From df1e93149456b4fd4c29a4398c5f4cabe2547bff Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Th=C3=A9ophile=20Bastian?= <contact@tobast.fr>
Date: Tue, 19 Jul 2016 23:26:19 +0100
Subject: [PATCH] Few more lines

---
 report.tex | 29 +++++++++++++++++++++++++++--
 1 file changed, 27 insertions(+), 2 deletions(-)

diff --git a/report.tex b/report.tex
index c0c41f7..d8bbfa4 100644
--- a/report.tex
+++ b/report.tex
@@ -46,6 +46,8 @@ 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{Informal approach}
+
 \subsection{Event structures}
 
 \begin{definition}[event structure]
@@ -67,13 +69,18 @@ game.
 The consistency relation is often replaced by a weaker \emph{conflict} binary
 relation $\confl$ indicating that two events cannot occur together.
 
+During this internship, my work was essentially carried on event structures
+without conflicts. Thus, the consistency set is not relevant and will be
+omitted in what follows, but one can refer to~\cite{castellan2016concurrent}
+for the corresponding constructions with consistency sets.
+
 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
+	structure $(E, \leq_E, \rho)$, where $\rho : E \to \set{+,-}$ is a
 	function associating a \emph{polarity} to each event.
 \end{definition}
 
@@ -81,7 +88,7 @@ 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 $X \subseteq \con_A$
+	A \emph{configuration} of an ESP $A$ is a subset $X \subseteq A$
 	that is \emph{down-closed}, \ie{} $\forall x \in X, \forall e \in E_A,
 	e \leq_A x \implies e \in X$.
 
@@ -177,6 +184,24 @@ 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.
 
+\subsection{Operations on games and strategies}
+
+\todo{intro}
+
+In this whole section, $E$ and $F$ denotes ESPs, $A$ and $B$ denotes games,
+$\sigma: S \to A$ and $\tau: T \to B$ denotes strategies.
+
+\begin{definition}[Parallel composition]
+	The \emph{parallel composition} $E \parallel F$ of two ESPs is an ESP
+	whose events are $\left(\set{0} \times E\right) \cup \left(\set{1} \times
+	F\right)$ (the disjoint tagged union of the events of $E$ and $F$), and
+	whose partial order is $\leq_E$ on $E$ and $\leq_F$ on $F$, with no
+	relation between elements of $E$ and $F$.
+
+	One can then naturally expand this definition to games (by preserving
+	polarities) and to strategies.
+\end{definition}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 \section{Implementation of deterministic concurrent games}