--- /dev/null
+\bibstyle{hdr}
+\gdef\numberofJ{0\xspace }
--- /dev/null
+%!PS-Adobe-3.0 EPSF-3.0
+%%Creator: Tk Canvas Widget
+%%For: couchot,,,
+%%Title: Window .f2.c
+%%CreationDate: Mon Mar 29 10:52:40 2010
+%%BoundingBox: 149 114 463 679
+%%Pages: 1
+%%DocumentData: Clean7Bit
+%%Orientation: Portrait
+%%DocumentNeededResources: font Courier-Bold
+%%EndComments
+
+%%BeginProlog
+/CurrentEncoding [
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quotesingle
+/parenleft/parenright/asterisk/plus/comma/hyphen/period/slash
+/zero/one/two/three/four/five/six/seven
+/eight/nine/colon/semicolon/less/equal/greater/question
+/at/A/B/C/D/E/F/G
+/H/I/J/K/L/M/N/O
+/P/Q/R/S/T/U/V/W
+/X/Y/Z/bracketleft/backslash/bracketright/asciicircum/underscore
+/grave/a/b/c/d/e/f/g
+/h/i/j/k/l/m/n/o
+/p/q/r/s/t/u/v/w
+/x/y/z/braceleft/bar/braceright/asciitilde/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/exclamdown/cent/sterling/currency/yen/brokenbar/section
+/dieresis/copyright/ordfeminine/guillemotleft/logicalnot/hyphen/registered/macron
+/degree/plusminus/twosuperior/threesuperior/acute/mu/paragraph/periodcentered
+/cedilla/onesuperior/ordmasculine/guillemotright/onequarter/onehalf/threequarters/questiondown
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/space/space/space/space/space/space/space/space
+/eth/ntilde/ograve/oacute/ocircumflex/otilde/odieresis/divide
+/oslash/ugrave/uacute/ucircumflex/udieresis/yacute/thorn/ydieresis
+] def
+
+50 dict begin
+% This is a standard prolog for Postscript generated by Tk's canvas
+% widget.
+% RCS: @(#) $Id: mkpsenc.tcl,v 1.3 2002/07/19 14:37:21 drh Exp $
+
+% The definitions below just define all of the variables used in
+% any of the procedures here. This is needed for obscure reasons
+% explained on p. 716 of the Postscript manual (Section H.2.7,
+% "Initializing Variables," in the section on Encapsulated Postscript).
+
+/baseline 0 def
+/stipimage 0 def
+/height 0 def
+/justify 0 def
+/lineLength 0 def
+/spacing 0 def
+/stipple 0 def
+/strings 0 def
+/xoffset 0 def
+/yoffset 0 def
+/tmpstip null def
+
+
+/cstringshow {
+ {
+ dup type /stringtype eq
+ { show } { glyphshow }
+ ifelse
+ }
+ forall
+} bind def
+
+
+
+/cstringwidth {
+ 0 exch 0 exch
+ {
+ dup type /stringtype eq
+ { stringwidth } {
+ currentfont /Encoding get exch 1 exch put (\001) stringwidth
+ }
+ ifelse
+ exch 3 1 roll add 3 1 roll add exch
+ }
+ forall
+} bind def
+
+% font ISOEncode font
+% This procedure changes the encoding of a font from the default
+% Postscript encoding to current system encoding. It's typically invoked just
+% before invoking "setfont". The body of this procedure comes from
+% Section 5.6.1 of the Postscript book.
+
+/ISOEncode {
+ dup length dict begin
+ {1 index /FID ne {def} {pop pop} ifelse} forall
+ /Encoding CurrentEncoding def
+ currentdict
+ end
+
+ % I'm not sure why it's necessary to use "definefont" on this new
+ % font, but it seems to be important; just use the name "Temporary"
+ % for the font.
+
+ /Temporary exch definefont
+} bind def
+
+% StrokeClip
+%
+% This procedure converts the current path into a clip area under
+% the assumption of stroking. It's a bit tricky because some Postscript
+% interpreters get errors during strokepath for dashed lines. If
+% this happens then turn off dashes and try again.
+
+/StrokeClip {
+ {strokepath} stopped {
+ (This Postscript printer gets limitcheck overflows when) =
+ (stippling dashed lines; lines will be printed solid instead.) =
+ [] 0 setdash strokepath} if
+ clip
+} bind def
+
+% desiredSize EvenPixels closestSize
+%
+% The procedure below is used for stippling. Given the optimal size
+% of a dot in a stipple pattern in the current user coordinate system,
+% compute the closest size that is an exact multiple of the device's
+% pixel size. This allows stipple patterns to be displayed without
+% aliasing effects.
+
+/EvenPixels {
+ % Compute exact number of device pixels per stipple dot.
+ dup 0 matrix currentmatrix dtransform
+ dup mul exch dup mul add sqrt
+
+ % Round to an integer, make sure the number is at least 1, and compute
+ % user coord distance corresponding to this.
+ dup round dup 1 lt {pop 1} if
+ exch div mul
+} bind def
+
+% width height string StippleFill --
+%
+% Given a path already set up and a clipping region generated from
+% it, this procedure will fill the clipping region with a stipple
+% pattern. "String" contains a proper image description of the
+% stipple pattern and "width" and "height" give its dimensions. Each
+% stipple dot is assumed to be about one unit across in the current
+% user coordinate system. This procedure trashes the graphics state.
+
+/StippleFill {
+ % The following code is needed to work around a NeWSprint bug.
+
+ /tmpstip 1 index def
+
+ % Change the scaling so that one user unit in user coordinates
+ % corresponds to the size of one stipple dot.
+ 1 EvenPixels dup scale
+
+ % Compute the bounding box occupied by the path (which is now
+ % the clipping region), and round the lower coordinates down
+ % to the nearest starting point for the stipple pattern. Be
+ % careful about negative numbers, since the rounding works
+ % differently on them.
+
+ pathbbox
+ 4 2 roll
+ 5 index div dup 0 lt {1 sub} if cvi 5 index mul 4 1 roll
+ 6 index div dup 0 lt {1 sub} if cvi 6 index mul 3 2 roll
+
+ % Stack now: width height string y1 y2 x1 x2
+ % Below is a doubly-nested for loop to iterate across this area
+ % in units of the stipple pattern size, going up columns then
+ % across rows, blasting out a stipple-pattern-sized rectangle at
+ % each position
+
+ 6 index exch {
+ 2 index 5 index 3 index {
+ % Stack now: width height string y1 y2 x y
+
+ gsave
+ 1 index exch translate
+ 5 index 5 index true matrix tmpstip imagemask
+ grestore
+ } for
+ pop
+ } for
+ pop pop pop pop pop
+} bind def
+
+% -- AdjustColor --
+% Given a color value already set for output by the caller, adjusts
+% that value to a grayscale or mono value if requested by the CL
+% variable.
+
+/AdjustColor {
+ CL 2 lt {
+ currentgray
+ CL 0 eq {
+ .5 lt {0} {1} ifelse
+ } if
+ setgray
+ } if
+} bind def
+
+% x y strings spacing xoffset yoffset justify stipple DrawText --
+% This procedure does all of the real work of drawing text. The
+% color and font must already have been set by the caller, and the
+% following arguments must be on the stack:
+%
+% x, y - Coordinates at which to draw text.
+% strings - An array of strings, one for each line of the text item,
+% in order from top to bottom.
+% spacing - Spacing between lines.
+% xoffset - Horizontal offset for text bbox relative to x and y: 0 for
+% nw/w/sw anchor, -0.5 for n/center/s, and -1.0 for ne/e/se.
+% yoffset - Vertical offset for text bbox relative to x and y: 0 for
+% nw/n/ne anchor, +0.5 for w/center/e, and +1.0 for sw/s/se.
+% justify - 0 for left justification, 0.5 for center, 1 for right justify.
+% stipple - Boolean value indicating whether or not text is to be
+% drawn in stippled fashion. If text is stippled,
+% procedure StippleText must have been defined to call
+% StippleFill in the right way.
+%
+% Also, when this procedure is invoked, the color and font must already
+% have been set for the text.
+
+/DrawText {
+ /stipple exch def
+ /justify exch def
+ /yoffset exch def
+ /xoffset exch def
+ /spacing exch def
+ /strings exch def
+
+ % First scan through all of the text to find the widest line.
+
+ /lineLength 0 def
+ strings {
+ cstringwidth pop
+ dup lineLength gt {/lineLength exch def} {pop} ifelse
+ newpath
+ } forall
+
+ % Compute the baseline offset and the actual font height.
+
+ 0 0 moveto (TXygqPZ) false charpath
+ pathbbox dup /baseline exch def
+ exch pop exch sub /height exch def pop
+ newpath
+
+ % Translate coordinates first so that the origin is at the upper-left
+ % corner of the text's bounding box. Remember that x and y for
+ % positioning are still on the stack.
