first commit

6 years ago · 8789447d8c
25 changed files with 988 additions and 0 deletions
--- a/.gitignore
+++ b/.gitignore
@ -0,0 +1,12 @@
+*.pdf
+*.bbl
+*.bcf
+*.blg
+*.log
+*.run.xml
+*.aux
+*.out
+*.cmi
+*.cmx
+*.o
+src/svg-inkscape/
--- a/13
+++ b/13
@ -0,0 +1,13 @@
+default: build clean
+
+build:
+	scripts/build.sh
+
+clean:
+	scripts/clean.sh
+
+test:
+	scripts/test.sh
+
+mrproper: clean
+	scripts/mrproper.sh
--- a/img/simple_bdd.dot
+++ b/img/simple_bdd.dot
@ -0,0 +1,4 @@
+digraph G {
+  P -> ⊥ [style="dashed"]
+  P -> ⊤
+}
--- a/img/simple_bdd2.dot
+++ b/img/simple_bdd2.dot
@ -0,0 +1,8 @@
+digraph G {
+  T1 [label="⊤"]
+  T2 [label="⊤"]
+  P -> Q [style="dashed"]
+  P -> T1
+  Q -> ⊥ [style="dashed"]
+  Q -> T2
+}
--- a/img/simple_bdd3.dot
+++ b/img/simple_bdd3.dot
@ -0,0 +1,8 @@
+digraph G {
+  T1 [label="⊤"]
+  T2 [label="⊤"]
+  Q -> P [style="dashed"]
+  Q -> T1
+  P -> ⊥ [style="dashed"]
+  P -> T2
+}
--- a/img/simple_bdd4.dot
+++ b/img/simple_bdd4.dot
@ -0,0 +1,6 @@
+digraph G {
+  P -> Q [style="dashed"]
+  P -> ⊤
+  Q -> ⊥ [style="dashed"]
+  Q -> ⊤
+}
--- a/img/simple_bdd5.dot
+++ b/img/simple_bdd5.dot
@ -0,0 +1,4 @@
+digraph G {
+  P -> ⊤
+  P -> ⊤ [style="dashed"]
+}
--- a/img/simple_bdd6.dot
+++ b/img/simple_bdd6.dot
@ -0,0 +1,3 @@
+digraph G {
+  ⊤;
+}
--- a/img/simple_bdd7.dot
+++ b/img/simple_bdd7.dot
@ -0,0 +1,19 @@
+digraph G {
+  P;
+  Q1 [label="Q"];
+  Q2 [label="Q"];
+  T  [label="⊤"];
+  B1 [label="⊥"];
+  B2 [label="⊥"];
+  B3 [label="⊥"];
+
+  P -> Q1
+  P -> Q2 [style="dashed"]
+  Q1 -> T
+  Q1 -> B1 [style="dashed"]
+  Q2 -> B2
+  Q2 -> B3 [style="dashed"]
+}
+
+
+
--- a/img/universite-paris-saclay.svg
+++ b/img/universite-paris-saclay.svg
@ -0,0 +1,542 @@
+<?xml version="1.0" encoding="utf-8"?>
+<!-- Generator: Adobe Illustrator 19.1.0, SVG Export Plug-In . SVG Version: 6.00 Build 0)  -->
+<!DOCTYPE svg PUBLIC "-//W3C//DTD SVG 1.1//EN" "http://www.w3.org/Graphics/SVG/1.1/DTD/svg11.dtd" [
+	<!ENTITY ns_extend "http://ns.adobe.com/Extensibility/1.0/">
+	<!ENTITY ns_ai "http://ns.adobe.com/AdobeIllustrator/10.0/">
+	<!ENTITY ns_graphs "http://ns.adobe.com/Graphs/1.0/">
+	<!ENTITY ns_vars "http://ns.adobe.com/Variables/1.0/">
+	<!ENTITY ns_imrep "http://ns.adobe.com/ImageReplacement/1.0/">
+	<!ENTITY ns_sfw "http://ns.adobe.com/SaveForWeb/1.0/">
+	<!ENTITY ns_custom "http://ns.adobe.com/GenericCustomNamespace/1.0/">
+	<!ENTITY ns_adobe_xpath "http://ns.adobe.com/XPath/1.0/">
+]>
+<svg version="1.1" id="Calque_1" xmlns:x="&ns_extend;" xmlns:i="&ns_ai;" xmlns:graph="&ns_graphs;"
+	 xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" x="0px" y="0px" viewBox="0 0 258.6 87.5"
+	 style="enable-background:new 0 0 258.6 87.5;" xml:space="preserve">
+<style type="text/css">
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+</style>
+<switch>
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+		<g>
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+</svg>
--- a/scripts/build.sh
