Ajout du diapo pour la soutenance.
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.gitignore
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build/
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svg-inkscape/
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DiaposSoutenance.tex
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782
DiaposSoutenance.tex
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% !TeX spellcheck = fr_FR
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\documentclass[12pt]{beamer}
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% Loading packages
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\usepackage[T1]{fontenc}
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\usepackage[utf8]{inputenc}
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\usepackage{amsmath}
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\usepackage{amssymb}
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\usepackage{bbm}
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\usepackage{stmaryrd}
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\usepackage[main=french,english]{babel}
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\usepackage{csquotes}
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\usepackage{listings}
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\usepackage{lstautogobble}
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\usepackage{algorithm2e}
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\usepackage{svg}
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\usepackage{tikz}
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\usepackage{multirow}
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\usepackage{multicol}
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\usepackage{tcolorbox}
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\usepackage{mdframed}
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\usepackage{proof}
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\usepackage{biblatex}
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\usepackage{xparse}
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\usepackage{cprotect}
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\usepackage{xpatch}
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\usepackage{enumitem}
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\usepackage{amsfonts}
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\usepackage{mathtools}
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\usepackage{geometry}
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\usepackage{subcaption}
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\usepackage{csquotes}
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\usepackage[lighttt]{lmodern}
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\usetikzlibrary{shapes.geometric,positioning}
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% Macros caractères globales
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\newcommand{\pgrph}{\P}
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\newcommand{\hsep}{\vspace{.2cm}\centerline{\rule{0.8\linewidth}{.05pt}}\vspace{.4cm}}
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\renewcommand{\P}{\mathbb{P}}
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\newcommand{\E}{\mathbb{E}}
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\newcommand{\1}{\scalebox{1.2}{$\mathbbm{1}$}}
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\newcommand{\floor}[1]{\left\lfloor#1\right\rfloor}
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\newcommand{\littleO}{o}
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\newcommand{\bigO}{\mathcal{O}}
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\newcommand{\longdash}{\:\textrm{---}\:}
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\newcommand\hole{\left[\raisebox{-0.25ex}{\scalebox{1.2}{$\cdot$}}\right]}
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\newcommand\bracket[1]{\!\left[#1\right]}
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\newcommand{\ssi}{\quad\text{\underline{ssi}}\quad}
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\newcommand\eng[1]{\textit{\foreignlanguage{english}{#1}}}
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\newcommand\spacebar{\;|\;}
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\def\nDownarrow{\not\mspace{1mu}\Downarrow}
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\let\pprec\preccurlyeq
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\DeclareUnicodeCharacter{03BB}{$\lambda$}
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\DeclareUnicodeCharacter{03B1}{$\alpha$}
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\DeclareUnicodeCharacter{03C0}{$\pi$}
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% Création des environnement globaux
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\newtheorem{property}{Propriété}
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\newcounter{rule}
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% Commandes logiques globales
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\newcommand{\ifnullthenelse}[3]{
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\ifnum\value{#1}=0
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#2
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\else
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#3
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\fi
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}
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%%% Commande \newtag permettant de changer le label d'une equation
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\makeatletter
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\newcommand\newtag[2]{#1\def\@currentlabel{#1}\label{#2}}
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\makeatother
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% Macros caractères spécifiques au document
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\newcommand{\methprec}[1]{\prec\mkern-8mu\left[\fj{#1}\right]}
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\DeclareMathOperator{\varnull}{varnull}
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\DeclareMathOperator{\nextop}{next}
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\DeclareMathOperator{\new}{\text{\textbf{new}}}
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\DeclareMathOperator{\CT}{CT}
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\DeclareRobustCommand{\leftsarrow}{\text{\reflectbox{$\;\rightarrow^*$}}}
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\newcommand\subclass{\mathrel{<:}}
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\newcommand\superclass{\mathrel{:>}}
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\newcommand\OKin{\text{\textnormal{\textsc{ OK in }}}}