+
+ translate
+ lineLength xoffset mul
+ strings length 1 sub spacing mul height add yoffset mul translate
+
+ % Now use the baseline and justification information to translate so
+ % that the origin is at the baseline and positioning point for the
+ % first line of text.
+
+ justify lineLength mul baseline neg translate
+
+ % Iterate over each of the lines to output it. For each line,
+ % compute its width again so it can be properly justified, then
+ % display it.
+
+ strings {
+ dup cstringwidth pop
+ justify neg mul 0 moveto
+ stipple {
+
+
+ % The text is stippled, so turn it into a path and print
+ % by calling StippledText, which in turn calls StippleFill.
+ % Unfortunately, many Postscript interpreters will get
+ % overflow errors if we try to do the whole string at
+ % once, so do it a character at a time.
+
+ gsave
+ /char (X) def
+ {
+ dup type /stringtype eq {
+ % This segment is a string.
+ {
+ char 0 3 -1 roll put
+ currentpoint
+ gsave
+ char true charpath clip StippleText
+ grestore
+ char stringwidth translate
+ moveto
+ } forall
+ } {
+ % This segment is glyph name
+ % Temporary override
+ currentfont /Encoding get exch 1 exch put
+ currentpoint
+ gsave (\001) true charpath clip StippleText
+ grestore
+ (\001) stringwidth translate
+ moveto
+ } ifelse
+ } forall
+ grestore
+ } {cstringshow} ifelse
+ 0 spacing neg translate
+ } forall
+} bind def
+
+%%EndProlog
+%%BeginSetup
+/CL 2 def
+%%IncludeResource: font Courier-Bold
+%%EndSetup
+
+%%Page: 1 1
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--- /dev/null
+\begin{xpl}
+ Without loss of generality, we consider Boolean domains. We have three
+ elements taking their value in the set $\{0,1\}$. Thus, a configuration is an
+ element of $\{0,1\}^3$, \textit{i.e.}, a binary number between 0 and 7. In
+ \Fig{fig:map} is given the function defining the dynamic and
+ \Fig{fig:xplgraph} depicts its associated connection graph. Operators
+ $\overline{a}$, sum and product are the classical Boolean operators. In the
+ incidence graph, or connection graph, the vertices are the elements of the
+ system and an edge from $i$ to $j$ indicates that $f_j$ depends on $i$. Notice
+ that the connection graph contains five cycles.
+
+\begin{figure}[ht]
+ \centering
+ \subfloat[Maps]{
+ \begin{minipage}%[h]
+{.6\linewidth}
+ \begin{center}
+ %\vspace{1.6em}
+ $ F(X)= \left \{
+% \begin{array}{rcl}
+% f_1(x_1,x_2,x_3) & = & x_1.(x_2 + x_3) \\
+% f_2(x_1,x_2,x_3) & = & \overline{x_1}.\overline{x_2}.x_3 \\
+% f_3(x_1,x_2,x_3) & = & x_1.x_2.x_3
+% \end{array}
+ \begin{array}{rcl}
+ f_1(x_1,x_2,x_3) & = & x_1.\overline{x_2} + x_3 \\
+ f_2(x_1,x_2,x_3) & = & x_1 + \overline{x_3} \\
+ f_3(x_1,x_2,x_3) & = & x_2.x_3
+ \end{array}
+ \right.
+ $
+ %\vspace{1.6em}
+ \end{center}
+ \end{minipage}
+ \label{fig:map}
+ }
+ \subfloat[Connexion Graph]{
+ \begin{minipage}%[h]
+{.35\linewidth}
+ \begin{center}
+ \includegraphics[width=4cm]{xplCnx.pdf}
+ \end{center}
+ \end{minipage}
+ \label{fig:xplgraph}
+ }
+% \vspace{-.5em}
+ \caption{Dynamic of the running example.}
+ %\vspace{-2em}
+\end{figure}
+
+% \begin{figure}
+% \begin{center}
+% %\includegraphics[width=3cm]{xplgraph.ps}
+% \includegraphics[width=3cm]{xplgraph.pdf}
+% \end{center}
+% \caption{Incidence graph of running example\label{fig:xplgraph}}
+% \end{figure}
+
+
+% \begin{figure}
+% \begin{center}
+% \includegraphics[width=3cm]{xplgraph.ps}
+% %\includegraphics[width=3cm]{xplgraph.pdf}
+% \end{center}
+% \caption{Incidence graph of running example\label{fig:xplgraph}}
+% \end{figure}
+
+
+% The prototype developed for~\cite{BCVC10:ir} has shown
+% that pure parallel iterations are convergent.
+% It has helped us to find pseudo-periodic strategies making chaotic
+% iterations (and then asynchronous iterations) diverging.
+% This example shows then a case when convergence is established
+% under some conditions whereas universal convergence is not.
+
+% \begin{figure}[ht]
+% \begin{center}
+% \includegraphics[width=12cm]{implementation/para_iterate_dec.ps}
+% \end{center}
+% \caption{Parallel iterations of running example\label{fig:xplparaFig}}
+% \end{figure}
+
+% In what follows and for brevity reasons, configurations are represented as
+% decimal numbers instead of binary numbers. The graph of parallel iterations is
+% given in Fig.~\ref{fig:xplparaFig}.
+% Starting from any configuration, the network
+% converges to the fixed point corresponding to the decimal number 19.
+
+% \begin{figure}[ht]
+% \begin{center}
+% \includegraphics[width=4cm]{chao_iterate_excerpt.ps}
+% \end{center}
+% \caption{Excerpt of chaotic iterations of running example
+% \label{fig:xplchaoFig}}
+% \end{figure}
+
+% An extract of the graph of chaotic iterations is given in Fig.~\ref{fig:xplchaoFig}.
+% Arc label is the characteristic function of activated elements
+% (i.e. the matrix $J^t$) expressed as a
+% decimal number. It allows us to shortly represent the strategy.
+% For instance,
+% in this graph we only consider the pseudo-periodic strategy that alternately activates both the
+% first two elements and the last four elements. This is formalized as
+% $ J^{2p} =
+% \left(
+% \begin{array}{lllll}
+% 1&0 &0 &0 &0 \\
+% 0&1 &0 &0 &0 \\
+% 0&0 &0 &0 &0 \\
+% 0&0 &0 &0 &0 \\
+% 0&0 &0 &0 &0
+% \end{array}
+% \right)
+% $
+% and
+% $J^{2p+1} =
+% \left(
+% \begin{array}{lllll}
+% 0&0 &0 &0 &0 \\
+% 0&1 &0 &0 &0 \\
+% 0&0 &1 &0 &0 \\
+% 0&0 &0 &1 &0 \\
+% 0&0 &0 &0 &1
+% \end{array}
+% \right)
+% $, $p=0,1,2,\ldots$.
+% The former matrix is represented as 24 and the later as 15.
+
+% Iterations do not converge for this strategy: for instance,
+% starting from the configuration 3 (resp. configuration 7), the network goes to the configuration 11 (resp. configuration 15) and
+% goes back to the configuration 3 (resp. configuration 7) by applying the strategy depicted above.
+
+
+% Since chaotic iterations do not converge for some strategies, it is obvious that
+% convergence cannot be observed for some asynchronous iterations with the
+% corresponding strategies. However, even if all the components
+% are activated, that is the asynchronous counterpart of pure parallel iterations,
+% delays may introduce divergence.
+
+% For instance, let $J^t$ be constantly equal to the identity matrix and
+% consider the matrix
+% \begin{equation*}
+% S^t = \left(
+% \begin{array}{lllll}
+% t & t' & t & t & t \\
+% t & t & t & t & t \\
+% t & t & t & t & t \\
+% t & t & t & t & t\\
+% t & t & t & t & t \\
+% \end{array}
+% \right)
+% \textrm{ where }
+% t' = \left\{
+% \begin{array}{ll}
+% t & \textrm{ if t is even;} \\
+% t-1 & \textrm{ otherwise.}
+% \end{array}
+% \right.
+% \end{equation*}
+
+% We have both
+% \begin{eqnarray*}
+% X^{2k+1} & =& F(X^{2k}) \\
+% X^{2k} & =& \left(
+% \begin{array}{l}
+% F_1(X_1^{2k-1},X_2^{2k-2},X_3^{2k-1},X_4^{2k-1},X_5^{2k-1}) \\
+% F_2(X^{2k-1})\\
+% \vdots \\
+% F_5(X^{2k-1})
+% \end{array}
+% \right).
+% \end{eqnarray*}
+
+
+% Starting from $X^0 = (0,0,0,1,1)$, the system reaches
+% $X^1 = (0,1,0,1,1)$ and enters in a cycle between these
+% two configurations (as in some chaotic iterations).
+
+% In
+% the next section, a particular execution mode is described which enables
+% asynchronism in iterations while guaranteeing the convergence.
+
+
+In what follows and for brevity reasons, configurations are represented as
+decimal numbers instead of binary numbers. It is not hard to establish that in
+parallel synchronous iterations, starting from any configuration different from
+7 (111) leads to the fixed point 2 (010), and starting from 7 leads to 7.
