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\author{Théophile \textsc{Bastian}, supervised by Glynn \textsc{Winskel}
	and Pierre \textsc{Clairambault} \\
	\begin{small}Cambridge University\end{small}}
\title{Internship report\\Concurrent games as event structures}
\date{June-July 2016}

\begin{document}
\maketitle

\todo{abstract}

\tableofcontents

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}

\paragraph{Game semantics} are a kind of denotational semantics in which a
program's behavior is abstracted as a two-players game, in which Player plays
for the program and Opponent plays for the environment of the program (the
user, the operating system, \ldots). The execution of a program, in this
formalism, is then represented as a succession of moves. For instance, the user
pressing a key on the keyboard would be a move of Opponent, to which Player
could react by triggering the corresponding action (\eg{} adding the
corresponding letter in a text field).

Game semantics emerged mostly with~\cite{hyland2000pcf}
and~\cite{abramsky2000pcf}, independently establishing a fully-abstract model
for PCF using game semantics, while ``classic'' semantics had failed to provide
a fully-abstract, reasonable and satisfying model. But this field mostly gained
in notoriety with the development of techniques to capture imperative
programming languages constructions, among which references
handling~\cite{abramsky1996linearity}, followed by higher-order
references~\cite{abramsky1998references}, allowing to model languages with side
effects; or exception handling~\cite{laird2001exceptions}. Since then, the
field has been deeply explored, providing a wide range of such constructions in
the literature.

A success of game semantics is to provide \emph{compositional} and
\emph{syntax-free} semantics. Syntax-free, because representing a program as a
strategy on a game totally abstracts it from the original syntax of the
programming language, representing only the behavior of a program reacting to
its execution environment, which is often desirable in semantics.
Compositional, because game semantics are usually defined by induction over the
syntax, thus easily composed. For instance, it is worth noting that the
application of one term to another is represented as the \emph{composition} of
the two strategies.

\paragraph{Concurrency in game semantics.} In the continuity of the efforts put
forward to model imperative primitives in game semantics, it was natural to
focus at some point on modelling concurrency. The problem was tackled by
\fname{Laird}~\cite{laird2001game}, introducing game semantics for a \lcalc{}
with a few additions, as well as a \emph{parallel execution} operator and
communication on channels. It is often considered, though, that \fname{Ghica}
and \fname{Murawski}~\cite{ghica2004angelic} really took the fundamental step
by defining game semantics for a fine-grained concurrent language including
parallel execution of ``threads'' and low-level semaphores --- a way more
realistic approach to the problem.

However, both of these constructions are based on \emph{interleavings}. That
is, they model programs on \emph{tree-like games}, games in which the moves
that a player is allowed to play at a given point are represented as a tree
(\eg, in a state $A$, Player can play the move $x$ by following an edge of the
tree starting from $A$, thus reaching $B$ and allowing Opponent to play a given
set of moves --- the outgoing edges of $B$). The concurrency is then
represented as the \emph{interleaving} of all possible sequences of moves, in
order to reach a game tree in which every possible ``unordered'' (\ie, that is
not enclosed in any kind of synchronisation block, as with semaphores)
combination of moves is a valid path.

However, this approach introduces non-determinism in the strategies: if two
moves are available to a player, the model states that they make a
non-deterministic uniform choice. Yet, one could reasonably argue that a
program should behave consistently with the environment, which would mean that
the semantics of a program --- even a concurrent one --- should still be
deterministic. This idea was explored outside of the game semantics context,
for instance by~\cite{reynolds1978syntactic}, establishing a type-checking
system to restrict concurrent programs to deterministic ones. Some recent work
makes use of linear logic~\cite{caires2010session} for similar purposes as
well.  Yet, the interleavings game semantics of these languages remains
non-deterministic.