+++ b/scripts/build.sh
@ -0,0 +1,13 @@
+#!/usr/bin/env sh
+
+set -eu
+
+( cd "$(dirname "$0")"/../src
+
+  xelatex -shell-escape main.tex
+  biber main
+  # xelatex -shell-escape src/main.tex
+  xelatex -shell-escape main.tex
+  xelatex -shell-escape main.tex
+  mv main.pdf ../main.pdf
+)
--- a/scripts/clean.sh
+++ b/scripts/clean.sh
@ -0,0 +1,10 @@
+#!/usr/bin/env sh
+
+set -eu
+
+( cd "$(dirname "$0")"/../src/
+
+  rm -f ./*.log ./*.tuc ./*.out ./*.aux ./*.toc ./*.pyg ./*.bbl ./*.bcf ./*.blg ./*.run.xml
+  rm -rf ./svg-inkscape/
+
+)
--- a/scripts/mrproper.sh
+++ b/scripts/mrproper.sh
@ -0,0 +1,9 @@
+#!/usr/bin/env sh
+
+set -eu
+
+( cd "$(dirname "$0")"/../
+
+  rm -f ./*.pdf
+
+)
--- a/scripts/test.sh
+++ b/scripts/test.sh
@ -0,0 +1,16 @@
+#!/usr/bin/env sh
+
+set -eu
+
+(
+  cd "$(dirname "$0")"/../
+
+  # Testing scripts
+  # shellcheck disable=SC2185
+  find -O3 . -type f -name '*.sh' -exec shellcheck {} \;
+  cd src/
+  # shellcheck disable=SC2185
+  find -O3 . -type f -name '*.tex' -exec chktex -q --nowarn 26 --nowarn 15 --nowarn 17 {} \;
+  # shellcheck disable=SC2185
+  find -O3 . -type f -name '*.tex' -exec lacheck {} \; # TODO: fix lacheck so it's possible to disable some warnings which are incorrects...
+)
--- a/src/abstract.tex
+++ b/src/abstract.tex
@ -0,0 +1,3 @@
+\begin{abstract}
+  Ce document est un rapport de mes travaux de recherche sur le sujet \emph{TODO}, effectué dans le cadre de ma formation en Master 1 Informatique à l'Université Paris-Saclay, j'ai été encadré par \href{https://www.lri.fr/~filliatr/}{Jean-Christophe \bsc{Filliâtre}}.
+\end{abstract}
--- a/src/app.tex
+++ b/src/app.tex
@ -0,0 +1 @@
+\section{Applications}
--- a/src/appendix.tex
+++ b/src/appendix.tex
@ -0,0 +1 @@
+\section{Annexes}
--- a/src/bdd.tex
+++ b/src/bdd.tex
@ -0,0 +1,91 @@
+\section{Les diagrammes de décision binaire}
+  \subsection{La logique propositionnelle}
+    \subsubsection{Les propositions}
+      \paragraph{}
+        Une proposition est un énoncé pour lequel il fait sens de juger de la valeur de vérité. Par exemple, la phrase \enquote{\bsc{OCaml} est le plus beau langage au monde} est soit vraie, soit fausse. En revanche, pour la phrase \enquote{Ça va ?}, il n'y a aucun sens à dire qu'elle est vraie ou fausse.
+      \paragraph{}
+      On notera $\bot$ la proposition correspondant à \emph{faux} et $\top$ celle correspondant à \emph{vrai}. Une proposition est donc un élément de l'ensemble $\{\top, \bot\}$, que l'on notera $\mathbb{B}$.
+    \subsubsection{Les fonctions}
+      \paragraph{}
+        Dès lors, il est possible de définir des fonctions de $\mathbb{B}$ vers $\mathbb{B}$. On a par exemple la \emph{négation}, qui est la fonction qui à $\top$ associe $\bot$ et qui à $\bot$ associe $\top$. La négation d'une proposition $P$ est notée $\neg P$. Par la suite, lorsque l'on écrira $P$ sans plus de précision, $P$ sera tenue pour vraie, et lorsque l'on écrira $\neg P$, elle sera tenue pour fausse.