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% Création des environnements spécifiques au document
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%%% Blocs colorés pour le code FeatherweightJava
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\newtcbox{\fjmaincodebox}{nobeforeafter,tcbox raise base, arc=0pt, outer arc=0pt, boxsep=0pt,left=2pt,right=2pt,top=2pt,bottom=2pt,boxrule=0pt,colback=white!85!black}
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\newtcbox{\fjind}{nobeforeafter,tcbox raise base, rounded corners=all, boxsep=0pt,left=1pt,right=1pt,top=1pt,bottom=1pt,boxrule=0pt,colback=white!52!orange}
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\newenvironment{fjlisting}[1][]
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{
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\begin{tcolorbox}[left=5pt,right=2pt,top=5pt,bottom=5pt,boxrule=1pt,rounded corners=all,colback=white!96!black,#1]
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\ttfamily
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}{
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\end{tcolorbox}
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}
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% Langage Featherweight Java
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\makeatletter
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\newcommand*\idstyle{%
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\expandafter\id@style\the\lst@token\relax
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}
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\def\id@style#1#2\relax{%
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\ifcat#1\relax\else
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\ifnum`#1=\lccode`#1%
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\ttfamily
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\fi
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\fi
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}
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\makeatother
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\lstdefinelanguage{FeatherweightJava}{
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keywords = {new, this, true, false},
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basicstyle=\ttfamily,
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identifierstyle=\idstyle,
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literate=%
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{à}{{\`a}}1
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{é}{{\'e}}1
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{è}{{\`e}}1
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{ù}{{\`u}}1
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}
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\lstset{literate=%
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{à}{{\`a}}1
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{é}{{\'e}}1
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{è}{{\`e}}1
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{ù}{{\`u}}1}
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%% Commande 4
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\NewDocumentEnvironment{textenv}{b}{\text{\textnormal{#1}}}{}
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%\lstMakeShortInline[language=FeatherweightJava]|
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\newcommand{\fj}[1]{\begin{textenv}\lstinline[language=FeatherweightJava,mathescape=true]~#1~\end{textenv}}
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% Définition des macros pour l'écriture des trucs Featherweight Java
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\newcommand\fjattr{\GenericWarning{$\backslash$ fjattr doit être utilisé dans une fjclass}}
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\newcounter{fjargcount}
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\newcommand{\fjparamattributes}[2][Object]{
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\null\qquad #1 #2;\newline
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}
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\newcommand{\printcomma}{,}
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\newcommand{\fjparamconstparameters}[2][Object]{
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\ifnullthenelse{fjargcount}{}{,$\text{ }$} \text{#1} \;\text{#2}
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\stepcounter{fjargcount}
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}
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\newcommand{\fjparamconstassign}[2][Object]{
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\null\qquad\qquad \textbf{this}.#2 = #2;\newline
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}
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\newcommand{\fjmethod}[4]{
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\null\qquad #1 #2(#3)\{\newline
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\null\qquad\qquad \textbf{return} \lstinline[language=FeatherweightJava]{#4};\newline
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\null\qquad\}\newline
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}
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\newcommand{\fjmain}[1]{
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\noindent\fjmaincodebox{\lstinline[language=Java]{#1}}
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}
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% Premier argument: Nom de la classe
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% Second argument: \fjattr{x}[Object] \fjattr{attr}[A]
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% Troisième argument \fjmethod{C}{metname}{Object w, B par}{((D)obj).apply(par)} plusieurs fois
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% Argument optionnel: la classe que l'on extend
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\newcommand{\fjclass}[4][Object]{%
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\noindent \textbf{class} #2 \textbf{extends} #1 \{\newline
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% Attributs
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\let\fjattr\fjparamattributes
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#3
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% Constructeur
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\let\fjattr\fjparamconstparameters
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\setcounter{fjargcount}{0}
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\null\qquad #2(\ifx&\else $#3$\fi) \{\newline
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\null\qquad\qquad \textbf{super}();\newline
