A lot of progress
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1 changed files with 276 additions and 6 deletions
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% vim: :spell spelllang=fr_fr
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% vim: :spell spelllang=fr
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\documentclass[11pt]{beamer}
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\usetheme{Warsaw}
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%\usecolortheme{wolverine}
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\usepackage[utf8]{inputenc}
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\usepackage[french]{babel}
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\usepackage[T1]{fontenc}
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%\usepackage{my_hyperref}
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\usepackage{my_listings}
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\usepackage{libertine}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{amssymb}
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\usepackage{graphicx}
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\usepackage{my_listings}
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\usepackage{my_hyperref}
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\usepackage{concurgames}
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\usepackage{theorems}
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\usepackage{todo}
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\usepackage{math}
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\usepackage{tikz}
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\usetikzlibrary{positioning}
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\setbeamertemplate{navigation symbols}{}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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@ -29,12 +37,274 @@ jeux}
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\begin{frame}
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\titlepage{}
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% \tableofcontents
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\end{frame}
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\begin{frame}
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\tableofcontents
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\end{frame}
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%\begin{frame}
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%\tableofcontents
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%\end{frame}
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\section{Langage étudié}
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\begin{frame}{\llccs~: syntaxe}
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\begin{columns}[t]
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\column{0.5\textwidth}
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\begin{center}Termes\end{center}\vspace{-1em}
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\begin{align*}
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t,u,\ldots ::=~&1 & \text{(succès)}\\
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\vert~&0 & \text{(erreur)}\\
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\vert~&t \parallel u & \text{(parallèle)}\\
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\vert~&t \cdot u & \text{(séquentiel)}\\
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\vert~&(\nu a) t & \text{(nouveau canal)} \\
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& & \\
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\vert~&x \in \mathbb{V} & \text{(variable)} \\
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\vert~&t\,u & \text{(application)}\\
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\vert~&\lambda x^A \cdot t & \text{(abstraction)}\\
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\end{align*}
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\column{0.5\textwidth}
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\begin{center}Types\end{center}\vspace{-1em}
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\begin{align*}
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A,B,\ldots ::=~&\proc & \text{(processus)} \\
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\vert~&\chan & \text{(canal)} \\
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\vert~&A \linarrow B & \text{(flèche linéaire)}
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\end{align*}
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\end{columns}
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\end{frame}
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\begin{frame}{\llccs~: règles de typage}
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\begin{align*}
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\frac{~}{\,\vdash 0:\proc} & (\textit{Ax}_0) &
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\frac{~}{\,\vdash 1:\proc} & (\textit{Ax}_1) &
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\alert{\frac{~}{t:A \vdash t:A}} & \alert{(\textit{Ax})} &
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\end{align*}\vspace{-1em}\begin{align*}
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\frac{\Gamma \vdash P : \proc \quad \Delta \vdash Q : \proc}
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{\Gamma,\Delta \vdash P \cdot Q : \proc}
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& (\cdot_\proc) &
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\frac{\Gamma \vdash P : \proc}
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{\Gamma,a:\chan \vdash a \cdot P: \proc}
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& (\cdot_\chan)
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\end{align*} \vspace{-1em} \begin{align*}
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\alert{\frac{\Gamma \vdash P : \proc \quad \Delta \vdash Q : \proc}
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{\Gamma,\Delta \vdash P \parallel Q : \proc}}
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& \alert{(\parallel)} &
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\frac{\Gamma, a:\chan, \bar{a}:\chan \vdash P : \proc}
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{\Gamma \vdash (\nu a) P : \proc}
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& (\nu)
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\end{align*}
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\begin{align*}
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\frac{~}{x : A \vdash x : A} & (\textit{Ax}) &
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\frac{\Gamma \vdash t : A \linarrow B \quad \Delta \vdash u : A}
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{\Gamma,\Delta \vdash t~u : B} & (\textit{App}) &
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\end{align*} \vspace{-1em} \begin{align*}
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\frac{\Gamma, x : A \vdash t : B}
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{\Gamma \vdash \lambda x^{A} \cdot t : A \linarrow B} & (Abs)
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\end{align*}
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\end{frame}
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\begin{frame}{\llccs~: sémantique opérationnelle}
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\begin{align*}
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\frac{~}{a \cdot P \redarrow{a} P} & &