+      \paragraph{}
+        De façon similaire, on peut définir des fonctions de $\mathbb{B}^2$ vers $\mathbb{B}$. Soient $P$ et $Q$ deux propositions, on a notamment la conjonction de $P$ et $Q$, notée $P \land Q$ ; leur disjonction, notée $P \lor Q$ ; l'implication, notée $P \Rightarrow Q$.
+      \paragraph{}
+      $P \land Q$ vaut $\top$ si et seulement si $P$ et $Q$ valent tous les deux $\top$. $P \lor Q$ vaut $\top$ si et seulement si au moins $P$ ou $Q$ vaut $\top$. $P \Rightarrow Q$ vaut $\bot$ si et seulement si $P$ vaut $\top$ et que $Q$ vaut $\bot$.
+    \subsubsection{Encore ?}
+      \paragraph{}
+        On ne détaillera pas plus la logique propositionnelle, et l'on considérera par la suite que le reste est connu. Le lecteur souhaitant en savoir plus peut se tourner vers \href{https://www.lri.fr/~paulin/Logique/}{ce cours de logique pour l'informatique}.
+  \subsection{Représentation graphique}
+    \paragraph{}
+      % TODO
+      En une phrase: les diagrammes de décision binaires permettent de représenter les fonctions de $\mathbb{B}^n$ dans $\mathbb{B}$. Les diagrammes de décision binaires peuvent être représentés graphiquement, ce par quoi on commencera avant toute définition formelle, afin d'en facilitier la compréhension\ldots
+    \paragraph{}
+      Représentons tout d'abord la formule $P$:
+        \begin{center}
+          \includesvg[width=100pt]{simple_bdd.svg}
+        \end{center}
+    \paragraph{}
+      Un diagramme de décision binaire a donc un sommet \enquote{de départ}, $P$ ici. Un sommet a soit deux arcs sortants, soit aucun. S'il en a deux, alors, il contient une \emph{variable propositionnelle} et dans ce cas, le \enquote{chemin à suivre} dépend de la valeur de vérité de la variable. Si la variable vaut $\bot$, on prend l'arc en pointillés, sinon, on prend l'autre. Si le sommet n'a aucun arc sortant, alors, il contient soit $\top$ soit $\bot$, ce qui correspond à la valeur de vérité de la formule représentée par ce diagramme dans le cas où on assigne aux variables les mêmes valeurs de vérités que celles que l'on a prises sur ce chemin.
+  \subsubsection{Ordre}
+    \paragraph{}
+      Prenons un autre exemple, voilà un diagramme représentant la proposition $P \lor Q$:
+        \begin{center}
+          \includesvg[width=100pt]{simple_bdd2.svg}
+        \end{center}
+    \paragraph{}
+      Il aurait aussi été possible de la représenter par celui-ci:
+        \begin{center}
+          \includesvg[width=100pt]{simple_bdd3.svg}
+        \end{center}
+    \paragraph{}
+      On va en fait se donner un \emph{ordre} sur les variables et dès que l'on \enquote{aura le choix} entre deux variables, on prendra la plus petite des deux selon cet ordre. Par exemple, si l'on décide d'utiliser l'ordre lexicographique, alors, on aura le premier des deux diagrammes ci-dessus.
+  \subsubsection{Non-redondance}
+    \paragraph{}
+    Une seconde chose à prendre en compte est le fait que l'on va éviter toute redondance dans nos diagrammes. Si deux sommets mènent au même résultat dans tous les cas, on va les fusionner. Sur notre exemple précédent, on peut par exemple fusionner les deux sommets $\top$:
+        \begin{center}
+          \includesvg[width=100pt]{simple_bdd4.svg}
+        \end{center}
+  \subsubsection{Simplification}
+    \paragraph{}
+    Enfin, si les deux arcs sortants d'un sommet mènent à la même chose, on va tout simplement supprimer ce sommet. Par exemple, la proposition $P \lor \neg P$ ne sera pas représentée comme ceci:
+        \begin{center}
+          \includesvg[width=50pt]{simple_bdd5.svg}
+        \end{center}
+    Mais comme cela:
+        \begin{center}
+          \includesvg[width=50pt]{simple_bdd6.svg}
+        \end{center}
+        % TODO: ssi tautologie...