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\let\fjattr\fjparamconstassign%
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#3
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\null\qquad\}\\
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% Méthodes
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#4
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\}
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}
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\addbibresource{Bilibibio.bib}
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\usetheme{Madrid}
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\hypersetup{pdfpagemode=FullScreen}
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% Transition en fade-in par défaut
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\addtobeamertemplate{background canvas}{\transfade[duration=0.4]}{}
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\addtobeamertemplate{frametitle}{
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\let\insertframetitle\insertsubsectionhead}{}
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\makeatletter
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\CheckCommand*\beamer@checkframetitle{\@ifnextchar\bgroup\beamer@inlineframetitle{}}
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\renewcommand*\beamer@checkframetitle{\global\let\beamer@frametitle\relax\@ifnextchar\bgroup\beamer@inlineframetitle{}}
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\makeatother
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\author{Samy Avrillon}
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\title{Définition d'un préordre sur les programmes Featherweight Java}
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\subtitle{Notes sur mon stage au LACL}
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\logo{\includesvg[width=1.5cm]{logoEns}}
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\institute{ENS de Lyon}
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\begin{document}
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\begin{frame}[plain]
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\maketitle
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\end{frame}
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\section*{Sommaire}
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\begin{frame}
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\begin{multicols}{2}
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\tableofcontents[pausesections,hideallsubsections]
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\end{multicols}
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\end{frame}
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\section{Introduction}
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\section{Featherweight Java}
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\subsection{Sa grammaire}
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\begin{frame}
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\begin{center}
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\refstepcounter{rule}
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\begin{tabular}{rll}
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$e$ $:=$ & \fj{x} & \qquad \textrm{\textsc{E-Var}} \\
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| & \fj{new C($\overline{e}$)} & \qquad \newtag{\textrm{\textsc{E-Cstr}}}{rule:expr:constructor} \\
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| & \fj{$e$.f} & \qquad \newtag{\textrm{\textsc{E-Field}}}{rule:expr:field} \\
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| & \fj{$e$.m($\overline{e}$)} & \qquad \newtag{\textrm{\textsc{E-Meth}}}{rule:expr:method} \\
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| & \fj{(C)$e$} & \qquad \newtag{\textrm{\textsc{E-Cast}}}{rule:expr:cast}
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\end{tabular}
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\end{center}
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\[\fj{new Paire((D)new B().get(), new C(new A())).snd.apply()}\]
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\end{frame}
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\subsection{Ses structures}
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\begin{frame}
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\begin{fjlisting}
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\textbf{class} A \textbf{extends} Object \{...\}\\
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\textbf{class} B \textbf{extends} Object \{...\}\\
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\fjclass{Paire}{\fjattr{fst} \fjattr{snd}}{}\\
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\pause
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\fjmain{new Paire(new A(), new B())}
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\end{fjlisting}
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\end{frame}
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\subsection{Sa réduction}
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\begin{frame}
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\begin{itemize}
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\item $\fj{new C(e1,e2,...,en).field} \rightarrow \fj{ek}$ (\textsc{R-Field})
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\item $\fj{new C(e1,e2,...,en).meth(f1,f2,...,fn)} \rightarrow e_{\fj{C.meth}}\bracket{\fj{new C(e1,e2,...,en)}\middle/ \fj{this},\fj{f1}\middle/\fj{x1},\fj{f2}\middle/\fj{x2},...,\fj{fn}\middle/\fj{xn}}$ (\textsc{R-Invk})
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\item $\fj{(C) expr} \rightarrow \fj{expr}$ (\textsc{R-Cast})
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\end{itemize}
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\end{frame}
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\subsection{Son typage}
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\begin{frame}
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\[\infer
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{x} : \Gamma(\fj{x})}
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{\fj{x} \in \Gamma}
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\]\[
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\infer
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{$e$.f} : \fj{C}}
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{Cc} \quad \fj{C}\;\fj{f} \in \operatorname{fields}(\fj{Cc})}
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\]\[
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\infer
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{$e$.m($\overline{e_x}$)} : \fj{C}}