+With other strategies,
+the behavior is quite different although the system always
+reaches a stabilization. Indeed, for the initial states 1, 3 and 5, the system
+converges towards anyone of the two fixed points 2 and 7. However, there is no
+infinite cycle and the convergence property defined in \Equ{eq:ltl:conv} is
+verified.
+Finally, due to the presence of
+cycle in the connexion graph,
+known sufficient conditions~\cite{Bah00} establishing convergence
+results for asynchronous modes cannot be applied here.
+We address this question in what follows.
+\end{xpl}
+
+
+%%% Local Variables:
+%%% mode: latex
+%%% fill-column: 80
+%%% ispell-dictionary: "american"
+%%% mode: flyspell
+%%% TeX-master: "main"
+%%% End:
+
--- /dev/null
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+ 3 -> 1 [label="+"]
+ 1 -> 2 [label="+"]
+ 3 -> 2 [label="-"]
+ 2 -> 3 [label="+"]
+ 3 -> 3 [label="+"]
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+\bibstyle{apalike}
+\bibdata{abbrev,biblioand}
+\bibcite{Bah00}{Bahi, 2000}
+\bibcite{BCV02}{Bahi and Contassot-Vivier, 2002}
+\bibcite{BCVC10:ir}{Bahi et~al., 2010}
+\bibcite{BM99}{Bahi and Michel, 1999}
+\bibcite{Cha06}{Chandrasekaran, 2006}
+\bibcite{CGK05}{Couchot et~al., 2005}
+\bibcite{Hol03}{Holzmann, 2003}
+\bibcite{RC07}{Richard and Comet, 2007}
+\bibcite{Wei97}{Weise, 1997}
--- /dev/null
+\begin{thebibliography}{}
+
+\bibitem[Bahi, 2000]{Bah00}
+Bahi, J.~M. (2000).
+\newblock Boolean totally asynchronous iterations.
+\newblock {\em International Journal of Mathematical Algorithms}, 1:331--346.
+
+\bibitem[Bahi and Contassot-Vivier, 2002]{BCV02}
+Bahi, J.~M. and Contassot-Vivier, S. (2002).
+\newblock Stability of fully asynchronous discrete-time discrete-state dynamic
+ networks.
+\newblock {\em IEEE Transactions on Neural Networks}, 13(6):1353--1363.
+
+\bibitem[Bahi et~al., 2010]{BCVC10:ir}
+Bahi, J.~M., Contassot-Vivier, S., and Couchot, J.-F. (2010).
+\newblock Convergence results of combining synchronism and asynchronism for
+ discrete-state discrete-time dynamic network.
+\newblock Research Report RR2010-02, LIFC - Laboratoire d’{I}nformatique de
+ l'{U}niversit\'{e} de {F}ranche {C}omt\'{e}.
+
+\bibitem[Bahi and Michel, 1999]{BM99}
+Bahi, J.~M. and Michel, C. (1999).
+\newblock Simulations of asynchronous evolution of discrete systems.
+\newblock {\em Simulation Practice and Theory}, 7:309--324.
+
+\bibitem[Chandrasekaran, 2006]{Cha06}
+Chandrasekaran, N. (2006).
+\newblock Verifying convergence of asynchronous iterative algorithms based on
+ lyapunov functions.
+
+\bibitem[Couchot et~al., 2005]{CGK05}
+Couchot, J.-F., Giorgetti, A., and Kosmatov, N. (2005).
+\newblock A uniform deductive approach for parameterized protocol safety.
+\newblock In Redmiles, D.~F., Ellman, T., and Zisman, A., editors, {\em ASE},
+ pages 364--367. ACM.
+
+\bibitem[Holzmann, 2003]{Hol03}
+Holzmann, G.~J. (2003).
+\newblock {\em The SPIN Model Checker: Primer and Reference Manual}.
+\newblock Addison-Wesley, Pearson Education.
+
+\bibitem[Richard and Comet, 2007]{RC07}
+Richard, A. and Comet, J.-P. (2007).
+\newblock Necessary conditions for multistationarity in discrete dynamical
+ systems.
+\newblock {\em Discrete Applied Mathematics}, 155(18):2403--2413.
+
+\bibitem[Weise, 1997]{Wei97}
+Weise, C. (1997).
+\newblock An incremental formal semantics for {PROMELA}.
+\newblock In {\em SPIN97, the Third SPIN Workshop}.
+
+\end{thebibliography}
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+\contentsline {theorem}{\numberline {1}Th\IeC {\'e}or\IeC {\`e}me\thmtformatoptarg {Correction}}{16}{theorem.1}
+\contentsline {theorem}{\numberline {2}Th\IeC {\'e}or\IeC {\`e}me\thmtformatoptarg {Compl\IeC {\'e}tude}}{16}{theorem.2}
+\contentsline {theorem}{\numberline {3}Th\IeC {\'e}or\IeC {\`e}me\thmtformatoptarg {Nombre d'\IeC {\'e}tats }}{17}{theorem.3}
+\contentsline {Proof}{\numberline {1}Preuve}{17}{Proof.1}
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+\contentsline {Proof}{\numberline {2}Preuve}{21}{Proof.2}
+\contentsline {lemma}{\numberline {2}Lemme}{21}{lemma.2}
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+\contentsline {Proof}{\numberline {4}Preuve}{21}{Proof.4}
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+\contentsline {theorem}{\numberline {4}Th\IeC {\'e}or\IeC {\`e}me\thmtformatoptarg {Soundness wrt universal convergence property}}{25}{theorem.4}
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+\contentsline {theorem}{\numberline {5}Th\IeC {\'e}or\IeC {\`e}me\thmtformatoptarg {Completeness wrt universal convergence property}}{26}{theorem.5}
+\contentsline {Proof}{\numberline {9}Preuve}{26}{Proof.9}
--- /dev/null
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+\contentsline {figure}{\numberline {3.3}{\ignorespaces Exemple pour SDD $\approx $ SPIN.}}{11}{figure.3.3}
+\contentsline {figure}{\numberline {3.4}{\ignorespaces Declaration des types de la traduction.}}{12}{figure.3.4}
+\contentsline {figure}{\numberline {3.5}{\ignorespaces Process init.}}{13}{figure.3.5}
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+\contentsline {figure}{\numberline {3.8}{\ignorespaces Sauvegarde de l'\IeC {\'e}tat courant}}{14}{figure.3.8}
+\contentsline {figure}{\numberline {3.9}{\ignorespaces Mise \IeC {\`a} jour des \IeC {\'e}l\IeC {\'e}ments.}}{14}{figure.3.9}
+\contentsline {figure}{\numberline {3.10}{\ignorespaces Application de la fonction $f$.}}{14}{figure.3.10}
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+\contentsline {figure}{\numberline {3.15}{\ignorespaces Contre exemple de convergence pour~\ref {fig:RC07CE}}}{19}{figure.3.15}
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+ []
+
+
+Underfull \hbox (badness 5504) in paragraph at lines 125--131
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+to
+ []
+
+
+Underfull \hbox (badness 5022) in paragraph at lines 154--164
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+r/m/n/10.95 ) = \OMS/txsy/m/n/10.95 f\OT1/txr/m/n/10.95 0\OMS/txsy/m/n/10.95 g$
+ \OT1/phv/m/sl/10.95 and $\OMS/txsy/m/n/10.95 9\OML/txmi/m/it/10.95 k : k \OMS/
+txsy/m/n/10.95 ^^U \OML/txmi/m/it/10.95 l \OMS/txsy/m/n/10.95 ^ \OML/txmi/m/it/
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+s-ta-bli-shed then
+ []
+
+
+Underfull \hbox (badness 1917) in paragraph at lines 154--164
+\OML/txmi/m/it/10.95 M[]\OT1/txr/m/n/10.95 (0)$ \OT1/phv/m/sl/10.95 is $\OT1/tx
+r/m/n/10.95 ([]\OML/txmi/m/it/10.95 ; l; c[]\OT1/txr/m/n/10.95 )$ \OT1/phv/m/sl
+/10.95 that is ad-ded in the []\OT1/txtt/m/n/10.95 diffuse_values \OT1/phv/m/sl
+/10.95 func-tion s.t.