+        En fait, on a en quelque sorte \enquote{calculé} la valeur de la proposition.
+  \subsection{Représentation formelle}
+    \subsubsection{Opérateur \texttt{if-then-else}}
+      On définit l'opérateur \texttt{if-then-else}, de $\mathbb{B}^3$ vers $\mathbb{B}$, et noté $P \rightarrow Q_1, Q_2$ ainsi:
+
+        \[P \rightarrow Q_1, Q_2 = (P \land Q_1) \lor (\neg P \land Q_2)\]
+
+      Ce qui se lit: \enquote{Si $P$, alors, $Q_1$, sinon, $Q_2$.}.
+    \subsubsection{Forme normale \texttt{if-then-else}}
+      On dit qu'une formule propositionnelle est en forme normale \texttt{if-then-else} (FNI) si elle est construite uniquement à partir de l'opérateur \texttt{if-then-else} et des constantes $\top$ et $\bot$ ; et de telle sorte que le premier opérande de l'opérateur \texttt{if-then-else} est toujours une variable.
+    \subsubsection{Expansion de \bsc{Shannon}}
+      On note $P[Q_1/Q_2]$  la proposition $P$ dans laquelle on a substitué toutes les occurences de $Q_1$ par $Q_2$. On a alors:
+        \[P = Q \rightarrow P[Q/\top], P[Q/\bot]\]
+        C'est ce que l'on appelle l'expansion de \bsc{Shannon} de $P$ par rapport à $Q$.
+    \subsubsection{Passage en FNI}
+      \paragraph{}
+        Pour n'importe quelle proposition, il est possible d'obtenir son équivalent en forme FNI. Il suffit pour cela d'appliquer l'expansion de Shannon de $P$ par rapport à n'importe quelle variable $Q$ de $P$, on aura alors $Q \rightarrow P[Q/\top], P[Q/\bot]$. Puis, on applique la même transformation à $P[Q/\top]$ et à $P[Q/\bot]$ et ce jusqu'à ce qu'il n'y ait plus de variables disponibles, on aura alors forcément soit $\top$ soit $\bot$.
+      \paragraph{}
+        Par exemple, la proposition $P \land Q$ peut être mise sous FNI ainsi:
+          % TODO: align
+          \[P \land Q = P \rightarrow (\top \land Q), (\bot \land Q) \]
+          \[= P \rightarrow (Q \rightarrow (\top \land \top), (\top \land \bot)), (Q \rightarrow (\bot \land \top), (\bot \land \bot)) \]
+          \[= P \rightarrow (Q \rightarrow \top, \bot), (Q \rightarrow \bot, \bot) \]
+
+    \subsubsection{Passage en arbre binaire}
+      ...
+        Ce qui donne le diagramme suivant:
+          \begin{center}
+            \includesvg[width=150pt]{simple_bdd7.svg}
+          \end{center}
+
+    \subsubsection{Passage en graphe}
+      Une proposition en FNI est en fait représentable sous forme d'arbre binaire et donc, sous forme de graphe. Un diagramme de décision binaire pour une proposition $P$ est donc simplement la représentation sous forme de graphe de la FNI obtenue à partir de $P$\ldots~après avoir appliqué quelques transformations à ce graphe ! % TODO: better explanation
--- a/src/bib.bib
+++ b/src/bib.bib
@ -0,0 +1,86 @@
+@article{Bryant86,
+ author = {Bryant, Randal E.},
+ title = {Graph-Based Algorithms for Boolean Function Manipulation},
+ journal = {IEEE Trans. Comput.},
+ issue_date = {August 1986},
+ volume = {35},
+ number = {8},
+ month = aug,
+ year = {1986},
+ issn = {0018-9340},
+ pages = {677--691},
+ numpages = {15},
+ url = {http://dx.doi.org/10.1109/TC.1986.1676819},
+ doi = {10.1109/TC.1986.1676819},
+ acmid = {6433},
+ publisher = {IEEE Computer Society},
+ address = {Washington, DC, USA},
+ keywords = {Boolean functions, binary decision diagrams, logic design
+verification, symbolic manipulation, symbolic manipulation, Boolean
+functions, binary decision diagrams, logic design verification},
+}
+
+@misc{Andersen97,
+  author = {Henrik Reif Andersen},