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{\begin{array}{c}\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{Cc} \quad \mathfrak{T},\mathcal{B},\Gamma \Vdash \overline{e_x} : \overline{\fj{CX}} \\[-1.4ex] \overline{\fj{CX}} \subclass \overline{\fj{D}} \quad \operatorname{mtype}(\fj{m},\fj{Cc}) = \overline{\fj{D}} \rightarrow \fj{C} \end{array}}
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\]\[
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\infer
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{new C(}\overline{e_x}\fj{)} : \fj{C}}
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash \overline{e_x} : \overline{\fj{CX}} \quad \overline{\fj{CX}} \subclass \overline{\fj{D}} \quad \operatorname{constructor}(\fj{C}) = \overline{\fj{D}}}
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{(C)$e$} : \fj{C}}
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{D} \quad \fj{C} \subclass \fj{D} \quad \fj{C} \neq \fj{D}}
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\]\[
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\infer
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{(C)$e$} : \fj{C}}
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{\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{D} \quad \fj{D} \subclass \fj{C}}
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\]
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\end{frame}
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\section{Notre problème}
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\subsection{Qu'est-ce qu'une équivalence}
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\begin{frame}
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\begin{itemize}
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\item Deux programmes «font la même chose»
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\item Assez stricte
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\item Assez large
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\end{itemize}
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\begin{itemize}
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\item Utiliser un autre algorithme (complexités)
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\item Utiliser une version plus sécurisée
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\item Effectuer une optimisation
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\end{itemize}
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\end{frame}
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\subsection{La méthode classique}
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\begin{frame}
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\[C\]
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\[C\bracket{P}\]
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\[P \prec Q \iff \forall C\quad C\bracket{P} \implies C\bracket{Q}\]
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\[P \equiv Q \iff P \prec Q \land Q \prec P\]
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\end{frame}
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\section{Équivalences simples}
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\subsection{Expressions à trou}
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\begin{frame}
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\[h := \hole \spacebar \fj{new C($\overline{e}$,$h$,$\overline{e}$)} \spacebar \fj{$h$.f} \spacebar \fj{$h$.m($\overline{e}$)} \spacebar \fj{$e$.m($\overline{e}$,$h$,$\overline{e}$)} \spacebar \fj{(C) $h$}\]
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\pause
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\begin{align*}
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\hole[e] &= e\\
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(\fj{new C($\overline{e_1}$,$h$,$\overline{e_2}$)})[e] &= \fj{new C($\overline{e_1}$,$h[e]$,$\overline{e_2}$)}\\
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(\fj{$h$.f})[e] &= \fj{$h[e]$.f}\\
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(\fj{$h$.m($\overline{e_1}$)})[e] &= \fj{$h[e]$.m($\overline{e_1}$)}\\
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(\fj{$e_1$.m($\overline{e_2}$,$h$,$\overline{e_3}$)})[e] &= \fj{$e_1$.m($\overline{e_2}$,$h[e]$,$\overline{e_3}$)}\\
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(\fj{(C) $h$})[e] &= \fj{(C) $h[e]$}
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\end{align*}
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\end{frame}
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\begin{frame}
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\[C = h \qquad C\bracket{P} = (\mathcal{A},h\bracket{e})\]
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\pause
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\begin{fjlisting}
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class A \{\}\\
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class B \{\}\\
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...\\
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\only<2>{\fjmain{expr}}
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\only<3>{\fjmain{expr.field}}
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\only<4>{\fjmain{expr.meth()}}
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\only<5>{\fjmain{new Z().meth(expr)}}
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\only<6>{\fjmain{new K(expr).snd}}
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\only<7>{\fjmain{new K().snd.get(expr.snd)}}
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\end{fjlisting}
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\end{frame}
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\subsection{Premier problème}
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\begin{frame}
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\begin{tcolorbox}
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\small
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\lstinputlisting[language=FeatherweightJava,breaklines=true,linerange=1-8]{NeedTestCT.java}
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\}
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\end{tcolorbox}
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\end{frame}
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\begin{frame}
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\begin{tcolorbox}
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\small
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\lstinputlisting[language=FeatherweightJava,breaklines=true,linerange={1-4,9-13}]{NeedTestCT.java}
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\end{tcolorbox}
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\end{frame}
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\subsection{Second problème}
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\begin{frame}
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\[\forall e \exists h \forall p \quad h\bracket{p} \rightarrow^*e\]