+ []
+
+[24]
+Overfull \hbox (237.9212pt too wide) has occurred while \output is active
+\OT1/phv/m/sl/10.95 A.3. PREUVE DE CORRECTION ET DE COMPL[]ETUDE DE L'APPROCHE
+ DE V[]ERIFICATION DE CONVERGENCE []A L'AIDE DE SPIN \OT1/phv/m/n/10.95 25
+ []
+
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--- /dev/null
+\select@language {french}
+\addvspace {10\p@ }
+\addvspace {10\p@ }
+\addvspace {10\p@ }
+\addvspace {10\p@ }
--- /dev/null
+\BOOKMARK [-1][]{part.1}{I Syst\350me Bool\351ens}{}% 1
+\BOOKMARK [0][]{chapter.1}{1 Iterations discr\350tes de Syst\350mes Dynamiques bool\351ens}{part.1}% 2
+\BOOKMARK [1][]{section.1.1}{1.1 Syst\350me dynamique bool\351en}{chapter.1}% 3
+\BOOKMARK [1][]{section.1.2}{1.2 Graphe d'it\351rations}{chapter.1}% 4
+\BOOKMARK [1][]{section.1.3}{1.3 Graphe d'interactions}{chapter.1}% 5
+\BOOKMARK [1][]{section.1.4}{1.4 Distance sur l'espace "487E912 1;n"587F913 NBn}{chapter.1}% 6
+\BOOKMARK [0][]{chapter.2}{2 Combinaisons Synchrones et Asynchrones de Syst\350mes Bool\351ens}{part.1}% 7
+\BOOKMARK [1][]{section.2.1}{2.1 G\351n\351ralisation au cadre asynchrone}{chapter.2}% 8
+\BOOKMARK [1][]{section.2.2}{2.2 Exemple jouet}{chapter.2}% 9
+\BOOKMARK [0][]{chapter.3}{3 Preuve de convergence de syst\350mes bool\351ens}{part.1}% 10
+\BOOKMARK [1][]{section.3.1}{3.1 Exemple jouet}{chapter.3}% 11
+\BOOKMARK [1][]{section.3.2}{3.2 Rappels sur le langage PROMELA}{chapter.3}% 12
+\BOOKMARK [1][]{section.3.3}{3.3 Du syst\350me bool\351en au mod\350le PROMELA}{chapter.3}% 13
+\BOOKMARK [2][]{subsection.3.3.1}{3.3.1 La strat\351gie}{section.3.3}% 14
+\BOOKMARK [2][]{subsection.3.3.2}{3.3.2 It\351rer la fonction f}{section.3.3}% 15
+\BOOKMARK [2][]{subsection.3.3.3}{3.3.3 Gestion des d\351lais}{section.3.3}% 16
+\BOOKMARK [2][]{subsection.3.3.4}{3.3.4 Propri\351t\351 de convergence universelle}{section.3.3}% 17
+\BOOKMARK [1][]{section.3.4}{3.4 Correction et compl\351tude de la d\351marche}{chapter.3}% 18
+\BOOKMARK [1][]{section.3.5}{3.5 Donn\351es pratiques}{chapter.3}% 19
+\BOOKMARK [1][]{section.3.6}{3.6 Conclusion}{chapter.3}% 20
+\BOOKMARK [0][]{appendix.A}{A Preuves sur les SDD}{part.1}% 21
+\BOOKMARK [1][]{section.A.1}{A.1 Preuve du th\351or\350me ??}{appendix.A}% 22
+\BOOKMARK [1][]{section.A.2}{A.2 Preuve de continuit\351 de Gf dans \(X,d\)}{appendix.A}% 23
+\BOOKMARK [1][]{section.A.3}{A.3 Preuve de Correction et de compl\351tude de l'approche de v\351rification de convergence \340 l'aide de SPIN}{appendix.A}% 24
\newcommand{\Bool}[0]{\ensuremath{\mathds{B}}}
\newcommand{\rel}[0]{\ensuremath{{\mathcal{R}}}}
\newcommand{\Gall}[0]{\ensuremath{\mathcal{G}}}
-\newcommand{\Sec}[1]{Sect.\,\ref{#1}}
-\newcommand{\Fig}[1]{Fig.\,\ref{#1}}
-\newcommand{\Alg}[1]{Algorithm~\ref{#1}}
-\newcommand{\Tab}[1]{Table~\ref{#1}}
+\newcommand{\Sec}[1]{Sect\,\ref{#1}}
+\newcommand{\Fig}[1]{{\sc Figure}~\ref{#1}}
+\newcommand{\Alg}[1]{Algorithme~\ref{#1}}
+\newcommand{\Tab}[1]{Tableau~\ref{#1}}
\newcommand{\Equ}[1]{(\ref{#1})}
\newcommand{\deriv}{\mathrm{d}}
\newcommand{\class}[1]{\ensuremath{\langle #1\rangle}}
\newtheorem{theorem}{Théorème}
\newtheorem{lemma}{Lemme}
-\newtheorem{xpl}{Exemple}
+\newtheorem*{xpl}{Exemple}
\newtheorem{Proof}{Preuve}
\begin{document}
\input{sdd}
+\chapter{Combinaisons Synchrones et Asynchrones de Systèmes Booléens}
+\input{mixage}
+
+
+
\chapter[Preuve de convergence de systèmes booléens]{Preuve automatique de convergence de systèmes booléens}\label{chap:promela}
\input{modelchecking}
--- /dev/null
+\JFC{Refaire le châpeau}
+\section{Généralisation au cadre asynchrone}
+Dans ce chapitre une stratégie $s=(s^{t})^{t \in \Nats}$ est une séquence
+\emph{des éléments} qui sont mis à jour au temps $t$. Pratiquement,
+on représente ceci comme un nombre décimal dont la représentation en
+binaire donne la liste des éléments modifiés. Par exemple, pour un système
+à 5 éléments la stratégie définie par
+\begin{equation}\label{eq:pseudo}
+s^{t}=24 \textrm{ si $t$ est pair et } s^{t}=15 \textrm{ sinon }
+\end{equation}
+\noindent active successivement les deux premiers éléments (24 est 11000)
+et les quatre derniers élements (15 est 01111).
+On dit que la stratégie est
+\emph{pseudo-periodique} si tous les éléments sont activés infiniment
+souvent.
+% , it is sufficient to establish that the set $\{t \mid t \in \mathbb{N}
+% \land \textit{bin}(s^t)[i] = 1\}$ is infinite for any $i$, $1 \le i \le n$,
+% where
+% The synchronous iterations modes are defined for any $i \in
+% \{1,\ldots,n\}$ and any time $t=0,1,2,...$ by:
+% \vspace{-.5em}
+% \begin{equation}\label{eq:sync}
+% x^{t+1}_i= \left\{
+% \begin{array}{l}
+% f_i(x^t) \textrm{ if } \textit{bin}(s^t)[i] = 1\\
+% x^{t}_i \textrm{ otherwise }
+% \end{array}
+% \right.
+% \end{equation}
+% \vspace{-.5em}
+% Notice that parallel iterations only constrain $s^t$ to be equal to $2^n-1$
+% for any $t$ whereas chaotic iterations do not constrain $s$.
+% for convenient reasons [[JFC : a affiner]], the set of components $\{1,
+% \ldots, n\}$ may be partitioned into $\alpha$ blocks $b_1, \ldots,
+% b_{\alpha}$.
+% %Elements of $b_i$ are ordered w.r.t. the component number.
+% For $1\le i \le \alpha$, let $B_i$ be the product-space of block $i$.
+% Formaly, $B_i = \Pi_{j \in b_{i} } E_j$.
+% To ease the reading, lowercase variable and upercase one represent
+% respectively an element of some $E_i$ and a matrix of elements in some $E_i$.
+% The components may be updated (in a random order) according to a
+% strategy $s$, as in the synchronous mode.
+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 dépent du temps de calcul et
+du temps d'acheminement de celles-ci. On parle de latence, de délai.
+
+Formalisons le mode les itérations asynchrones.
+Soit $x^0 =(x_1^0, \ldots, x_n^0)$ une configuration initiale.
+Soit $(D^{t})^{t \in \Nats}$ la suite de matrice de taille $n \times n$
+dont chaque élément $D_{ij}^{t}$ represente la date (inférieure ou égale à $t$)
+à laquelle la valeur $x_j$ produite par le composant $j$ devient
+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 asynchrones} est défini pour chaque $i
+\in \{1,\ldots,n\}$ et chaque $t=0,1,2,...$ par:
+
+\vspace{-.5em}
+\begin{equation}\label{eq:async}
+ 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\\
+ x^{t}_i \textrm{ sinon }
+ \end{array}
+ \right.
+\end{equation}
+
+\noindent où $\textit{bin}$ convertit un entier en un nombre binaire.
+Les itérations de $f$ sont \emph{convergentes} modulo une configuration
+initiale $x^0$, une stratégi $s$ et une matrice de dates $(D^{t})^{t \in
+ \Nats}$, si la fonction atteint un point fixe.
+Cela revient à vérifier la propriété suivante:
+\begin{equation}\label{eq:conv}
+\exists t_0 \,.\,
+(\forall t \,.\,
+t \geq t_0 \Rightarrow x^{t}=x^{t_0}).
+\end{equation}
+Sinon les itérations sont dites \emph{divergentes}.
