From 9a1fddaf8e9752c2c3bd4d68690f73a72794ae70 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Th=C3=A9ophile=20Bastian?= <contact@tobast.fr>
Date: Wed, 10 Aug 2016 18:45:28 +0100
Subject: [PATCH] Bit more on operations

---
 math.sty   |  2 ++
 report.tex | 44 ++++++++++++++++++++++++++++++++++++++++----
 2 files changed, 42 insertions(+), 4 deletions(-)

diff --git a/math.sty b/math.sty
index 554dc69..5ea14a9 100644
--- a/math.sty
+++ b/math.sty
@@ -10,6 +10,8 @@
 
 \newcommand{\eqdef}{{~\coloneqq~}}
 
+\newcommand{\id}{\operatorname{id}}
+
 % Intervalle discret.
 \newcommand{\discrIv}[1]{\llbracket #1 \rrbracket}
 
diff --git a/report.tex b/report.tex
index 3366d06..cf390d8 100644
--- a/report.tex
+++ b/report.tex
@@ -21,6 +21,7 @@
 \newcommand{\qtodo}[1]{\colorbox{orange}{\textcolor{blue}{#1}}}
 \newcommand{\todo}[1]{\colorbox{orange}{\qtodo{\textbf{TODO:} #1}}}
 \newcommand{\qnote}[1]{\colorbox{Cerulean}{\textcolor{Sepia}{[#1]}}}
+\newcommand{\note}[1]{\qnote{\textbf{NOTE:} #1}}
 
 \author{Théophile \textsc{Bastian}, supervised by Glynn \textsc{Winskel}
 	and Pierre \textsc{Clairambault} \\
@@ -533,14 +534,18 @@ then $x$.
 %%%%%
 \subsubsection{Operations on games and strategies}
 
-\todo{intro}
+In order to manipulate strategies and define them by induction over the syntax,
+the following operations will be extensively used. It may also be worth noting
+that in the original formalism~\cite{castellan2016concurrent}, games,
+strategies and maps between them form a bicategory in which these operations
+play special roles.
 
-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.
+In this whole section, unless stated otherwise, $E$ and $F$ denotes ESPs, $A$,
+$B$ and $C$ denotes games, $\sigma: A$ and $\tau: B$ 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
+	whose events are $\left(\set{0} \times E\right) \sqcup \left(\set{1} \times
 	F\right)$ (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 $F$.
@@ -549,6 +554,37 @@ $\sigma: S \to A$ and $\tau: T \to B$ denotes strategies.
 	polarities) and to strategies.
 \end{definition}
 
+Given two strategies on dual games $A$ and $A^\perp$, it is interesting to
+compute their \emph{interaction}, that is, ``what will happen if one strategy
+plays against the other''.
+
+\note{Are the following names clear enough?}
+\begin{definition}[Interaction]
+	Given two strategies $\sigma : A$ and $\tau : A^\perp$, their
+	\emph{interaction} $\sigma \wedge \tau$ is the ESP
+	$\sigma \cup \tau \subseteq A$ from which causal loops has been removed.
+
+	More precisely, $\sigma \cup \tau$ is a set adjoined with a \emph{preorder}
+	($\leq_\sigma \cup \leq_\tau$) that may not respect antisymmetry, that is,
+	may have causal loops. $\sigma \wedge \tau$ is then obtained by removing
+	all the elements contained in such loops from $\sigma \cup \tau$.
+\end{definition}
+\textit{Note: this can be interpreted as a pullback in the category mentioned
+above.\\
+This construction, even though it is equivalent to the construction
+of~\cite{castellan2016concurrent} when considering deterministic strategies, is
+no longer valid when adding a consistency set.}
+
+\begin{definition}[Compositional interaction]
+	Given two strategies $\sigma : A^\perp \parallel B$ and $\tau : B^\perp
+	\parallel C$, their \emph{compositional interaction} $\tau \strInteract
+	\sigma$ is defined as $(\sigma \parallel \id_C) \wedge (\id_A \parallel
+	\tau)$. \qtodo{Tell me more?}
+\end{definition}
+
+\begin{definition}[Strategies composition]
+\end{definition}
+
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 \subsection{Interpretation of \llccs}