \label{fig:mix:xplchaoFig} + } + \end{center} + \caption{Graphes des itérations de $f$ définie à la figure~\ref{fig:mix:map}} \end{figure} +\end{xpl} + Dans ce qui suit, les configurations sont représentées à l'aide d'entiers -plutôt que nombres binaires. Le graphe des itérations parallèles est donné +plutôt que nombres binaires. Le graphe des itérations synchrones est donné en~\Fig{fig:mix:xplparaFig}. Depuis n'importe quelle configuration, on constate qu'il converge vers le point fixe correspondant à l'entier 19. -Un extrait du graphe des itérations chaotiques est donné à +Un extrait du graphe des itérations unaires est donné à la~\Fig{fig:mix:xplchaoFig}. Les libellés des arcs correspondent aux éléments -activés. Les itérations chaotiques ne convergent pas pour la stratégie +activés. Les itérations unaires ne convergent pas pour la stratégie pseudo périodique donnée à l'équation~\Equ{eq:pseudo}: le système peut infiniment boucler entre 11 et 3, entre 15 et 7. -Comme les itérations chaotiques ne convergent pas pour certaines stratégies, +Comme les itérations unaires ne convergent pas pour certaines stratégies, les itérations asynchrones basées sur les même stratégies peuvent ne pas converger aussi. Cependant, même si l'on considère que tous les composants sont activés à chaque itération, c'est à dire si $s^t$ est -constamment égal à $2^n-1$, le délais peut introduire de la divergence. +constamment égal à $2^n-1$, le délai peut introduire de la divergence. On considère par exemple la matrice $D^t$ dont chaque élément vaut $t$ sauf $D^t_{12}$ qui vaut $t-1$ si $t$ est impair. On a ainsi $x^{t+1}= f(x^{t})$ si $t$ est pair et @@ -206,7 +205,7 @@ Les itérations mixes d'un système discret suit l'équation \Equ{eq:async} où de plus $bin(s^t)[i]=bin(s^t)[j]$ et $D_{ij}^t=D_{ji}^t=t$ si $i \eqNode j$. \end{Def} -Dans ce contexte, il n'y a plus de délais entre deux noeuds de la même CFC +Dans ce contexte, il n'y a plus de délai entre deux noeuds de la même CFC et leurs mises à jour sont synchronisées. Cependant, pour $p_0$ et $p_1$ dans la même classe \class{p}, et $q$ dans une autre classe \class{q}, ce mode opératoire autorise @@ -224,12 +223,6 @@ Pour gommer cette distinction, on définit le mode suivant: \bigwedge_{p_k \in \class{p}, q_k \in \class{q} } D_{p_{k}q_{k}}^{t} = d_{pq}^t \end{equation*} - % Je me demande si cette formalisation n'a pas un souci par rapport à une - % formalisation avec un seul q ?! (car deux éléments supposent qu'ils sont distincts) - % car si on a deux éléments dans chaque classe, on peut alors avoir deux - % groupes de délais : supposons (a,b) dans p et (c,d) dans q - % a,c et b,d sont égaux et a,d et b,c sont égaux mais rien n'implique que a,c et - % a,d soient égaux !!! \end{Def} On a alors le théorème suivant. @@ -252,7 +245,7 @@ La preuve de ce théorème est donnée en section~\ref{anx:mix}. Cette section donne des bornes supérieures et inférieures des durées globales de convergence pour les modes synchrones, mixes et asynchrones. Pour simplifier le discours, on considère que les itérations -convergent en in $I$ étapes dans le mode synchrone et que le graphe +convergent en $I$ étapes dans le mode synchrone et que le graphe d'interaction ne contient qu'une seule composante connexe. Les durées de convergence prennent en compte les temps de calcul et les temps de communication, ce depuis l'initialisation et jusqu'à la stabilisation. @@ -265,11 +258,13 @@ un entier $\delta_0$ tel que $0 \le t-D_{ij}^t \le \delta_0$ est établi pour tout couple de n{\oe}uds $(i,j)$. Les notations utilisées sont les suivantes: \begin{description} -\item [Taille pour coder l'information:] elle représente le nombre de nécessaire de bits +\item [Taille pour coder l'information] elle représente le nombre + de bits + nécessaires pour représenter l'état courant du composant $i$ et est notée $\textit{cs}_i$; -\item [Temps de calcul:] le composant $i$ a besoins de $\textit{cp}_i$ unités de temps +\item [Temps de calcul] le