Useful custom libs

concurgames.sty

@@ -0,0 +1,14 @@
+%% By Théophile Bastian
+%% Useful commands for concurrent games as event structures
+
+\usepackage[normalem]{ulem}
+
+\newcommand{\con}{\operatorname{Con}}
+\newcommand{\confl}{\raisebox{0.5em}{\uwave{\hspace{2em}}}}
+
+\newcommand{\forkover}[1]{[\qtodo{fork}~#1]}
+\newcommand{\fork}{[\qtodo{fork}]}
+\newcommand{\edgeArrow}{\rightarrowtriangle}
+
+\newcommand{\config}{\mathscr{C}}
+

leftrule_theorems.sty

@@ -0,0 +1,41 @@
+\usepackage[dvipsnames]{xcolor}
+\usepackage{ifthen}
+\usepackage[framemethod=tikz]{mdframed}
+
+\newcommand{\linedenvTop}[2]
+	{\begin{samepage}\qquad\textbf{#1}
+	\ifthenelse{\equal{#2}{}}
+		{}
+		{~\textbf{(}\emph{#2}\textbf{)}}
+	\trivlist\item[]\ignorespaces}
+\newcommand{\linedenvBot}{\endtrivlist\end{samepage}}
+
+\newenvironment{lemma}[1][]{\linedenvTop{Lemma}{#1}}{\linedenvBot}
+\surroundwithmdframed[linewidth=1.5pt,
+	linecolor=BurntOrange,
+	bottomline=false,topline=false,rightline=false]{lemma}
+
+\newenvironment{definition}[1][]{\linedenvTop{Definition}{#1}}{\linedenvBot}
+\surroundwithmdframed[linewidth=1.5pt,
+	linecolor=Plum,
+	bottomline=false,topline=false,rightline=false]{definition}
+
+\newenvironment{theorem}[1][]{\linedenvTop{Theorem}{#1}}{\linedenvBot}
+\surroundwithmdframed[linewidth=2.5pt,
+	linecolor=Red,
+	bottomline=false,topline=false,rightline=false]{theorem}
+
+%\newenvironment{proof}{\textbf{Proof}\\}{}
+\surroundwithmdframed[linewidth=1.0pt,
+	linecolor=Blue,
+	bottomline=false,topline=false,rightline=false]{proof}
+
+\newenvironment{notation}[1][]{\linedenvTop{Notation}{#1}}{\linedenvBot}
+\surroundwithmdframed[linewidth=1.5pt,
+	linecolor=Brown,
+	bottomline=false,topline=false,rightline=false]{notation}
+
+\newenvironment{example}[1][]{\linedenvTop{Example}{#1}}{\linedenvBot}
+\surroundwithmdframed[linewidth=1.5pt,
+	linecolor=LimeGreen,
+	bottomline=false,topline=false,rightline=false]{example}