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\begin{tcolorbox}
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\begin{center}
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$\fj{new Paire($\hole$,$e$).snd} \rightarrow e$
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\end{center}
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\end{tcolorbox}
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\[\fj{new C(new D(...), ..., new E(...))}\]
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\end{frame}
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\subsection{Tentatives de correction}
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||||
\begin{frame}
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||||
\begin{itemize}
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\item Ajout d'une \eng{class table} de test
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\item Restreindre les \eng{class tables}
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\end{itemize}
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\[\mathcal{A} \prec \mathcal{B} \iff \forall e \forall v \quad (\mathcal{A},e)\Downarrow v \implies (\mathcal{B},e)\Downarrow v\]
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\[C\bracket{P} = (\mathcal{C}, h\bracket{e})\qquad \underline{\text{si } \mathcal{A} \prec \mathcal{C}}\]
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\begin{tcolorbox}
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\[P = (\mathcal{A}, e_P) \prec Q = (\mathcal{B}, e_Q) \ssi\]
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\[\forall C=(\mathcal{C}, e_C)\quad
|
||||
\begin{dcases}\mathcal{A} \prec \mathcal{C}\\ \mathcal{B} \prec \mathcal{C} \\
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\forall v\quad C\bracket{P}\Downarrow v \implies C\bracket{Q}\Downarrow v\end{dcases} \]
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\end{tcolorbox}
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\end{frame}
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\begin{frame}
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\begin{multicols}{2}
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\begin{fjlisting}
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\fjclass{Get}{
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\fjattr[A]{a}
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\fjattr[B]{b}
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}{
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\fjmethod{Object}{get}{}{this.a}
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}
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\end{fjlisting}
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||||
\columnbreak
|
||||
\begin{fjlisting}
|
||||
\fjclass{Get}{
|
||||
\fjattr[A]{a}
|
||||
\fjattr[B]{b}
|
||||
}{
|
||||
\fjmethod{Object}{get}{}{this.b}
|
||||
}
|
||||
\end{fjlisting}
|
||||
\end{multicols}
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
\[\begin{array}{c}
|
||||
e_A = \fj{(A)(new Get(new A(), new B()).get())} \\
|
||||
e_B = \fj{(B)(new Get(new A(), new B()).get())}
|
||||
\end{array}\]
|
||||
\[\begin{array}{cc}
|
||||
(CL_A,e_A)\Downarrow \new A()& (CL_A,e_B)\nDownarrow \\
|
||||
(CL_B,e_A)\nDownarrow & (CL_B,e_B)\Downarrow \new B()
|
||||
\end{array}\]
|
||||
\end{frame}
|
||||
\section{Idée d'une nouvelle structure}
|
||||
|
||||
\subsection{Comparer uniquement les CT}
|
||||
\begin{frame}
|
||||
\begin{center}
|
||||
\begin{multicols}{2}
|
||||
\null\vfill
|
||||
\begin{fjlisting}[width=.49\textwidth]
|
||||
\textbf{class} XXX \textbf{extends} XXX \{\}\\
|
||||
\vdots\\
|
||||
\fjmain{expression}
|
||||
\end{fjlisting}
|
||||
\vfill\null
|
||||
\columnbreak
|
||||
\begin{fjlisting}[width=.49\textwidth]
|
||||
\textbf{class} XXX \textbf{extends} XXX \{\\
|
||||
\}\\
|
||||
\vdots\\
|
||||
\textbf{class} Main \{\\
|
||||
\fjmethod{Object}{main}{}{expr}
|
||||
\}
|
||||
\end{fjlisting}
|
||||
\end{multicols}
|
||||
\end{center}
|
||||
\[\fj{new Main().main()}\rightarrow \fj{expr}\]
|
||||
\end{frame}
|
||||
\subsection{Idées d'une structure}
|
||||
\begin{frame}
|
||||
\begin{itemize}
|
||||
\item Besoin de restreindre les contextes
|
||||
\item Empêcher au test d'accéder à tout
|
||||
\item Utiliser un mécanisme déjà là
|
||||
\item Inspiration: \eng{header file} en C
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
\subsection{Exemple et théorème}
|
||||
\begin{frame}
|
||||
\begin{fjlisting}
|
||||
Number \{\}\\\null
|
||||
|
||||
Int \{\}\\
|
||||
Int : Int suivant(Int)\\
|
||||
Int : Int add(Int,Int)\\
|
||||
Int <: Number\\\null
|
||||
\end{fjlisting}
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
\begin{fjlisting}
|
||||
Frac(Int numerateur, Int denominateur)\\
|
||||
Frac : Frac inverted()\\
|
||||
Frac : Int floor()\\
|
||||
Frac <: Number\\\null
|
||||
|
||||
RichInt \{Int value\}\\
|
||||
RichInt : Int getInt()\\
|
||||
RichInt <: Int
|
||||
\end{fjlisting}
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
|
||||
\begin{itemize}
|
||||
\item Opérateur d'implémentation ($\triangleright$)
|
||||
\item Typage avec la TI ($\Vdash$)
|
||||
\item Validation d'une CT dans une TI
|
||||
\end{itemize}
|
||||
|
||||
\begin{theorem}
|
||||
\begin{itemize}
|
||||
\item $\mathcal{A} \triangleright \mathfrak{T}$
|
||||
\item $\mathcal{B} \OKin \mathfrak{T}$
|
||||
\item $\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{D}$
|
||||
\end{itemize}
|
||||
|
||||
\[\mathcal{A} \oplus \mathcal{B},\Gamma \vdash e : \fj{C}\quad\text{et}\quad\fj{C} \subclass \fj{D}\]
|
||||
\end{theorem}
|
||||
\end{frame}
|
||||
|
||||
\section{Formalisation des \eng{test interfaces}}
|
||||
|
||||
\subsection{Leur grammaire}
|
||||
\begin{frame}
|
||||
\begin{tcolorbox}[left=5pt,right=2pt,top=5pt,bottom=5pt,boxrule=1pt,rounded corners=all,colback=white!92!blue]
|
||||
\begin{center}
|
||||
\refstepcounter{rule}
|
||||
\begin{tabular}{rll}
|
||||
TR := & C : C m($\overline{\fj{C}}$) & \qquad \newtag{\textrm{\textsc{TIG-Meth}}}{rule:tsg:method} \\
|
||||
| & C \{$\overline{\fj{C}}$ $\overline{\fj{f}}$\} & \qquad \newtag{\textrm{\textsc{TIG-Attr}}}{rule:tsg:attributes} \\
|
||||
| & C ($\overline{\fj{C}}$ $\overline{\fj{?}\fj{f}}$) & \qquad \newtag{\textrm{\textsc{TIG-Cstr}}}{rule:tsg:constructor} \\
|
||||
| & C <: C & \qquad \newtag{\textrm{\textsc{TIG-Cast}}}{rule:tsg:cast}
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{tcolorbox}
|
||||
\begin{itemize}
|
||||
\item Chaque nom de classe est défini au plus une fois par ou bien une \autoref{rule:tsg:attributes} ou bien une \autoref{rule:tsg:constructor}.
|
||||
\item Chaque nom de methode est défini au plus une fois par nom de classe par une \autoref{rule:tsg:method}.
|
||||
\item Les noms de classe utilisés sont définis (sauf éventuellement \fj{Object}).