+  title = {An Introduction to Binary Decision Diagrams},
+  howpublished = {Lectures notes from Technical University of Denmark},
+  year = 1997,
+  note = {\url{http://www.cs.utexas.edu/~isil/cs389L/bdd.pdf}}
+}
+
+@article{Hayes97,
+ author = {Hayes, Barry},
+ title = {Ephemerons: A New Finalization Mechanism},
+ journal = {SIGPLAN Not.},
+ issue_date = {Oct. 1997},
+ volume = {32},
+ number = {10},
+ month = oct,
+ year = {1997},
+ issn = {0362-1340},
+ pages = {176--183},
+ numpages = {8},
+ url = {http://doi.acm.org/10.1145/263700.263733},
+ doi = {10.1145/263700.263733},
+ acmid = {263733},
+ publisher = {ACM},
+ address = {New York, NY, USA},
+ keywords = {finzlization, garbage collection, resource management, weak pointers},
+}
+
+@article{Leroy00,
+  author = {Xavier Leroy},
+  title = {A modular module system},
+  journal = {Journal of Functional Programming},
+  volume = 10,
+  number = 3,
+  year = 2000,
+  pages = {269--303},
+  urllocal = {http://xavierleroy.org/publi/modular-modules-jfp.pdf},
+  urlpublisher = {http://journals.cambridge.org/action/displayAbstract?aid=54525},
+  abstract = {A simple implementation of a SML-like module system is presented as a
+module parameterized by a base language and its type-checker. This
+demonstrates constructively the applicability of that module system to
+a wide range of programming languages. Full source code available
+in the Web appendix \url{http://gallium.inria.fr/~xleroy/publi/modular-modules-appendix/}},
+  xtopic = {modules}
+}
+
+@InProceedings{ConchonFilliatre06,
+  author = 	 {Sylvain Conchon and Jean-Christophe Filli\^atre},
+  title = 	 {{Type-Safe Modular Hash-Consing}},
+  booktitle =    {ACM SIGPLAN Workshop on ML},
+  address =      {Portland, Oregon},
+  month =	 sep,
+  year =	 2006,
+  url = {http://www.lri.fr/~filliatr/ftp/publis/hash-consing2.pdf},
+}
+
+@book{Knuth11,
+  author = {Knuth, Donald E.},
+  title = {The Art of Computer Programming, volume 4A: Combinatorial
+Algorithms, Part 1},
+  year = 2011,
+  edition = {1st},
+  publisher = {Addison-Wesley Professional}
+}
--- a/src/conclusion.tex
+++ b/src/conclusion.tex
@ -0,0 +1 @@
+\section{Conclusion}
--- a/src/implem.tex
+++ b/src/implem.tex
@ -0,0 +1 @@
+\section{Implémentation}
--- a/src/introduction.tex
+++ b/src/introduction.tex
@ -0,0 +1,8 @@
+\section{Introduction}
+\cite{Leroy00}
+\cite{Knuth11}
+\cite{Andersen97}
+\cite{ConchonFilliatre06}
+\cite{Bryant86}
+\cite{Hayes97}
+
--- a/src/main.tex
+++ b/src/main.tex
@ -0,0 +1,27 @@
+\documentclass[a4paper,12pt,titlepage]{article}
+
+\input{packages.tex}
+
+\bibliography{bib}
+
+\begin{document}
+
+\input{title.tex}
+
+\input{abstract.tex}
+
+\tableofcontents
+
+\input{introduction.tex}
+\input{bdd.tex}
+\input{implem.tex}
+\input{app.tex}
+\input{conclusion.tex}
+
+\appendix
+
+\input{appendix.tex}
+
+\printbibliography{}
+
+\end{document}
--- a/src/packages.tex
+++ b/src/packages.tex
@ -0,0 +1,75 @@
+\usepackage[french]{babel}
+
+\usepackage{euler}
+
+\usepackage{fontspec}                 % encoding
+  \setmainfont{Linux Libertine O}
+  \setsansfont{Linux Biolinum O}
+  %\setmonofont{Source Code Pro}
+  %\newfontfamily\unicodemonofont{Everson Mono}
+
+\usepackage{graphicx}                                 % to include img
+\usepackage{enumitem}                                 % powerful lists
+\usepackage{xspace}                                   % correct quotes printing
+\usepackage[np]{numprint}                             % correct numbers printing
