Bit more on operations

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Théophile Bastian 2016-08-10 18:45:28 +01:00
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\newcommand{\eqdef}{{~\coloneqq~}}
\newcommand{\id}{\operatorname{id}}
% Intervalle discret.
\newcommand{\discrIv}[1]{\llbracket #1 \rrbracket}

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\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}