|
||||
\end{itemize}
|
||||
\end{frame}
|
||||
|
||||
\subsection{L'opérateur d'implémentation}
|
||||
|
||||
\begin{frame}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\mathcal{A} \triangleright \left[\fj{C : G m($\overline{\texttt{F}}$)}\right]}
|
||||
{
|
||||
\operatorname{mtype}_\mathcal{A}(\fj{m},\fj{C}) = \fj{$\overline{\texttt{E}}$} \rightarrow \fj{H}
|
||||
\quad \overline{\fj{F}}\subclass_\mathcal{A}\overline{\fj{E}}
|
||||
\quad \fj{H}\subclass_\mathcal{A}\fj{G}
|
||||
}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\mathcal{A} \triangleright \left[\fj{ C \{ $\overline{\texttt{E}}\;\overline{\texttt{f}}$\}}\right]}
|
||||
{
|
||||
\overline{\fj{F}}\;\overline{\fj{f}} \subset \operatorname{fields}_\mathcal{A}(C)
|
||||
\quad \overline{\fj{F}}\subclass_\mathcal{A}\overline{\fj{E}}
|
||||
}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\mathcal{A} \triangleright \left[\fj{ C ( $\overline{\texttt{E}}\;\overline{\texttt{f}}$)}\right]}
|
||||
{\begin{array}{l}
|
||||
\exists \overline{\fj{f'}}\quad \overline{\fj{f0}} = \overline{\fj{f}}\boxplus\overline{\fj{f'}} \\
|
||||
\overline{\fj{F}} = \overline{\fj{E}} \quad\text{où \fj{f} est défini} \\
|
||||
\overline{\fj{E}} \subclass \overline{\fj{F}} \quad \land \quad
|
||||
\overline{\fj{F}}\;\overline{\fj{f0}} = \operatorname{fields}_\mathcal{A}(C)
|
||||
\end{array}
|
||||
}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\mathcal{A} \triangleright \left[\fj{C} \subclass \fj{D}\right]}
|
||||
{\fj{C} \subclass_\mathcal{A} \fj{D}}
|
||||
\end{exampleblock}
|
||||
|
||||
\end{frame}
|
||||
|
||||
\subsection{Nouvelles applications de \eng{lookup}}
|
||||
\begin{frame}
|
||||
\ttfamily
|
||||
\begin{center}
|
||||
\begin{multicols}{2}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{construct}(\fj{C}) = \overline{\fj{D}}}
|
||||
{\left[\fj{C($\overline{\texttt{D}}\;\overline{\texttt{?f}}$)}\right] \in \mathfrak{T}}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{construct}(\fj{C}) = \overline{\fj{D}}}
|
||||
{\fj{class C\{C($\overline{\texttt{D}}\;\overline{\texttt{f}}$)\{...\} ... \}}\in \mathcal{B}}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{fields}(\fj{C}) = \overline{\fj{D}}\;\overline{\fj{f}} + \bigcup_{C \subclass E}\operatorname{fields}(\fj{E})}
|
||||
{\left[\fj{ C \{ $\overline{\texttt{D}}\;\overline{\texttt{f}}$\}}\right] \in \mathfrak{T}}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{fields}(\fj{C}) = \underbrace{\overline{\fj{D}}\;\overline{\fj{f}}}_{\text{\textrm{pour \fj{f} défini}}} + \bigcup_{C \subclass E}\operatorname{fields}(\fj{E})}
|
||||
{\left[\fj{ C ( $\overline{\texttt{D}}\;\overline{\texttt{f}}$)}\right] \in \mathfrak{T}}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{fields}(\fj{C}) = \overline{\fj{D}}\;\overline{\fj{f}} + \bigcup_{C \subclass E}\operatorname{fields}(\fj{E})}
|
||||
{\fj{class C \{$\overline{\texttt{D}}\;\overline{\texttt{f}}$; ...\}} \in \mathcal{B}}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{mtype}(\fj{m},\fj{C}) = \overline{\fj{D}}\rightarrow\fj{E}}
|
||||
{\left[\fj{ C : E m( $\overline{\texttt{D}}$)}\right] \in \mathfrak{T}}
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{mtype}(\fj{m},\fj{C}) = \overline{\fj{D}}\rightarrow\fj{E}}
|
||||
{\fj{class C \{...; E m($\overline{\texttt{D}}\;\overline{\texttt{x}}$)\}} \in \mathcal{B}}
|
||||
|
||||
\end{exampleblock}
|
||||
\begin{exampleblock}{}
|
||||
\infer
|
||||
{\operatorname{mtype}(\fj{m},\fj{C}) = \overline{\fj{D}}\rightarrow\fj{E}}
|
||||
{\operatorname{mtype}(\fj{m},\fj{C'}) = \overline{\fj{D}}\rightarrow\fj{E}\quad \fj{C} \subclass \fj{C'}}
|
||||
\end{exampleblock}
|
||||
\end{multicols}
|
||||
\end{center}
|
||||
\end{frame}
|
||||
|
||||