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\frac{~}{1 \parallel P \redarrow{\tau} P} & &
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\frac{~}{1 \cdot P \redarrow{\tau} P} & &
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\end{align*}\begin{align*}
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\frac{P \longrightarrow_\beta Q}
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{P \redarrow{\tau} Q} & &
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\frac{P \redarrow{\tau_c} Q}
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{(\nu a) P \redarrow{\tau} Q} & (c \in \set{a,\bar{a}})&
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\end{align*}\begin{align*}
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\alert{\frac{P \redarrow{a} P'\quad Q \redarrow{\bar{a}} Q'}
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{P \parallel Q \redarrow{\tau_a} P' \parallel Q'}} & &
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\frac{P \redarrow{x} P'}
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{(\nu a)P \redarrow{x} (\nu a)P'} & (x \not\in \set{a,\tau_a}) &
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\end{align*}\begin{align*}
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\frac{P \redarrow{x} P'}
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{P \parallel Q \redarrow{x} P' \parallel Q} & &
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\frac{Q \redarrow{x} Q'}
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{P \parallel Q \redarrow{x} P \parallel Q'} & &
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\frac{P \redarrow{x} P'}
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{P \cdot Q \redarrow{x} P' \cdot Q} & &
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\end{align*}
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\end{frame}
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\begin{frame}{Quelques exemples}
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\begin{itemize}
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\item $(1 \parallel 1) \cdot 1$~: succès
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\pause%
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\item $\newch{a} (a \cdot 1 \parallel \bar{a} \cdot 1)$
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\pause%
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\item $\newch{a} (a \cdot \bar{a} \cdot 1)$~: \textit{deadlock}
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\pause%
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\item $F := \lambda x^\chan \cdot \lambda y^\chan \cdot x \cdot y \cdot
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1$
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\pause%
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\item $\newch{a} F\,a\,\bar{a}$~: \textit{deadlock}
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\pause%
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\item $\newch{a} \newch{b} (F\,a\,b \parallel \bar{a} \cdot \bar{b}
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\cdot 1)$~: termine
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\end{itemize}
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\end{frame}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\section{Structures d'événements}
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\begin{frame}{Sémantique des jeux}
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\begin{itemize}
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\item Programme $\longrightarrow$ jeu à deux joueurs
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\pause\vspace{1em}
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\item \textit{Player}~: joue pour le programme
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\item \textit{Opponent}~: joue pour l'environnement (OS, utilisateur,
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\ldots)
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\pause\vspace{1em}
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\item \textit{Jeu}~: structure imposée (architecture physique, \ldots)
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\item \textit{Stratégie}~: modélisation du programme
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\end{itemize}
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\end{frame}
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\subsection{Sémantique par entrelacements}
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\begin{frame}{Sémantique usuelle~: par entrelacements}
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\begin{itemize}
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\item Jeux en arbres\\
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\begin{columns}
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\column{0.5\textwidth}
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\begin{center}\begin{tikzpicture}[node distance=0.5cm]
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\node (1) {a};
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\node (2) [below left=of 1] {b};
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\node (3) [below right=of 1] {c};
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\path [->]
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(1) edge (2)
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edge (3);
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\end{tikzpicture}
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\end{center}
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\column{0.5\textwidth}
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États du jeu~: $\varepsilon,~a,~a \cdot b,~a \cdot c$
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\end{columns}
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\pause%
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\item \og{}Exécution parallèle~\fg{}:
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\begin{columns}
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\column{0.5\textwidth}
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\begin{center}\begin{tikzpicture}[node distance=0.5cm]
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\node (1) {a};
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\node (2) [below left=of 1] {b};
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\node (3) [below right=of 1] {c};
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\node (5) [right=of 3] {e};
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\node (4) [above right=of 5] {d};
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\node (6) [below right=of 4] {f};
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\path [->]
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(1) edge (2)
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edge (3)
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(4) edge (5)
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edge (6);
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\end{tikzpicture}
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\end{center}
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\column{0.5\textwidth}
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États du jeu~: $\varepsilon,~a,~d,~a \cdot b,~a \cdot d \cdot e
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\cdot b, \ldots$
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\end{columns}
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\pause%
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\item Comment représenter \og{}proprement~\fg{} le jeu suivant~?