+\usepackage{amsmath}                                  % maths and better typo
+\usepackage{amssymb}                                  % more maths
+\usepackage{dsfont}                                   % and even more maths
+\usepackage{color}                                    % color power
+\usepackage[dvipsnames, table]{xcolor}                  % more color power
+  \definecolor{darkBackground}{HTML}{282c34}            % source code background
+  \definecolor{darkBackgroundHighlight}{HTML}{2c323c}   % source code background
+  \definecolor{lightBackground}{HTML}{D7D3CB}           % source code background
+\usepackage{microtype}                                % better typo
+
+\usepackage{hyperref} % internal and external links (e.g. mail, www)
+\usepackage{cleveref}
+
+\newcommand\myshade{85}
+\colorlet{mylinkcolor}{violet}
+\colorlet{mycitecolor}{YellowOrange}
+\colorlet{myurlcolor}{Aquamarine}
+
+\usepackage[cache=false, draft=false]{minted}         % source code
+  \usemintedstyle{paraiso-dark}
+  \setminted{bgcolor=lightBackground}
+  \setminted{linenos, autogobble, breaklines, breakanywhere, breakautoindent, fontseries=m, fontsize=\fontsize{12pt}{12pt}, frame=none, stepnumber=2}
+
+\usepackage{placeins} % float displaying
+\usepackage{tikz}     % .tex figures (e.g. from Dia)
+
+\hypersetup{linkcolor  = mylinkcolor!\myshade!black,
+citecolor  = mycitecolor!\myshade!black,
+urlcolor   = myurlcolor!\myshade!black,
+colorlinks = true,
+pdfstartview=FitH
+}
+
+
+\usepackage{csquotes}
+\usepackage{ellipsis} % correct ...
+\usepackage{svg}
+
+\usepackage[
+  autolang=other,
+  backend=biber,         % choix de l'outil de traitement
+  backref=true,          % liens dans la bibliographie pour remonter dans le texte
+  backrefstyle=none,     % afficher toutes les utilisations de la référence
+  bibstyle=alphabetic,   % style pour les clés des références dans la bibliographie : [initialesAnnée]
+  citestyle=alphabetic,  % style pour les clés des références dans le texte : [initialesAnnée]
+  sorting=none,          % bibliographie triée par ordre d'utilisation des références
+]{biblatex}                  % support des bibliographies
+
+\usepackage{calligra}
+\usepackage{tikz}
+\usetikzlibrary{matrix, fit, chains, calc, scopes, positioning}
+\usepackage{auto-pst-pdf} %To compile psvectorian directly
+\usepackage{psvectorian}
+\renewcommand*{\psvectorianDefaultColor}{Red}%
+
+\graphicspath{{./../img/}}
+
+\usepackage[bottom]{footmisc}
+\usepackage{wrapfig}
+
+\definecolor{processblue}{cmyk}{0.96,0,0,0}
--- a/src/title.tex
+++ b/src/title.tex
@ -0,0 +1,27 @@
+\begin{titlepage}
+  \begin{center}
+
+    %\vspace*{0.1\textheight}
+    \fontsize{25}{25}\scshape Partage d'implémentation, implémentation du partage:\\
+    une bibliothèque fonctorisée de diagrammes de décision binaires
+    \vspace*{0.1\textheight}
+
+    {\fontsize{18}{18}\calligra{Rapport de travaux de recherche par\\}}
+    {\fontsize{28}{28}\scshape Léo \bsc{Andrès}\\}
+    \vspace*{0.01\textheight}
+    {\fontsize{18}{18}\calligra{\today\\}}
+    \vspace*{0.03\textheight}
+    {\fontsize{18}{18}\calligra{Master 1 Informatique\\}}
+    \vspace*{0.03\textheight}
+    {\includesvg{universite-paris-saclay.svg}}\\
+    \vspace*{0.1\textheight}
+
+    \begin{tikzpicture}[start chain=main going right]
+
+      \node[on chain,align=center,draw=none](a1)
+      {{\fontsize{18}{18}\calligra Sous la direction de} \\
+        {\Large {Dr.~Jean-Christophe~\bsc{Filliâtre}}}};
+
+    \end{tikzpicture}
+  \end{center}
+\end{titlepage}