+De plus, si $ (x^{(t)})^{t \in \mathbb{N}}$ défini selon l'équation
+\Equ{eq:async} satisfait \Equ{eq:conv} pour tous les $x^{(0)}
+\in E$, pour toutes les stratégies pseudo périodiques
+$s$ et pour toutes les matrices de dates,
+$(D^{(t)})^{t \in \Nats}$, alors les itérations de $f$ sont
+\emph{universellement convergentes}.
+
+
+\section{Exemple jouet}
+On considère cinq éléments prenant 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'intéraction associé à cette fonction.
+On note que le graphe d'intéraction contient cinq cycles. Les résultats
+connus~\cite{Bah00} de conditions suffisantes établissant la convergencedu système pour les itérations synchrones et asynchrones 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{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} \\
+ f_3(x_1,x_2,x_3,x_4,x_5) & = & x_3.\overline{x_1} \\
+ 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]{images/xplgraphmix.eps}
+ \end{center}
+ \caption{Graphe d'interaction associé à $f$.}
+ \label{fig:mix:xplgraph}
+\end{minipage}
+\end{figure}
+
+
+\begin{figure}
+\begin{minipage}{0.56\linewidth}
+ \includegraphics[scale=0.55]{images/para_iterate_dec.eps}
+ \caption{Itérations parallèlles de $f$.}\label{fig:mix:xplparaFig}
+\end{minipage}
+\hfill
+\begin{minipage}{0.39\linewidth}
+ \includegraphics[scale=0.55]{images/chao_iterate_excerpt.eps}
+ \caption{Extrait d'itérations chaotiques.}
+ \label{fig:mix:xplchaoFig}
+\end{minipage}
+\end{figure}
+
+Dans ce qui suit, les configurations sont representées à l'aide d'entiers
+plutôt que nombres binaires. Le graphe des itérations parallèles 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
+la~\Fig{fig:mix:xplchaoFig}. Les libélés des arcs correspondent aux éléments
+activés. Les itérations chaotiques 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,
+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.
+
+
+
+
+
+
+ For instance, consider the matrix $D^t$ to be equal to $(t)$ except
+in $D^t_{12}$ where it is equal to $t-1$ if $t$ is odd. Then if $t$ is even,
+$x^{t+1}$ is $f(x^{t})$; if $t$ is odd we have
+$$
+x^{t+1} = \left(
+f_1(x_1^{t},x_2^{t-1},x_3^{t},x_4^{t},x_5^{t}), f_2(x^{t}), \ldots,
+f_5(x^{t})
+\right).
+$$
+\noindent Starting from $x^0=00011$, the system reaches $x^1 = 01011$ and enters
+in a cycle between these two configurations. We are then confronted to
+divergent asynchronous iterations whereas the synchronous ones converge. In the
+next section, a particular execution mode is described which enables
+asynchronism in iterations while guaranteeing the convergence.
+
+
-\JFC{donner dans les rappels les délais et les propriétés de convergence uniforme}
+\JFC{donner dans les rappels les délais et les propriétés de convergence universelle}
+
+\JFC{Statuer sur la taille des exemples traitables par la démarche, cf données pratiques}
+
+\section{Exemple jouet}
+
+\begin{xpl}
+ On considère dans ce chapitre l'exemple où trois éléments dans $\Bool$.
+ Chaque configuration est ainsi un élement de $\{0,1\}^3$, \textit{i.e.},
+ un nombre entre 0 et 7.
+ La \Fig{fig:map} précise la fonction $f$ considérée et
+ la \Fig{fig:xplgraph} donne son graphe d'intéraction.
+
+\begin{figure}[ht]
+ \centering
+ \begin{minipage}%[h]
+ {.6\linewidth}
+ \begin{center}
+ $ F(x)= \left \{
+ \begin{array}{rcl}
+ f_1(x_1,x_2,x_3) & = & x_1.\overline{x_2} + x_3 \\
+ f_2(x_1,x_2,x_3) & = & x_1 + \overline{x_3} \\
+ f_3(x_1,x_2,x_3) & = & x_2.x_3
+ \end{array}
+ \right.
+ $
+ \end{center}
+ \caption{Fonction à itérer} \label{fig:map}
+ \end{minipage}
+ \begin{minipage}%[h]
+ {.35\linewidth}
+ \begin{center}
+ \includegraphics[width=4cm]{images/xplCnxMc.eps}
+ \end{center}
+ \caption{Graphe d'intéraction}
+ \label{fig:xplgraph}
+ \end{minipage}
+ \caption{Exemple pour SDD $\approx$ SPIN.}
+\end{figure}
+
+
+
+
+
+
+On peut facilement vérifier que toutes les itérations parallèles initialisées
+avec $x^0 \neq 7$ soit $(111)$
+convergent vers $2$ soit $(010)$; celles initialisées avec
+$x^0=7$ restent en 7.
+Pour les autres modes synchrones avec une
+stratégie pseudo périodique, les comportements selon la configuration initiale:
+\begin{itemize}
+\item initialisée avec 7, les itérations restent en 7;
+\item initialisée avec 0, 2, 4 ou 6 les itérations convergent vers 2;
+\item initialisées avec 1, 3 ou 5, les itérations convergent vers un des
+deux points fixes 2 ou 7.
+\end{itemize}
+\end{xpl}
+
+
+
\section{Rappels sur le langage PROMELA}
on peut déclarer des tableaux à une dimension de taille constante
ou des nouveaux types de données (introduites par le mot clef
\verb+typedef+). Ces derniers sont utilisés pour définir des tableaux à deux
-dimension.
+dimensions.
\begin{xpl}
Le programme donné à la {\sc Figure}~\ref{fig:arrayofchannels} correspond à des
\item les constantes \verb+N+ et \verb+d_0+ qui précisent respectivement le nombre
$n$ d'éléments et le délais maximum $\delta_0$;
\item les deux tableaux (\verb+X+ et \verb+Xp+) de \verb+N+ variables booléennes;
-les cellules \verb+X[i]+ et \verb+Xp[i]+ sont associées à la variables $X_{i+1}$
+les cellules \verb+X[i]+ et \verb+Xp[i]+ sont associées à la variables $x_{i+1}$
d'un système dynamique discret
(le décalages d'un entier est dû à l'indexation à partir de zéro des cellules d'un tableau);
Elles mémorisent les valeurs de $X_{i+1}$ respectivement avant et après sa mise à jour;
-il suffit ainsi de comparer \verb+X+ et \verb+Xp+ pour constater si $X$ à changé ou pas;
+il suffit ainsi de comparer \verb+X+ et \verb+Xp+ pour constater si $x$ à changé ou pas;
\item le tableau \verb+mods+ contient les éléments qui doivent être modifiés lors de l'itération
-en cours; cela correspond naturellement à l'ensemble des éléments $S^t$;
+en cours; cela correspond naturellement à l'ensemble des éléments $s^t$;
\item le type de données structurées \verb+vals+ et le tableau de tableaux
- \verb+Xd[+$i$\verb+].v[+$j$\verb+]+ qui vise à mémoriser $X_{j+1}^{D^{t-1}_{i+1j+1}}$
- pour l'itération au temps $t$ (en d'autres termes, utile lors du calcul de $X^{t}$).
+ \verb+Xd[+$i$\verb+].v[+$j$\verb+]+ qui vise à mémoriser $x_{j+1}^{D^{t-1}_{i+1j+1}}$
+ pour l'itération au temps $t$ (en d'autres termes, utile lors du calcul de $x^{t}$).
\end{itemize}
Puisque le décalage d'un indices ne change pas fondamentalement
le comportement de la version PROMELA par rapport au modèle initial
et pour des raisons de clarté, on utilisera par la suite la même
-lettre d'indice entre les deux niveaux (pour le modèle: $X_i$ et pour PROMELA:
+lettre d'indice entre les deux niveaux (pour le modèle: $x_i$ et pour PROMELA:
\texttt{X[i]}). Cependant, ce décalage devra être conservé mémoire.
Une donnée de type \texttt{channel} permet le
\begin{itemize}
\item un canal \verb+sent+ qui vise à mémoriser\verb+d_0+ messages de type
\verb+bool+; le tableau nommé \verb+channels+ de \verb+N+*\verb+N+
- éléments de type \verb+a_send+ est utilisé pour mémoriser les valeurs intermédiaires $X_j$;
+ éléments de type \verb+a_send+ est utilisé pour mémoriser les valeurs intermédiaires $x_j$;
Il permet donc de temporiser leur emploi par d'autres elements $i$.
\item les deux canaux \verb+unlock_elements_update+ et \verb+sync_mutex+ contenant
chacun un message booléen et utilisé ensuite comme des sémaphores.
Les instructions d'affectation sont interprétées usuellement.
-Les canaux sont cernés par des instructions particulières d'envoi et de
+Les canaux sont concernés par des instructions particulières d'envoi et de
réception de messages. Pour un canal
-\verb+ch+, ces instruction sont respectivement notées
-\verb+ch ! m+ et \verb+ch ? m+.
+\verb+ch+, ces instructions sont respectivement notées
+\verb+ch ! m+ et \verb+ch ? m+.