\paragraph{The purpose of this internship} was to try to take a first step
towards the reunification of those two developments. For that purpose, my
objective was to give a \emph{deterministic} game semantics to a linear
lambda-calculus enriched with parallel and sequential execution operators, as
well as synchronization on channels. In order to model this, I used the games
as \emph{event structures} formalism, described later on and introduced
in~\cite{rideau2011concurrent} by S. \fname{Rideau} and G. \fname{Winskel}.
Roughly, event structures represent a strategy as a \emph{partial order} on the
moves, stating that move $x$ can only be played after move $y$, which is more
flexible than tree-like game approaches. Although a full-abstraction result
could not be reached --- but is not so far away ---, I have proved the
\emph{adequacy} of the operational and denotational semantics, and have
obtained an implementation of the (denotational) game semantics, that is, code
that translates a term of the language into its corresponding strategy.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A linear \lcalc{} with concurrency primitives: \llccs}

The language on which my internship was focused was meant to be simple, easy to
parse and easy to work on both in theory and on the implementation. It should
of course include concurrency primitives. For these reasons, we chose to
consider a variant of CCS~\cite{milner1980ccs} --- a simple standard language
including parallel and sequential execution primitives, as well as
synchronization of processes through \emph{channels} ---, lifted up to the
higher order through a \lcalc. The language was then restricted to a
\emph{linear} one --- that is, each identifier declared must be referred to
exactly once ---, partly to keep the model simple, partly to meet the
determinism requirements.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{A linear variant of CCS~: \linccs}

The variant of CCS we chose to use has two base types:
\emph{processes}~($\proc$) and \emph{channels}~($\chan$). It has two base
processes, $0$ (failure) and $1$ (success), although a process can be
considered ``failed'' without reducing to $0$ (in case of deadlock).

\begin{figure}[h]
	\begin{minipage}[t]{0.60\textwidth}
		\begin{center}Terms\end{center}\vspace{-1em}
		\begin{align*}
			t,u,\ldots ::=~&1 & \text{(success)}\\
			\vert~&0 & \text{(error)}\\
			\vert~&t \parallel u & \text{(parallel)}\\
			\vert~&t \cdot u & \text{(sequential)}\\
			\vert~&(\nu a) t & \text{(new channel)}
		\end{align*}
	\end{minipage} \hfill \begin{minipage}[t]{0.35\textwidth}
		\begin{center}Types\end{center}\vspace{-1em}
		\begin{align*}
			A,B,\ldots ::=~&\proc & \text{(process)} \\
			\vert~&\chan & \text{(channel)}
		\end{align*}
	\end{minipage}
	\caption{\linccs{} terms and types}\label{fig:lccs:def}
\end{figure}

The syntax is pretty straightforward to understand: $0$ and $1$ are base
processes; $\parallel$ executes in parallel its two operands; $\cdot$ executes
sequentially its two operands (or synchronizes on a channel if its left-hand
operand is a channel); $(\nu a)$ creates a new channel $a$ on which two
processes can be synchronized. Here, the ``synchronization'' simply means that
a call to the channel is blocking until its other end has been called as well.

The language is simply typed as in figure~\ref{fig:lccs:typing}. Note that
binary operators split their environment between their two operands, ensuring
that each identifier is used at most once, and that no rules (in particular the
axiom rules) ``forget'' any part of the environment, ensuring that each
identifier is used at least once.

\begin{figure}[h]
	\begin{align*}
		\frac{~}{\,\vdash 0:\proc} & (\textit{Ax}_0) &
		\frac{~}{\,\vdash 1:\proc} & (\textit{Ax}_1) &
		\frac{~}{t:A \vdash t:A} & (\textit{Ax}) &
		\frac{\Gamma, a:\chan, \bar{a}:\chan \vdash P : \proc}
			{\Gamma \vdash (\nu a) P : \proc}
			& (\nu)
	\end{align*}\vspace{-1.5em}\begin{align*}
		\frac{\Gamma \vdash P : \proc \quad \Delta \vdash Q : \proc}
			{\Gamma,\Delta \vdash P \parallel Q : \proc}
			& (\parallel) &
		\frac{\Gamma \vdash P : \proc \quad \Delta \vdash Q : \proc}
			{\Gamma,\Delta \vdash P \cdot Q : \proc}
			& (\cdot_\proc) &
		\frac{\Gamma \vdash P : \proc}
			{\Gamma,a:\chan \vdash a \cdot P: \proc}
			& (\cdot_\chan)
	\end{align*} \vspace{-1.5em}
	\caption{\linccs{} typing rules}\label{fig:lccs:typing}
\end{figure}