math.sty

@@ -0,0 +1,67 @@
+\usepackage{stmaryrd}
+\usepackage{amsmath}
+\usepackage{amsfonts}
+\usepackage{amssymb}
+\usepackage{amsthm}
+\usepackage{mathtools}
+\usepackage{fancybox}
+\usepackage{mathrsfs}
+
+% Intervalle discret.
+\newcommand{\discrIv}[1]{\llbracket #1 \rrbracket}
+
+% ensembliste
+\newcommand{\set}[1]{\left\{ #1 \right\}}
+\newcommand{\card}[1]{\left\vert{} #1 \right\vert}
+\newcommand{\abs}[1]{\left\vert{} #1 \right\vert}
+\newcommand{\interior}[1]{\left({#1}\right)^\circ}
+\newcommand{\floor}[1]{\left\lfloor{} #1 \right\rfloor}
+\newcommand{\ceil}[1]{\left\lceil{} #1 \right\rceil}
+
+% Abréviations courrantes
+\newcommand{\ie}{\textit{ie.}}
+\newcommand{\eg}{\textit{eg.}}
+\newcommand{\wrt}{\textit{wrt.}}
+\newcommand{\st}{\textit{st.}}
+
+% Notations à polices étranges
+\newcommand{\domain}{\mathcal{D}}
+\newcommand{\bigO}{\mathcal{O}}
+\newcommand{\calA}{\mathcal{A}}
+\newcommand{\calC}{\mathcal{C}}
+\newcommand{\calG}{\mathcal{G}}
+\newcommand{\calV}{\mathcal{V}}
+\newcommand{\calT}{\mathcal{T}}
+\newcommand{\calP}{\mathcal{P}}
+
+% Ensembles
+\newcommand{\realset}{\mathbb{R}}
+\newcommand{\natset}{\mathbb{N}}
+\newcommand{\relset}{\mathbb{Z}}
+\newcommand{\powerset}{\mathcal{P}}
+
+% Probas
+\newcommand{\prob}{\mathbb{P}}
+\newcommand{\probP}[1]{\mathbb{P}\left(#1\right)}
+\newcommand{\expec}{\mathbb{E}}
+\newcommand{\expecP}[1]{\mathbb{E}\left[#1\right]}
+\newcommand{\variance}{\mathbb{V}}
+\newcommand{\ber}{\mathcal{B}er}
+\newcommand{\bin}{\mathcal{B}in}
+\newcommand{\poi}{\mathcal{P}oi}
+
+% Suppression des points
+\newcommand{\ibar}{\overline{\imath}}
+\newcommand{\jbar}{\overline{\jmath}}
+
+% Fonctions
+%\newcommand{\functiondef}[4]{\left\lbrace \begin{tabular}{r l} #1 & \rightarrow #2 \\ #3 & \mapsto #4\end{tabular} \right.}
+\newcommand{\functiondef}[4]{\begin{cases}
+#1 & \to #2 \\
+#3 & \mapsto #4
+\end{cases}}
+
+
+% Preuve par équivalence - puces
+\newcommand{\impliesbullet}{\ovalbox{$\implies$}}
+\newcommand{\impliedbybullet}{\ovalbox{$\impliedby$}}

my_hyperref.sty

@@ -0,0 +1,25 @@
+\usepackage{hyperref}
+\usepackage{xcolor}
+
+\definecolor{link_blue}{RGB}{0,0,97}
+
+\hypersetup{
+%    bookmarks=true,         % show bookmarks bar?
+%    unicode=false,          % non-Latin characters in Acrobat’s bookmarks
+%    pdftoolbar=true,        % show Acrobat’s toolbar?
+%    pdfmenubar=true,        % show Acrobat’s menu?
+%    pdffitwindow=false,     % window fit to page when opened
+%    pdfstartview={FitH},    % fits the width of the page to the window
+%    pdftitle={My title},    % title
+%    pdfauthor={Author},     % author
+%    pdfsubject={Subject},   % subject of the document
+%    pdfcreator={Creator},   % creator of the document
+%    pdfproducer={Producer}, % producer of the document
+%    pdfkeywords={keyword1} {key2} {key3}, % list of keywords
+%    pdfnewwindow=true,      % links in new PDF window
+    colorlinks=true,       % false: boxed links; true: colored links
+    linkcolor=link_blue,          % color of internal links (change box color with linkbordercolor)
+    citecolor=green,        % color of links to bibliography
+    filecolor=magenta,      % color of file links
+    urlcolor=link_blue           % color of external links
+}

report.tex

@@ -5,12 +5,20 @@
 \usepackage{amsfonts}
 \usepackage{amssymb}
 \usepackage{graphicx}
+\usepackage{indentfirst}
+\usepackage{enumerate}
 \usepackage[left=2cm,right=2cm,top=2cm,bottom=2cm]{geometry}
 
 % Custom packages
+\usepackage{leftrule_theorems}
+\usepackage{concurgames}
 \usepackage{my_listings}
+\usepackage{my_hyperref}
 \usepackage{math}
 
+\newcommand{\qtodo}[1]{\colorbox{orange}{\textcolor{blue}{#1}}}
+\newcommand{\todo}[1]{\colorbox{orange}{\qtodo{\textbf{TODO:} #1}}}
+
 \author{Théophile \textsc{Bastian}}
 \title{Internship report --- Concurrent games as event structures\\
 	\begin{small}Cambridge University --- Glynn Winskel\end{small}}
@@ -19,5 +27,108 @@
 \begin{document}
 \maketitle
 
