From a5abbd2680bd37d900968b412eca6eca39ed7ae2 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Th=C3=A9ophile=20Bastian?= <contact@tobast.fr>
Date: Thu, 21 Jul 2016 18:10:02 +0100
Subject: [PATCH] Started informal intro

---
 report.tex | 64 ++++++++++++++++++++++++++++++++++++++++++++++++++++++
 1 file changed, 64 insertions(+)

diff --git a/report.tex b/report.tex
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--- a/report.tex
+++ b/report.tex
@@ -48,6 +48,70 @@ Opponent plays $B$, thus reaching the configuration $A \cdot B$'').
 
 \subsection{Informal approach}
 
+The traditional approach to concurrent games is to represent them as
+\emph{tree-like games}. If the considered game consists in three moves, namely
+$A$, $B$ and $C$, where $A$ can be played by Opponent and the others by Player
+\emph{after} Opponent has played $A$, that means that the states of the game
+will be $\epsilon$, $A$, $A \cdot B$ and $A \cdot C$, which corresponds to the
+game tree
+
+\[
+	\begin{tikzpicture}
+		\node (1) [ellipse] {A} ;
+		\node (2) [below left of=1, ellipse] {B};
+		\node (3) [below right of=1, ellipse] {C};
+
+		\path [->]
+			(1) edge (2)
+				edge (3);
+	\end{tikzpicture}
+\]
+
+This can of course be used to describe much larger games, and is often useful
+to reason concurrently, since the causal histories appear clearly: the possible
+states of the game can be read easily by concatenating the events that are on a
+same branch.
+
+But it also has the major drawback of growing exponentially in size: let us
+consider a game in which Opponent must play $A$ and $B$ in no particular order
+before Player can play $C$. The corresponding tree-like game would be
+\[
+	\begin{tikzpicture}
+		\node (11) {$A_1$};
+		\node (21) [below of=11] {$B_1$};
+		\node (31) [below of=21] {$C_1$};
+		\node (22) [right of=11] {$B_2$};
+		\node (12) [below of=22] {$A_2$};
+		\node (32) [below of=12] {$C_2$};
+
+		\path [->]
+			(11) edge (21)
+			(21) edge (31)
+			(22) edge (12)
+			(12) edge (32);
+	\end{tikzpicture}
+\]
+
+This problem motivated the (still marginal) introduction of \emph{event
+structures} as a formalism to describe such games. Informally, an event
+structure is a partial order $\leq$ on \emph{events} (here, the game's moves),
+alongside with a \emph{consistency} relation.
+
+The relation $e_1 \leq e_2$ means that $e_1$ must have been played before $e_2$
+can be played, while the consistency relation states which events can occur
+together in a same game. For instance, the previous game would have all its
+events consistent with one another and its Hasse diagram would look like
+\[
+	\begin{tikzpicture}
+		\node (1) {A};
+		\node (2) [right of=1] {B};
+		\node (3) [below left of=1, below right of=2] {C};
+		\path[->]
+			(1) edge (3)
+			(2) edge (3);
+	\end{tikzpicture}
+\]
+
 \subsection{Event structures}
 
 \begin{definition}[event structure]