L'instruction de réception consomme la valeur en tête du canal \verb+ch+
et l'affecte à la variable \verb+m+ (pour peu que \verb+ch+ soit initialisé et non vide).
-De manière similaire,l'instruction d'envoi ajoute la valeur de \verb+m+ à la queue du canal
+De manière similaire, l'instruction d'envoi ajoute la valeur de \verb+m+ à la queue du canal
\verb+ch+ (pour peu que celui-ci soit initialisé et non rempli).
Dans les cas problématiques, canal non initialisé et vide pour une réception ou bien rempli pour un envoi,
le processus est bloqué jusqu'à ce que les conditions soient remplies.
Dans le process \verb+init+ détaillé à la {\sc Figure}~\ref{fig:spin:init},
une boucle de taille $N$ initialise aléatoirement la variable globale de type tableau \verb+Xp+.
Ceci permet par la suite de vérifier si les itérations sont convergentes pour n'importe
-quelle configuration initiale $X^{(0)}$.
+quelle configuration initiale $x^{(0)}$.
Pour chaque élément $i$, si les itérations sont asynchrones
\begin{itemize}
\item on stocke d'abord la valeur de \verb+Xp[i]+ dans chaque \verb+Xd[j].v[i]+
-puisque la matrice $S^0$ est égale à $(0)$,
+puisque la matrice $s^0$ est égale à $(0)$,
\item puis, la valeur de $i$ (représentée par \verb+Xp[i]+) devrait être transmise
à $j$ s'il y a un arc de $i$ à $j$ dans le graphe d'incidence. Dans ce cas,
- c'est la fonction \verb+hasnext+ (non détaillée ici)
- \JFC{la détailler}
+ c'est la fonction \verb+hasnext+ (détaillée à la~\Fig{fig:spin:hasnext})
qui mémorise ce graphe
en fixant à \texttt{true} la variable \verb+is_succ+, naturellement et à
\texttt{false} dans le cas contraire.
\end{itemize}
\begin{figure}[t]
- \begin{minipage}[h]{.52\linewidth}
+ \begin{minipage}[h]{.32\linewidth}
\begin{tiny}
\begin{lstlisting}
init{
\end{tiny}
\caption{Process init.}\label{fig:spin:init}
\end{minipage}\hfill
- \begin{minipage}[h]{.42\linewidth}
+ \begin{minipage}[h]{.32\linewidth}
\begin{tiny}
\begin{lstlisting}
active proctype scheduler(){
\caption{Process scheduler pour la stratégie pseudo pérodique.
\label{fig:scheduler}}
\end{minipage}
+\begin{minipage}[h]{.30\linewidth}
+\begin{tiny}
+\begin{lstlisting}
+inline hasnext(i,j){
+ if
+ :: i==0 && j ==0 -> is_succ = 1
+ :: i==0 && j ==1 -> is_succ = 1
+ :: i==0 && j ==2 -> is_succ = 0
+ :: i==1 && j ==0 -> is_succ = 1
+ :: i==1 && j ==1 -> is_succ = 0
+ :: i==1 && j ==2 -> is_succ = 1
+ :: i==2 && j ==0 -> is_succ = 1
+ :: i==2 && j ==1 -> is_succ = 1
+ :: i==2 && j ==2 -> is_succ = 1
+ fi
+}
+\end{lstlisting}
+\end{tiny}
+\caption{Codage du graphe d'intéraction de $f$.
+ \label{fig:spin:hasnext}}
+\end{minipage}
+
\end{figure}
\subsection{La stratégie}\label{sub:spin:strat}
Regardons comment une stratégie pseudo périodique peut être représentée en PROMELA.
Intuitivement, un process \verb+scheduler+ (comme représenté à la {\sc Figure}~\ref{fig:scheduler})
-est iterrativement appelé pour construire chaque $S^t$ représentant
+est iterrativement appelé pour construire chaque $s^t$ représentant
les éléments possiblement mis à jour à l'itération $t$.
Basiquement, le process est une boucle qui est débloquée lorsque la valeur du sémaphore
aléatoirement (grâce à $n$ choix successifs) et sont mémorisés dans le tableau
\verb+mods+, dont la taille est \verb+ar_len+.
Dans la séquence d'exécution, le choix d'un élément mis à jour est directement
-suivi par des mis à jour: ceci est réalisé grâce à la modification de la valeur du sémaphore
+suivi par des mises à jour: ceci est réalisé grâce à la modification de la valeur du sémaphore
\verb+unlock_elements_updates+.
-\subsection{Itérer la fonction $F$}\label{sub:spin:update}
-La mise à jour de l'ensemble $S^t=\{s_1,\ldots, s_m\}$ des éléments qui constituent la stratégie
-$(S^t)^{t \in \Nats}$ est implanté à l'aide du process \verb+update_elems+ fourni à la
+\subsection{Itérer la fonction $f$}\label{sub:spin:update}
+La mise à jour de l'ensemble $s^t=\{s_1,\ldots, s_m\}$ des éléments qui constituent la stratégie
+$(s^t)^{t \in \Nats}$ est implantée à l'aide du process \verb+update_elems+ fourni à la
{\sc Figure}~\ref{fig:proc}.
Ce process actif attend jusqu'à ce qu'il soit débloqué par le process
\verb+scheduler+ à l'aide du sémaphore \verb+unlock_elements_update+.
-L'implantation contient donc cinq étapes:
-
-\begin{enumerate}
-\item elle commence en mettant à jour la variable \texttt{X} avec les valeurs de \texttt{Xp}
- dans la fonction \texttt{update\_X} (non détaillée ici);
-\item elle mémorise dans \texttt{Xd} la valeurs disponibles des éléments grâce à
- la fonction \texttt{fetch\_values} (cf. \Sec{sub:spin:vt});
-\item une boucle sur les \texttt{ar\_len} éléments qui peuvent être modifiés
- met à jour iterrativement la valeur de $j$ (grâce à l'appel de fonction \texttt{F(j)})
- pour peu que celui-ci doive être modifié, \textit{i.e.}, pour peu qu'il soit renseigné dans
- \texttt{mods[count]}; le code source de \texttt{F} est donné en {\sc Figure}~\ref{fig:p} et est une
- traduction directe de l'application $F$;
-\item les nouvelles valeurs des éléments \texttt{Xp} sont symboliquement envoyés aux
- autres éléments qui en dépendent grâce à la fonction
- \texttt{diffuse\_values(Xp)} (cf. \Sec{sub:spin:vt});
-\item finalement, le process informe le scheduler de la fin de la tâche
- (au travers du sémaphore \texttt{sync\_mutex}).
-\end{enumerate}
-
+L'implantation se déroule en cinq étapes:
\begin{figure}[t]
- \begin{minipage}[h]{.475\linewidth}
+\begin{minipage}[b]{.32\linewidth}
+\begin{tiny}
+\begin{lstlisting}
+inline update_X(){
+ int countu;
+ countu = 0;
+ do
+ :: countu == N -> break ;
+ :: countu != N ->
+ X[countu] = Xp[countu];
+ countu ++ ;
+ od
+}
+\end{lstlisting}
+\end{tiny}
+\caption{Sauvegarde de l'état courant}\label{fig:spin:sauve}
+\end{minipage}\hfill%
+\begin{minipage}[b]{.32\linewidth}
\begin{tiny}
\begin{lstlisting}
active proctype update_elems(){
\end{minipage}\hfill%
%\end{figure}
%\begin{figure}
- \begin{minipage}[h]{.45\linewidth}
+ \begin{minipage}[b]{.33\linewidth}
\begin{tiny}
\begin{lstlisting}
inline F(){
}
\end{lstlisting}
\end{tiny}
-\caption{Application de la fonction $F$.}\label{fig:p}
+\caption{Application de la fonction $f$.}\label{fig:p}
\end{minipage}
\end{figure}
+\begin{enumerate}
+\item elle commence en mettant à jour la variable \texttt{X} avec les valeurs de \texttt{Xp} dans la fonction \texttt{update\_X},~\Fig{fig:spin:sauve}
+\item elle mémorise dans \texttt{Xd} la valeurs disponible pour chaque élément grâce à la fonction \texttt{fetch\_values}; cette fonction est détaillée
+dans la section suivante;
+\item une boucle %sur les \texttt{ar\_len} éléments qui peuvent être modifiés
+ met à jour iterrativement la valeur de $j$ (grâce à l'appel de fonction \texttt{f(j)})
+ pour peu que celui-ci doive être modifié, \textit{i.e.}, pour peu qu'il soit renseigné dans
+ \texttt{mods[count]}; le code source de \texttt{F} est donné en {\sc Figure}~\ref{fig:p} et est une
+ traduction directe de l'application $f$;
+\item les nouvelles valeurs des éléments \texttt{Xp} sont symboliquement
+ envoyés aux autres éléments qui en dépendent grâce à la fonction
+ \texttt{diffuse\_values(Xp)}; cette dernière fonction est aussi détaillée
+ dans la section suivante;
+\item finalement, le process informe le scheduler de la fin de la tâche
+ (au travers du sémaphore \texttt{sync\_mutex}).