composant $i$ a besoins de $\textit{cp}_i$ unités de temps pour faire une mise à jour locale de son état; -\item [Temps de communication:] On utilise le modèle classique de communication +\item [Temps de communication] on utilise le modèle classique de communication $\beta+L\tau$ où $L$ est le nombre de bits transférés. On définit $\beta_{ij}$ et $\tau_{ij}$ comme la latence et la bande passante du lien entre $i$ et $j$. @@ -299,7 +294,7 @@ graphe d'interaction, le temps global de convergence est Intuitivement la convergence se propage selon les dépendances internes au système: un n{\oe}uds se stabilise lorsque ceux dont il dépend sont eux aussi stables. Cette stabilisation progressive est illustrée à la \Fig{fig:evalsync} qui - représente des exécutions parallèles dans le cas d'une initialisation avec la + représente des exécutions synchrones dans le cas d'une initialisation avec la valeur (00100). Dans cette figure et les suivantes, les blocs doublement hachurés indiquent la stabilisation du composant. @@ -309,7 +304,7 @@ graphe d'interaction, le temps global de convergence est \centering \begin{minipage}{1\linewidth} \includegraphics[scale=0.4]{eval_sync} - \caption{Itérations parallèles} + \caption{Itérations synchrones} \label{fig:evalsync} \end{minipage} @@ -390,38 +385,14 @@ graphe d'interaction, le temps global de convergence est \subsection{le mode mixe} \label{sec:evalmixed} -% As detailed in Sect.~\ref{sec:mdn}, the mixed case asynchronously combines -% subsets of synchronized components (the different classes). The double interest -% of that approach is to ensure the convergence of the system while using -% asynchronism. - -% The part of asynchronism often reduces the global execution time as the -% communications between subgroups are implicitly overlapped by computations. -% However, the iterative scheme is no more the same as the synchronous one and its -% number of iterations to reach the convergence will be greater or equal. - -% Le nombre d'itérations requises pour obtenir la convergence en mode mixe -% dépend des arangements entre les délais de communication et les durées de -% calcul. - -% number directly depends on the arrangement of delays during the execution and -% then on the communication times. But it also depends on the evolution functions -% which influence the way each part of the system stabilizes itself. -% In fact, according to its evolution function, a component may reach its fixed -% point state even with a part of its input data not recently updated. In -% addition, as mentioned earlier, the set of components in any system does not -% stabilize at the same time and there is often a propagation of the stabilization -% through the system. -% Also, the previously mentioned phenomenon of stabilization propagation through -% the system is still present in mixed mode. On considère $|\mathcal{K}|$ classes de composants synchronisés. -(comme donné en équation~(\ref{eq:I}). +(comme donné en équation~(\ref{eq:I})). Soit $I_k$ le nombre d'itérations suffisants pour que la classe -$k \in \mathcal{K}$ se stabilise +$\class{k} \in \mathcal{K}$ se stabilise sachant toutes ses dépendances ont déjà convergé. -Ainsi $I$ vaut $\sum_{k \in \mathcal{K}} I_k$. -La borne inférieur pour la durée de convergence des itérations asynchrones est +Ainsi $I$ vaut $\sum_{\class{k} \in \mathcal{K}} I_k$. +La borne inférieure pour la durée de convergence des itérations asynchrones est \begin{equation} \label{eq:mixtelow} T(\textit{Mixed})\ge \sum_{k\in \mathcal{K}} I_k(\max_{l\in k}\textit{cp}_{l}) @@ -446,7 +417,7 @@ ascendants pour converger. On a dans ce cas: Une exécution du mode mixe est donnée à la~\Fig{fig:evalmixte}. On peut constater que le temps d'exécution peut être plus petit que pour le - mode parallèle. + mode synchrone. \end{xpl} \subsection{Le mode généralisé asynchrone} @@ -467,11 +438,12 @@ La borne supérieure quant à elle est donnée par: \label{eq:asyncup} T(\textit{Async})\le\sum_{i=1}^{n}\left(I_i\times \textit{cp}_{i}+\max_{1\le k \le n}B_{ki}(\beta_{ik}+\textit{cs}_{i}\tau_{ik})\right) \end{equation} -et apparaît lorsque chaque élément dépend des autres et que les calcules +et apparaît lorsque chaque élément dépend des autres et que les calculs ne recouvrent nullement les communications. \begin{xpl} - La \Fig{fig:evalasync} présente un exemple d'exécution du mode asynchrone. + La \Fig{fig:evalasync} présente un exemple d'exécution du mode généralisé + asynchrone. Certaines communications issues de l'élément $4$ n'ont pas été représentées pour des raisons de clarté. On constate que le temps global de convergence est plus petit que celui des @@ -500,6 +472,6 @@ ne recouvrent nullement les communications. %%% Local Variables: %%% mode: latex %%% TeX-master: "main" -%%% ispell-dictionary: "american" +%%% ispell-dictionary: "french" %%% mode: flyspell %%% End: diff --git a/sdd.tex b/sdd.tex index 3755295..530ef34 100644 --- a/sdd.tex +++ b/sdd.tex @@ -119,18 +119,26 @@ schémas suivants : \item \textbf{Schéma parallèle synchrone :} basé sur la relation de récurrence $x^{t+1}=f(x^t)$. Tous les $x_i$, $1 \le i \le n$, sont ainsi mis à jour à chaque itération en utilisant l'état global précédent du système $x^t$. -\item \textbf{Schéma unaire :} cette terminologie a plusieurs - interprétations - dans la littérature, mais celle que nous - retenons ici consiste à modifier la valeur +\item \textbf{Schéma unaire :} ce schéma est parfois + qualifié de chaotique + dans la littérature. + Il consiste à modifier la valeur d'un unique élément $i$, $1 \le i \le n$, à chaque itération. Le choix de l'élément qui est modifié à chaque itération est défini par une suite $S = \left(s^t\right)^{t \in \mathds{N}}$ qui est une séquence d'indices dans $[n]$. Cette suite est appelée \emph{stratégie unaire}. -% Lorsque cette suite est strictement cyclique (sans - % occurrences multiples dans le motif) sur l'ensemble des éléments $\{1,\ldots - % n\}$, alors on retrouve le comportement du mode séquentiel synchrone. + Il est basé sur la relation définie pour tout $i \in [n]$ par + $$ + x^{t+1}_i= + \left\{ \begin{array}{l} + f_i(x^t) \textrm{ si } i=s^t, \\ + x^t_i\textrm{ sinon.} + \end{array} + \right.$$ + + + \item \textbf{Schéma généralisé:} dans ce schéma, ce sont les valeurs d'un ensemble d'éléments de $[n]$ qui sont modifiées à chaque itération. Dans le cas particulier où c'est la valeur d'un singleton @@ -146,6 +154,17 @@ schémas suivants : jour. La suite $S = \left(s^t\right)^{t \in \mathds{N}}$ est une séquence de sous-ensembles de $[n]$ appelée \emph{stratégie généralisée}. + Il est basé sur la relation définie pour tout $i \in [n]$ par + $$ + x^{t+1}_i= + \left\{ \begin{array}{l} + f_i(x^t) \textrm{ si } i \in s^t, \\ + x^t_i\textrm{ sinon.} + \end{array} + \right.$$ + + + \end{itemize} @@ -168,7 +187,7 @@ sont les éléments de $\Bool^n$ (voir \textsc{Figure}~\ref{fig:xpl:graphs}). \item Le \emph{graphe des itérations synchrones} de $f$, noté $\textsc{gis}(f)$ est le graphe orienté de $\Bool^n$ qui contient un arc $x \rightarrow y$ si et seulement si $y=f(x)$. -\item Le \emph{graphe des itérations unaires} de $f$, noté $\textsc{gia}(f)$ +\item Le \emph{graphe des itérations unaires} de $f$, noté $\textsc{giu}(f)$ est le graphe orienté de $\Bool^n$ qui contient un arc $x \rightarrow y$ si et seulement s'il existe $x \in \Delta f(x)$ tel que $y = \overline{x}^i$. \item Le \emph{graphe des itérations généralisées} de $f$, noté $\textsc{gig}(f)$ @@ -200,7 +219,7 @@ associés à $f$. \end{minipage} \label{fig:fsig} } - \subfigure[$\textsc{gia}(f)$]{ + \subfigure[$\textsc{giu}(f)$]{ \begin{minipage}{0.33\textwidth} \begin{center} \includegraphics[scale=0.4]{faig} @@ -237,9 +256,6 @@ Les points fixes sont particulièrement intéressants car ils correspondent aux états stables: dans chaque graphe d'itérations, le point $x$ est un point fixe si et seulement si il est son seul successeur. -Dans le contexte des réseaux de régulation de gènes, -ces points