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\begin{center}\begin{tikzpicture}[node distance=0.5cm]
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\node (3) {c};
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\node (1) [above left=of 3] {a};
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\node (2) [above right=of 3] {b};
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\path [->]
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(1) edge (3)
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(2) edge (3);
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\end{tikzpicture}
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\end{center}
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\end{itemize}
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\end{frame}
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\subsection{Structures d'événements}
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\begin{frame}{Structures d'événements}
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\begin{definition}{structure d'événement (déterministe)}
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$(E, \leq_E)$ ensemble d'\emph{événements} partiellement ordonné.
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\end{definition}
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-\pause%
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\begin{definition}{structure d'événement polarisée (SEP) / \alert{jeu}}
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$(E, \leq_E, \rho_E)$ où $(E, \leq_E)$ est une structure d'événements
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et $\rho_E : E \to \set{\ominus, \oplus}$.
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$E^\perp$~: SEP $E$ avec $\rho^\perp$.
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\end{definition}
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$\qquad\longrightarrow$ DAG
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\pause%
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\begin{definition}{configuration}
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$x \subseteq E$ tel que $\forall e \in E, e' \in x, e \leq e' \implies
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e \in x$.\\
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$\config(E)$~: ensemble des configurations de $E$.\\
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Pour $e \in E$, $[e]$ configuration induite par $e$.
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\end{definition}
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\end{frame}
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\begin{frame}{Structures d'événements (suite)}
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\begin{definition}{stratégie}
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$\left(\sigma : A\right)$~: stratégie sur $A$ si $\sigma$ SEP
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\textit{tq.}\\
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\begin{itemize}
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\item $\sigma \subseteq A$
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\item $\config(\sigma) \subseteq \config(A)$
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\item $\rho_\sigma = {\rho_A}_{\vert \sigma}$
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\item $\cc_A \strComp \sigma = \sigma$ ($\cc_A$~:
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\og{}identité~\fg)
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\end{itemize}
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\end{definition}
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\end{frame}
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\begin{frame}{Opérations sur structures d'événements}
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\begin{definition}{parallèle}
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$A \parallel B := \set{0} \times A \cup \set{1} \times B$~: mise
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en parallèle de deux SE\@.
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\end{definition}
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\begin{definition}{interaction}
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Pour $\sigma : A$, $\tau : A^\perp$, $\sigma \wedge \tau$~: $\sigma
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\cap \tau$ où les boucles causales sont retirées.
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\end{definition}
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\begin{definition}{composition}
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Pour $\sigma : A^\perp \parallel B$, $\tau : B^\perp \parallel C$,
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\[ \tau \strComp \sigma := \left(\sigma \parallel C^\perp\right) \wedge
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\left(A \parallel \tau\right) \]
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\end{definition}
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\begin{definition}{copycat}
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$\cc_A : A^\perp \parallel A$~: $(A^\perp \parallel \emptyset) \cup
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(\emptyset \parallel A)$, plus les $\ominus \rightarrow \oplus$ d'une
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composante à l'autre.
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\end{definition}
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\end{frame}
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\section{Interprétation de \llccs}
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\begin{frame}{Interprétation de \llccs}
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\begin{columns}[t]
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\column{0.5\textwidth}
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\begin{align*}
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\seman{\proc} &\eqdef
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\end{align*}
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\column{0.5\textwidth}
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\begin{align*}
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\seman{x^A} &\eqdef \cc_{\seman{A}} \\
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\seman{A \linarrow B} &\eqdef \seman{A}^\perp \parallel \seman{B}\\
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\seman{t^{A \linarrow B}~u^{A}} &\eqdef
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\cc_{A \linarrow B} \strComp \left( \seman{t} \parallel
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\seman{u} \right) \\
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\seman{\lambda x^A \cdot t} &\eqdef \seman{t}
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\end{align*}
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\end{columns}
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\end{frame}
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\end{document}
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