+\end{enumerate}
+
+
+
+
+
+
+
+
\subsection{Gestion des délais}\label{sub:spin:vt}
Cette section montre comment les délais inhérents au mode asynchrone sont
traduits dans le modèle PROMELA grâce à deux
qui doivent être modifiés.
Pour chaque élément dans \verb+mods+, identifié par la variable
$j$, la fonction récupère les valeurs des autres éléments (dont le libellé est $i$)
-dont $j$ dépend. \JFC{vérifier si c'est ce sens ici}
+dont $j$ dépend.
Il y a deux cas.
\begin{itemize}
\item puisque $i$ connaît sa dernière valeur (\textit{i.e.}, $D^t_{ii}$ est toujours $t$)
\item depuis la perspective de $j$ la valeur de $i$ peut ne pas avoir changé (
c'est l'instruction \verb+skip+) ou n'est pas utile; ce dernier cas apparaît
lorsqu'il n'y a pas d'arc de $i$ à $j$ dans le graphe d'incidence, \textit{i.e.}, lorsque
- la valeur de \verb+is_succ+ qui est calculée par \verb+hasnext(i,j)+ is 0;
+ la valeur de \verb+is_succ+ qui est calculée par \verb+hasnext(i,j)+ est 0;
dans ce cas, la valeur de \verb+Xd[j].v[i]+ n'est pas modifiée;
\item sinon, on affecte à \verb+Xd[j].v[i]+ la valeur mémorisée
dans le canal \verb+channels[i].sent[j]+ (pour peu que celui-ci ne soit pas vide).
- Les valeurs des éléments sont ajoutées dans ce canal au travers de la fonction \verb+diffuse_values+
- donnée juste après.
\end{itemize}
\end{itemize}
-L'objectif de la fonction \verb+diffuse_values+ est de stocker les valeurs de $X$ représenté
-dans le modèle par \verb+Xp+ dans le canal \verb+channels+.
+Les valeurs des éléments sont ajoutées dans ce canal au travers de la fonction \verb+diffuse_values+. L'objectif de cette fonction
+est de stocker les valeurs de $x$ (représenté
+dans le modèle par \verb+Xp+) dans le canal \verb+channels+.
Il permet au modèle-checker SPIN d'exécuter
le modèle PROMELA comme s'il pouvait y avoir des délais entre processus
Il y a deux cas différents pour la valeur de $X_{j}$:
L'introduction de l'indéterminisme à la fois dans les fonctions \verb+fetch_values+
et \verb+diffuse_values+ est nécessaire dans notre contexte. Si celui-ci n'était
-présent que dans la fonction \verb+fetch_values+, nous ne pourrions pas par exemple récupérer
-la valeur $X_i^{(t)}$ sans considérer la valeur $X_i^{(t-1)}$.
+présent que dans la fonction \verb+fetch_values+, nous ne pourrions pas par exemple récupérer *
+la valeur $x_i^{(t)}$ sans considérer la valeur $x_i^{(t-1)}$.
De manière duale, si le non déterminisme était uniquement
utilisé dans la fonction \verb+diffuse_values+, alors chaque fois qu'une valeur serait
-mise dans le canal, elle serait immédiatement consommé, ce qui est contradictoire avec la notion de
+mise dans le canal, elle serait immédiatement consommée, ce qui est contradictoire avec la notion de
délai.
% \subsection{Discussion}
% instructions, making the PROMELA code and the DDN as closed as possible.
\subsection{Propriété de convergence universelle}
-Il reste à formaliser dans le model checker SPIN que les itérations d'un système
+Il reste à formaliser dans le model checker SPIN le fait que les
+itérations d'un système
dynamique à $n$ éléments est universellement convergent.
Rappelons tout d'abord que les variables \verb+X+ et \verb+Xp+
-contiennent respectivement la valeur de $X$ avant et après la mise à jour.
+contiennent respectivement la valeur de $x$ avant et après la mise à jour.
Ainsi, si l'on effectue une initialisation non déterministe de
\verb+Xp+ et si l'on applique une stratégie pseudo périodique,
il est nécessaire et suffisant
\diamond (\Box \verb+Xp+ = \verb+X+)
\label{eq:ltl:conv}
\end{equation}
-où les opérateur $\diamond$ et $\Box$ on la sémantique usuelle \textit{i.e.}, à savoir
+où les opérateur $\diamond$ et $\Box$ ont
+la sémantique usuelle, à savoir
respectivement {\em éventuellement} et {\em toujours} dans les chemins suivants.
-On note que cette propriété assure seulement la stabilisation du système,
-mais ne donne aucune métrique quant à la manière dont celle-ci est obtenue.
+On note que cette propriété, si elle est établie, garantit
+la stabilisation du système.
+Cependant elle ne donne aucune métrique quant à
+la manière dont celle-ci est obtenue.
En particulier, on peut converger très lentement ou le système peut même
disposer de plusieurs points fixes.
-vers plusieurs
+
\section{Correction et complétude de la démarche}\label{sec:spin:proof}
-Cette section présente les théorème de correction et de complétude de l'approche.
+Cette section présente les théorèmes
+de correction et de complétude de l'approche.
(Théorèmes~\ref{Theo:sound} et~\ref{Theo:completeness}).
Toutes les preuves sont déplacées en
annexes~\ref{anx:promela}.
puis présente ensuite les expérimentations issues de ce travail.
\begin{theorem}[Nombre d'états ]
- Soit $\phi$ un modèle de système dynamique discret à $n$ éléments, $m$ arc dans le graphe d'incidence
+ Soit $\phi$ un modèle de système dynamique discret à $n$ éléments, $m$
+ arcs dans le graphe d'incidence
et $\psi$ sa traduction en PROMELA. Le nombre de configurations
- de l'exécution en SPIN de $\psi$ est bornée par $2^{m\times(\delta_0+1)+n(n+2)}$.
+ de l'exécution en SPIN de $\psi$ est bornée par $2^{m(\delta_0+1)+n(n+2)}$.
\end{theorem}
\begin{Proof}
Une configuration est une valuation des variables globales.
Chaque canal de \verb+array_of_channels+
peut engendrer $1+2^1+\ldots+2^{\delta_0}= 2^{\delta_0+1}-1$ états.
Puisque le nombre d'arêtes du graphe d'incidence est $m$,
- il y a $m$ canaux non constants, ce qui génère approximativement $2^{m\times(\delta_0+1)}$ états.
- Le nombre de configurations est donc borné par $2^{m\times(\delta_0+1)+n(n+2)}$.
+ il y a $m$ canaux non constants, ce qui génère approximativement $2^{m(\delta_0+1)}$ états.
+ Le nombre de configurations est donc borné par $2^{m(\delta_0+1)+n(n+2)}$.
On remarque que cette borne est traitable par SPIN pour des valeurs raisonnables de $n$,
$m$ et $\delta_0$.
\JFC{Donner un ordre de grandeur de cet ordre de grandeur}
\end{Proof}
La méthode détaillée ici a pu être appliquée sur l'exemple jouet
-pour prouver formellement sa convergence uniforme.
+pour prouver formellement sa convergence universelle.
On peut remarquer que SPIN n'impose l'équité faible qu'entre les process
alors que les preuves des deux théorèmes précédentes reposent sur le fait que
celle-ci est établie dès qu'un choix indéterministe est effectué.
Naïvement, on pourrait considérer comme hypothèse la formule suivante
chaque fois qu'un choix indéterministe se produit entre $k$ événements
-respectivement notés $l1$, \ldots $lk$:
+respectivement notés $l_1$, \ldots $l_k$:
$$
-[] <> (l == l0) \Rightarrow
-(([] <> (l== l1)) \land \ldots \land ([] <> (l == lk)))
+\Box \diamond (l == l_0) \Rightarrow
+((\Box \diamond (l== l_1)) \land \ldots \land (\Box \diamond (l == l_k)))
$$
-où le libellé $l0$ dénote le libellé de la ligne précédent le choix indéterministe.
+où le libellé $l_0$ dénote le libellé de la ligne précédent
+le choix indéterministe.
Cette formule traduit exactement l'équité faible.
Cependant en raison de l'explosion de la taille du produit entre
l'automate de Büchi issu de cette formule et celui issu du programme PROMELA,
-SPIN n'arrive pas à vérifier si la convergence uniforme est établie ou non sur des exemples
+SPIN n'arrive pas à vérifier si la convergence universelle est établie
+ou non sur des exemples
simples\JFC{faire référence à un tel exemple}.
-Ce problème a été pratiquement résolu en laissant SPIN générer toutes les traces d'exécution,
-même les non équitables, puis en le laissant vérifier la propriété de convergence dessus.