\subsection{Nouveau typage}
|
||||
\begin{frame}
|
||||
\begin{tcolorbox}[left=5pt,right=2pt,top=5pt,bottom=5pt,boxrule=1pt,rounded corners=all,colback=white!92!blue]
|
||||
\def\arraystretch{1.5}
|
||||
\begin{tabular}{c@{\hskip 2em}l}
|
||||
\infer
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{x} : \Gamma(\fj{x})}
|
||||
{\fj{x} \in \Gamma}
|
||||
& \newtag{\textrm{\textsc{(TI-Var)}}}{rule:ti:variable}
|
||||
\\[3ex]
|
||||
\infer
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{$e$.f} : \fj{C}}
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{Cc} \quad \fj{C}\;\fj{f} \in \operatorname{fields}(\fj{Cc})}
|
||||
& \newtag{\textrm{\textsc{(TI-Field)}}}{rule:ti:field}
|
||||
\\[3ex]
|
||||
\infer
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{$e$.m($\overline{e_x}$)} : \fj{C}}
|
||||
{\begin{array}{c}\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{Cc} \quad \mathfrak{T},\mathcal{B},\Gamma \Vdash \overline{e_x} : \overline{\fj{CX}} \\[-1.4ex] \overline{\fj{CX}} \subclass \overline{\fj{D}} \quad \operatorname{mtype}(\fj{m},\fj{Cc}) = \overline{\fj{D}} \rightarrow \fj{C} \end{array}}
|
||||
& \newtag{\textrm{\textsc{(TI-Invk)}}}{rule:ti:invoke}
|
||||
\\[3ex]
|
||||
\infer
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{new C(}\overline{e_x}\fj{)} : \fj{C}}
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash \overline{e_x} : \overline{\fj{CX}} \quad \overline{\fj{CX}} \subclass \overline{\fj{D}} \quad \operatorname{constructor}(\fj{C}) = \overline{\fj{D}}}
|
||||
& \newtag{\textrm{\textsc{(TI-New)}}}{rule:ti:new}
|
||||
\\[3ex]
|
||||
\infer
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{(C)$e$} : \fj{C}}
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{D} \quad \fj{C} \subclass \fj{D} \quad \fj{C} \neq \fj{D}}
|
||||
& \newtag{\textrm{\textsc{(TI-UCast)}}}{rule:ti:upCast}
|
||||
\\[3ex]
|
||||
\infer
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash \fj{(C)$e$} : \fj{C}}
|
||||
{\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{D} \quad \fj{D} \subclass \fj{C}}
|
||||
& \newtag{\textrm{\textsc{(TI-DCast)}}}{rule:ti:downCast}
|
||||
\end{tabular}
|
||||
\end{tcolorbox}
|
||||
\end{frame}
|
||||
\subsection{Théorème de cohérence}
|
||||
\begin{frame}
|
||||
\begin{theorem}
|
||||
\label{thm:tiTyp}
|
||||
Soit une \eng{test interface} $\mathfrak{T}$, une \eng{class table} $\mathcal{A}$, un couple $(\mathcal{B},e)$ \eng{class table} $\times$ expression fermée appelée \enquote{test}, et un environnement de typage $\Gamma$ vérifiant
|
||||
\begin{itemize}
|
||||
\item $\mathcal{A} \triangleright \mathfrak{T}$
|
||||
\item $\mathcal{B} \OKin \mathfrak{T}$
|
||||
\item $\mathfrak{T},\mathcal{B},\Gamma \Vdash e : \fj{D}$
|
||||
\end{itemize}
|
||||
|
||||
Alors il existe \fj{C} tel que $\mathcal{A} \oplus \mathcal{B},\Gamma \vdash e : \fj{C}$ et $\fj{C} \subclass \fj{D}$.
|
||||
\end{theorem}
|
||||
\end{frame}
|
||||
|
||||
\subsection{Définition du préordre}
|
||||
\begin{frame}
|
||||
\begin{definition}
|
||||
Soit une \eng{test interface} $\mathfrak{T}$.
|
||||
|
||||
Soient deux \eng{class tables} $\mathcal{A}$ et $\mathcal{B}$ qui implémentent toutes deux $\mathfrak{T}$
|
||||
|
||||
Alors
|
||||
\[\mathcal{A} \prec_\mathfrak{T} \mathcal{A'} \ssi \forall (\mathcal{B},e) \left|\begin{array}{l}
|
||||
\mathcal{B} \OKin \mathfrak{T}\\
|
||||
\mathfrak{T},\mathcal{B} \Vdash e : \fj{D}\\
|
||||
\mathcal{A}\oplus\mathcal{B} \Downarrow v
|
||||
\end{array}\right. \implies \mathcal{A'}\oplus\mathcal{B}\Downarrow v\]
|
||||
|
||||
où $\mathcal{A} \oplus \mathcal{B}$ dénote l'ensemble des classes de $\mathcal{A}$ auquel on a ajouté les classes de $\mathcal{B}$ dont les noms n'étaient pas déjà dans $\mathcal{A}$.