fixes correspondent aux configurations stables pour l'expression de -gènes. @@ -272,7 +288,7 @@ Ainsi $\Gamma$ contient toujours au moins un attracteur. \begin{xpl} -Les attracteurs de $\textsc{gia}(f)$ et de $\textsc{gig}(f)$ sont +Les attracteurs de $\textsc{giu}(f)$ et de $\textsc{gig}(f)$ sont le point fixe $000$ et l'attracteur cyclique $\{001, 101,111, 011 \}$. Les attracteurs de $\textsc{gis}(f)$ sont le point fixe $000$ @@ -316,7 +332,7 @@ $f'_{ij}(x)=0$. Ainsi $B(f)_{ij}$ est nulle. La réciproque est aussi vraie. En outre, les interactions peuvent se représenter à l'aide d'un -graphe $G(f)$ orienté et signé défini ainsi: +graphe $\Gamma(f)$ orienté et signé défini ainsi: l'ensemble des sommets est $[n]$ et il existe un arc de $j$ à $i$ de signe $s\in\{-1,1\}$, noté $(j,s,i)$, si $f_{ij}(x)=s$ pour au moins @@ -400,7 +416,7 @@ La \textsc{Figure}~(\ref{fig:f:incidence}) donne la matrice d'incidence complèt \begin{figure}[ht] \begin{center} - \subfigure[Matrice jacobienne de $f$.]{ + \subfigure[Matrice jacobienne]{ \begin{minipage}{0.90\textwidth} \begin{center} $ @@ -440,8 +456,8 @@ La \textsc{Figure}~(\ref{fig:f:incidence}) donne la matrice d'incidence complèt \end{minipage} \label{fig:f:jacobienne} } - - \subfigure[Graphe d'interaction de $f$.]{ + ~ + \subfigure[Graphe d'interaction]{ \begin{minipage}{0.45\textwidth} \begin{center} \includegraphics[scale=0.5]{gf} @@ -450,7 +466,7 @@ La \textsc{Figure}~(\ref{fig:f:incidence}) donne la matrice d'incidence complèt \end{minipage} } - \subfigure[Matrice d'incidence de $f$.]{ + \subfigure[Matrice d'incidence]{ \begin{minipage}{0.45\textwidth} \begin{center} $ @@ -468,24 +484,30 @@ La \textsc{Figure}~(\ref{fig:f:incidence}) donne la matrice d'incidence complèt \end{minipage} } \end{center} -\caption{Représentations des dépendances entre les éléments de la fonction $f$ de l'exemple illustratif.} +\caption{Représentations des dépendances entre les éléments +de la fonction +$f:\Bool^3 \rightarrow \Bool^3$ telle que +$(x_1, x_2, x_3) \mapsto +((\overline{x_1} + \overline{x_2}).x_3, +x_1.x_3, +x_1 + x_2 + x_3)$} \end{figure} \end{xpl} -Soit $P$ une suite d'arcs de $G(f)$ de la forme +Soit $P$ une suite d'arcs de $\Gamma(f)$ de la forme \[ (i_1,s_1,i_2),(i_2,s_2,i_3),\ldots,(i_r,s_r,i_{r+1}). \] -Alors, $P$ est dit un chemin de $G(f)$ de longueur $r$ et de signe +Alors, $P$ est dit un chemin de $\Gamma(f)$ de longueur $r$ et de signe $\Pi_{i=1}^{r}s_i$ et $i_{r+1}$ est dit accessible depuis $i_1$. $P$ est un {\emph{circuit}} si $i_{r+1}=i_1$ et si les sommets $i_1$,\ldots $i_r$ sont deux à deux disjoints. -Un sommet $i$ de $G(f)$ a une {\emph{boucle}} -positive (resp. négative) , si $G(f)$ a un +Un sommet $i$ de $\Gamma(f)$ a une {\emph{boucle}} +positive (resp. négative) , si $\Gamma(f)$ a un arc positif (resp. un arc négatif) de $i$ vers lui-même. @@ -522,9 +544,9 @@ François Robert~\cite{Rob95} a énoncé en 1995 le théorème suivant de convergence dans le mode des itérations unaires. -\begin{theorem}\label{Th:conv:GIA} -Si le graphe $G(f)$ n'a pas de cycle et si la stratégie unaire est -pseudo-périodique, alors tout chemin de $\textsc{GIA}(f)$ atteint +\begin{theorem}\label{Th:conv:GIU} +Si le graphe $\Gamma(f)$ n'a pas de cycle et si la stratégie unaire est +pseudo-périodique, alors tout chemin de $\textsc{giu}(f)$ atteint l'unique point fixe $\zeta$ en au plus $n$ pseudo-périodes. \end{theorem} @@ -536,9 +558,10 @@ dont l'union est $[n]$, cette stratégie est {pseudo-périodique}. J. Bahi~\cite{Bah00} a démontré le théorème suivant: \begin{theorem}\label{Th:Bahi} -Si le graphe $G(f)$ n'a pas de cycle et si la stratégie généralisée +Si le graphe $\Gamma(f)$ n'a pas de cycle et si la stratégie généralisée est pseudo-périodique alors -tout chemin de $\textsc{gig}(f)$ finit par atteindre +tout chemin de $\textsc{gig}(f)$ (et donc de $\textsc{giu}(f)$) +finit par atteindre l'unique point fixe $\zeta$. \end{theorem} -- 2.39.5

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