+Ce problème a été pratiquement résolu en laissant SPIN
+générer toutes les traces d'exécution,
+même celles qui ne sont pas équitables,
+puis ensuite vérifier la propriété de convergence sur toutes celles-ci.
Il reste alors à interpréter les résultats qui peuvent être de deux types. Si la convergence est
établie pour toutes les traces, elle le reste en particulier pour les traces équitables.
Dans le cas contraire on doit analyser le contre exemple produit par SPIN.
La méthode détaillée ici a été appliquée sur des exemples pour prouver formellement
-leur convergence ou leur divergence (Fig.~\ref{fig:async:exp}).
-Dans ces expériences, les délais on été bornés par $\delta_0=10$.
-Dans ce tableau, $P$ est vrai ($\top$) si et seulement si la convergence uniforme
-est établie et faux ($\bot$) sinon. Le nombre $M$ et la taille de la mémoire consommée (en MB) et
+leur convergence ou leur divergence (\Fig{fig:async:exp}).
+Dans ces expériences, les délais ont été bornés par $\delta_0=10$.
+Dans ce tableau, $P$ est vrai ($\top$) si et seulement si la convergence
+universelle
+est établie et faux ($\bot$) sinon. Le nombre $M$ est
+la taille de la mémoire consommée (en MB) et
$T$ est le temps d'exécution sur un Intel Centrino Dual Core 2 Duo @1.8GHz avec 2GB de mémoire vive
pour établir un verdict.
-L'exemple RE est l'exemple jouet de ce chapitre,
-AC2D est un automate cellulaire avec 9 elements prenant des valeurs booléennes en fonction de
-de 4 voisins et BM99, issu de~\cite{BM99} consiste en 10 process
-qui modifient leur valeur booléennes dans un graphe d'adjacence proche du graphe complet.
+
+\begin{figure}
+\begin{center}
+\begin{tiny}
+\begin{tabular}{|*{7}{c|}}
+\cline{2-7}
+\multicolumn{1}{c|}{ }
+&\multicolumn{3}{|c|}{Parallèles} & \multicolumn{3}{|c|}{Chaotiques} \\
+\cline{2-7}
+\multicolumn{1}{c|}{ }&
+P & M & T&
+P & M & T \\
+\hline %\cline{2-7}
+\textit{RE} &
+$\top$ & 2.7 & 0.01s &
+$\bot$ & 369.371 & 0.509s \\
+\hline %\cline{2-7}
+\cite{RC07} &
+$\bot$ & 2.5 & 0.001s & % RC07_sync.spin
+$\bot$ & 2.5 & 0.01s \\ % RC07_sync_chao_all.spin
+\hline
+\cite{BM99} &
+$\top$ & 36.7 & 12s & % BM99_sync_para.spin
+$\top$ & & \\ % BM99_sync_chao.spin
+\hline
+\end{tabular}
+\end{tiny}
+\end{center}
+\caption{Expérimentations avec des itérations synchrones}\label{fig:sync:exp}
+\end{figure}
+
\begin{figure}
P & M & T&
P & M & T \\
\hline %cline{2-13}
-Running Example &
+\textit{RE} &
$\top$ & 409 & 1m11s&
$\bot$ & 370 & 0.54 &
$\bot$ & 374 & 7.7s&
&$\bot$ & 2.5 & 0.01s % RC07_async.spin
&$\bot$ & 2.5 & 0.01s \\ % RC07_async_all.spin
\hline %\cline{2-13}
-BM99
+\cite{BM99}
&$\top$ & & %BM99_mixed_para.spin
&$\top$ & & % RC07_async_mixed_all.spin
&$\bot$ & & % RC07_async.spin
-L'exemple~\cite{RC07} concerne un réseau composé de deux gènes à valeur dans $\{0,1,2\}$.
-Comme la convergence n'est déjà pas établie pour les itérations parallèles, il en est donc
+L'exemple \textit{RE} est l'exemple jouet de ce chapitre,
+\cite{RC07} concerne un réseau composé de deux gènes
+à valeur dans $\{0,1,2\}$,
+AC2D est un automate cellulaire avec 9 elements prenant des
+valeurs booléennes en fonction de
+de 4 voisins et
+\cite{BM99} consiste en 10 process
+qui modifient leur valeur booléennes dans un graphe d'adjacence proche
+du graphe complet.
+
+
+L'exemple jouet \textit{RE} a été prouvé comme universellement convergent.
+\JFC{statuer sur AC2D}
+Comme la convergence n'est déjà pas établie pour les itérations parallèles
+de~\cite{RC07}, il en est donc
de même pour les itérations asynchrones.
-LA {\sc Figure}~\ref{fig:RC07CE} donne une trace de la sortie de SPIN de menant à la violation
+La {\sc Figure}~\ref{fig:RC07CE} donne une trace de la sortie de SPIN de menant à la violation
de la convergence. Celle-ci correspond à une stratégie périodique qui répète
-$\{1,2\};\{1,2\};\{1\};\{1,2\};\{1,2\}$ et débute avec $X=(0,0)$.
-
-
-Dans l'exemple issu de~\cite{BM99}, nous avons 10 process
-à valeur booléennes. En raison de la dépendance forte entre les éléments,
+$\{1,2\};\{1,2\};\{1\};\{1,2\};\{1,2\}$ et débute avec $x=(0,0)$.
+En raison de la dépendance forte entre les éléments
+de~\cite{BM99},
$\delta_0$ est réduit à 1. Cela aboutit cependant à $2^{100}$
configurations dans le mode des itérations asynchrones.
+
+\JFC{Quid de ceci?}
La convergence des itérations asynchrones de l'exemple~\cite{BCVC10:ir} n'est pas établie
-lorsque pour $\delta_0$ vaut 1. Il ne peut donc y avoir convergence uniforme.
+lorsque pour $\delta_0$ vaut 1. Il ne peut donc y avoir convergence universelle.
\begin{figure}
-\begin{center}
-\begin{tiny}
-\begin{tabular}{|*{7}{c|}}
-\cline{2-7}
-\multicolumn{1}{c|}{ }
-&\multicolumn{3}{|c|}{Parallel} & \multicolumn{3}{|c|}{Pseudo-Periodic} \\
-\cline{2-7}
-\multicolumn{1}{c|}{ }&
-P & M & T&
-P & M & T \\
-\hline %\cline{2-7}
-Running &
-$\top$ & 2.7 & 0.01s &
-$\bot$ & 369.371 & 0.509s \\
-\hline %\cline{2-7}
-\cite{RC07} exemples &
-$\bot$ & 2.5 & 0.001s & % RC07_sync.spin
-$\bot$ & 2.5 & 0.01s \\ % RC07_sync_chao_all.spin
-\hline
-\cite{BM99} exemple &
-$\top$ & 36.7 & 12s & % BM99_sync_para.spin
-$\top$ & & \\ % BM99_sync_chao.spin
-\hline
-\end{tabular}
-\end{tiny}
-\end{center}
-\caption{Expérimentations avec des itérations synchrones}\label{fig:sync:exp}
-\end{figure}
+\centering
+\includegraphics[scale=0.6]{images/RC07ce.eps}
+\caption{Contre exemple de convergence pour~\ref{fig:RC07CE}} \label{fig:RC07CE}
+\end{figure}
+
Cependant, comme ces implantations ne sont pas exhaustives, elles ne donnent un résultat
formel que lorsqu'elles fournissent un contre-exemple. Lorsqu'elles exhibent une convergence,
cela ne permet que donner une intuition de convergence, pas une preuve.
-Autant que nous sachions, aucune démarche de preuve formelle de convergence n'a jamais été établie.
+Autant que nous sachions, aucune démarche de preuve formelle automatique
+de convergence n'a jamais été établie.
Dans le travail théorique~\cite{Cha06}, Chandrasekaran a montré que les itérations asynchrones sont convergentes
si et seulement si on peut construire une fonction de Lyaponov décroissante, mais il ne donne pas de méthode
automatique pour construire cette fonction.
realistic in practice for close systems.
However, in real large scale distributed systems where bandwidth is weak,
this restriction is too strong. In that case, one should only consider that
-matrix $S^{t}$ follows the iterations of the system, \textit{i.e.},
+matrix $s^{t}$ follows the iterations of the system, \textit{i.e.},
for all $i$, $j$, $1 \le i \le j \le n$, we have$
-\lim\limits_{t \to \infty} S_{ij}^t = + \infty$.
+\lim\limits_{t \to \infty} s_{ij}^t = + \infty$.
One challenge of this work should consist in weakening this constraint.
We plan as future work to take into account other automatic approaches
to discharge proofs notably by deductive analysis~\cite{CGK05}.
de $d_S(s,s')$
n'est pas nulle, alors $s_l$ est différent de $s'_l$.
-On peut démontrer que pour toute fonction booléenne $f$,
+On a démontré que pour toute fonction booléenne $f$,
$G_f$ est continue sur $\mathcal{X}$ (cf annexe~\ref{anx:cont}).