+\todo{abstract}
+
+\section{Existing work}
+
+My work is set in the context of a wider theory, the basics of which are
+necessary to understand properly what follows. It is the purpose of this
+section to bring light upon this theory.
+
+The general work of the team I was working in could be described as
+``concurrent games as event structures'', that is, using the \textit{event
+structures} formalism to describe concurrent games, instead of the more
+traditional approach of \emph{tree-like games} (``Player plays $A$, then
+Opponent plays $B$, thus reaching the configuration $A \cdot B$'').
+
+\subsection{Event structures}
+
+\begin{definition}[event structure]
+	An \emph{event structure} is a pair $(E, \leq_E, \con_E)$, where $E$ is a
+	set of \emph{events}, $\leq_E$ is a partial order on $E$ and
+	$\con_E \subseteq \powerset(E)$ is a set of \emph{consistent events}.
+
+	The partial order $\leq_E$ naturally induces a binary relation $\edgeArrow$
+	over $E$ that is defined as the transitive reduction of $\leq_E$.
+\end{definition}
+
+In this context, the right intuition of event structures is a set of events
+that can occur, the players' moves, alongside with a partial order stating that
+a given move cannot occur before another move, and a consistency relation
+indicating whether a given set of moves can occur in the same instance of the
+game.
+
+The consistency relation is often replaced by a weaker \emph{conflict} binary
+relation $\confl$ indicating that two events cannot occur together.
+
+Event structures are often represented as a directed acyclic graph (DAG)
+where the vertices are the elements of $E$ and the edges are the transitive
+reduction of $\leq_E$; onto which the conflict relation is superimposed.
+
+\begin{definition}[event structure with polarities]
+	An \emph{event structure with polarities} (\textit{ESP}) is an event
+	structure $(E, \leq_E, \con_E, \rho)$, where $\rho : E \to \set{+,-}$ is a
+	function associating a \emph{polarity} to each event.
+\end{definition}
+
+In games theory, this is used to represent whether a move is to be played by
+Player or Opponent.
+
+\begin{definition}[configuration]
+	A \emph{configuration} of an ESP $A$ is a subset $X \subseteq \con_A$
+	that is \emph{down-closed}, \ie{} $\forall x \in X, \forall e \in E_A,
+	e \leq_A x \implies e \in X$.
+
+	$\config(A)$ is the set of configurations of $A$.
+\end{definition}
+
+\begin{notation}
+	For $x,y \in \config(A)$, $x \forkover{e} y$ states that $y = x \cup
+	\set{e}$ (and that both are valid configurations). It is also possible to
+	write $x \forkover{e}$, stating that $x \cup \set{e} \in \config(A)$, or $x
+	\fork y$.
+\end{notation}
+
+\subsection{Concurrent games}
+
+\begin{definition}[game]
+	A \emph{game} $A$ is an event structure with polarities. \\
+	The dual game $A^\perp$ is the game $A$ where all the polarities in
+	$\rho$ have been reversed.
+\end{definition}
+
+\begin{definition}[pre-strategy]
+	A \emph{pre-strategy} $\sigma: S \to \calG$ is a total map of ESPs, where
+	$\calG$ is the game on which the strategy plays, such that
+	\begin{enumerate}[(i)]
+		\item $\forall x \in \config(S), \sigma(x) \in \config(\calG)$;
+		\item $\forall s,s' \in \config(S), \sigma(s) = \sigma(s') \implies
+			s = s'$ (local injectivity);
+		\item $\forall s \in S, \rho_\calG(\sigma(s)) = \rho_S(s)$
+	\end{enumerate}
+\end{definition}
+
+\begin{definition}[strategy]
+	A \emph{strategy} is a pre-strategy $\sigma : S \to A$ that
+	``behaves well'', \ie{} that is
+	\begin{enumerate}[(i)]
+		\item\label{def:receptive}
+			\textit{receptive}: for all $x \in \config(A)$ \st{}
+			$\sigma(x) \forkover{e^-}$, $\exists! s \in S : \sigma(s) = a$;
+
+		\item\label{def:courteous}
+			\textit{courteous}: $\forall x \edgeArrow x' \in S,
+			(\rho(x),\rho(x')) \neq (-,+) \implies
+			\sigma(x) \edgeArrow \sigma(x')$.
+	\end{enumerate}
+\end{definition}
+
+(\ref{def:receptive}) captures the idea that we should not force Opponent not to
+play one of its moves, while~(\ref{def:courteous}) states that unless a
+dependency relation is imposed by the games' rules, one can only make one of
+its moves depend on an Opponent move.
+
+\section{Implementation of non-concurrent games}
+
 \end{document}