|
||||
\end{definition}
|
||||
\end{frame}
|
||||
|
||||
\section{Les valeurs infinies}
|
||||
|
||||
\subsection{Qu'est-ce qu'un context lemma}
|
||||
|
||||
\begin{frame}
|
||||
\[(\mathcal{A},e)\prec(\mathcal{B},f) \implies (\mathcal{A},h\bracket{e})\prec(\mathcal{A},h\bracket{f})\]
|
||||
\end{frame}
|
||||
|
||||
\subsection{Pourquoi est-il faux ici}
|
||||
\begin{frame}
|
||||
\begin{fjlisting}
|
||||
public Static \{\\
|
||||
\fjmethod{IList}{intList}{Int i}{new IList(i, intList(i+1))}
|
||||
\fjmethod{IList}{zeroList}{}{new IList(0, zeroList())}
|
||||
\}
|
||||
\end{fjlisting}
|
||||
\[e_i = \fj{intList(0)} \quad\text{et}\quad e_0 = \fj{zeroList()}\]
|
||||
\end{frame}
|
||||
\begin{frame}
|
||||
\[
|
||||
\begin{array}{rcl}
|
||||
h\bracket{e_i} &=& \fj{intList(0).next.obj}\\
|
||||
&\rightarrow& \fj{new IList(0, intList(0+1)).next.obj}\\
|
||||
&\rightarrow& \fj{intList(0+1).obj}\\
|
||||
&\rightarrow& \fj{new IList(0+1, intList(0+1+1)).obj}\\
|
||||
&\rightarrow& \fj{0+1}\\
|
||||
&\rightarrow^{\!*}& \fj{1}\\
|
||||
|
||||
h\bracket{e_0} &=& \fj{zeroList().next.obj}\\
|
||||
&\rightarrow& \fj{new IList(0, zeroList()).next.obj}\\
|
||||
&\rightarrow& \fj{zeroList().obj}\\
|
||||
&\rightarrow& \fj{new IList(0, zeroList()).obj}\\
|
||||
&\rightarrow& \fj{0}
|
||||
\end{array}
|
||||
\]
|
||||
\end{frame}
|
||||
\subsection{Idée de valeurs infinies}
|
||||
|
||||
\subsection{Explication de la formalisation}
|
||||
\begin{frame}
|
||||
\textsc{E-Var} et \textsc{E-Cstr}
|
||||
\[\fj{new C(x,new S(new S(y,new G()),z))}\]
|
||||
\[(\mathcal{A},e)\pprec(\mathcal{B},f) \ssi \forall \hbar \forall \alpha \quad \left(e \rightarrow_\mathcal{A}^* \hbar\bracket{\alpha} \implies \exists \beta\quad f \rightarrow_\mathcal{B}^* \hbar\bracket{\beta}\right)\]
|
||||
\end{frame}
|
||||
|
||||
\subsection{Redéfinition des préordres}
|
||||
\begin{frame}
|
||||
\[\mathcal{A}\pprec\mathcal{B} \ssi \forall e \quad (\mathcal{A},e) \pprec (\mathcal{B},e)\]
|
||||
\end{frame}
|
||||
\subsection{Propriétés}
|
||||
\begin{frame}
|
||||
\begin{property}
|
||||
\[\forall \hbar \forall \alpha \quad \hbar\bracket{\alpha}\rightarrow^* e' \implies \exists \beta\quad e' = \hbar\bracket{\beta}\]
|
||||
\end{property}
|
||||
|
||||
\begin{property}
|
||||
\[\mathcal{A} \pprec \mathcal{B} \implies \mathcal{A} \prec \mathcal{B}\]
|
||||
\end{property}
|
||||
|
||||
\begin{property}[Context Lemma]
|
||||
\begin{gather*}
|
||||
\forall \mathcal{A} \quad\forall e,f \quad\forall (h,\mathcal{B})\\
|
||||
(e,\mathcal{A})\pprec(f,\mathcal{B}) \implies (h[e],\mathcal{A} \oplus \mathcal{B})\pprec(h[f],\mathcal{A}\oplus\mathcal{B})
|
||||
\end{gather*}
|
||||
\end{property}
|
||||
\end{frame}
|
||||
\section{Exemple concret}
|
||||
\begin{frame}
|
||||
\begin{tcolorbox}
|
||||
\lstinputlisting[language=Java,breaklines=true,linerange={2-26}]{FinalExample.java}
|
||||
\lstinputlisting[language=Java,breaklines=true,linerange={29-62}, texcl]{FinalExample.java}
|
||||
\lstinputlisting[language=FeatherweightJava,breaklines=true,linerange={65-78}]{FinalExample.java}
|
||||
\end{tcolorbox}
|
||||
\end{frame}
|
||||
|
||||
\section{Conclusion}
|
||||
\begin{frame}
|
||||
\begin{tcolorbox}
|
||||
\begin{itemize}
|
||||
\item On a créé un préordre
|
||||
\item Assez puissant (sémantique)
|
||||
\item Assez tolérant
|
||||
\item Context Lemma
|
||||
\item Utilisable autre part
|
||||
\end{itemize}
|
||||
\end{tcolorbox}
|
||||
\begin{tcolorbox}
|
||||
\begin{itemize}
|
||||
\item Comparer uniquement certains types de retour
|
||||
\item Théorèmes pour simplifier l'équivalence
|
||||
\end{itemize}
|
||||
\end{tcolorbox}
|
||||
\end{frame}
|
||||
|
||||
\begin{frame}
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\LARGE\[\mathfrak{FIN}\]
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19
logoEns.svg
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19
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Reference in New Issue
Block a user