We also equip this language with operational semantics, in the form of a
labeled transition system (LTS), as described in figure~\ref{fig:lccs:opsem},
where $a$ denotes a channel and $x$ denotes any possible label.

\begin{figure}[h]
	\begin{align*}
		\frac{~}{a \cdot P \redarrow{a} P} & &
		\frac{~}{1 \parallel P \redarrow{\tau} P} & &
		\frac{~}{1 \cdot P \redarrow{\tau} P} & &
		\frac{P \redarrow{\tau_c} Q}
			{(\nu a) P \redarrow{\tau} Q} & (c \in \set{a,\bar{a}})&
		\frac{P \redarrow{a} P'\quad Q \redarrow{\bar{a}} Q'}
			{P \parallel Q \redarrow{\tau_a} P' \parallel Q'}
	\end{align*}\begin{align*}
		\frac{P \redarrow{x} P'}
			{P \parallel Q \redarrow{x} P' \parallel Q} & &
		\frac{Q \redarrow{x} Q'}
			{P \parallel Q \redarrow{x} P \parallel Q'} & &
		\frac{P \redarrow{x} P'}
			{P \cdot Q \redarrow{x} P' \cdot Q} & &
		\frac{P \redarrow{x} P'}
			{(\nu a)P \redarrow{x} (\nu a)P'} & (a \not\in \set{x,\tau_a})
	\end{align*}
	\caption{\linccs{} operational semantics}\label{fig:lccs:opsem}
\end{figure}

We consider that a term $P$ \emph{converges} whenever $P \redarrow{\tau}^\ast
1$, and we write $P \Downarrow$.

The $\tau_a$ reduction scheme may sound a bit unusual. It is, however,
necessary. Consider the reduction of $(\nu a) (a \cdot 1 \parallel \bar{a}
\cdot 1)$: the inner term $\tau_a$-reduces to $1$, thus allowing the whole term
to reduce to $1$; but if we replaced that $\tau_a$ with a $\tau$, the whole
term would reduce to $(\nu a) 1$, which has no valid types since $a$ and
$\bar{a}$ are not consumed. Our semantics would then not satisfy subject
reduction.

Switching to an \emph{affine} \linccs{} while keeping it wrapped in a
\emph{linear} \lcalc{} was considered, but yielded way too much problems, while
switching to a fully-affine model would have modified the problem too deeply.

\subsection{Lifting to the higher order: linear \lcalc}

In order to reach the studied language, \llccs, we have to lift up \linccs{} to
a \lcalc. To do so, we add to the language the constructions of
figure~\ref{fig:llam:syntax}, which are basically the usual \lcalc{}
constructions slightly transformed to be linear (which is mostly reflected by
the typing rules). In particular, the only base types are only $\proc$ and
$\chan$.

\begin{figure}[h]
	\begin{minipage}[t]{0.55\textwidth}
		\begin{center}Terms\end{center}\vspace{-1em}
		\begin{align*}
			t,u,\ldots ::=~&x \in \mathbb{V} & \text{(variable)}\\
			\vert~&t~u & \text{(application)}\\
			\vert~&\lambda x^A \cdot t & \text{(abstraction)}\\
			\vert~&\text{\linccs}\textit{ constructions} &
		\end{align*}
	\end{minipage} \hfill \begin{minipage}[t]{0.40\textwidth}
		\begin{center}Types\end{center}\vspace{-1em}
		\begin{align*}
			A,B,\ldots ::=~&A \linarrow B & \text{(linear arrow)}\\
			\vert~&\proc~\vert~\chan & \text{(\linccs)}
		\end{align*}
	\end{minipage}
	\caption{Linear \lcalc{} terms and types}\label{fig:llam:syntax}
\end{figure}

To enforce the linearity, the only typing rules of the usual \lcalc{} that
have to be changed are the $(\textit{Ax})$ and $(\textit{App})$ presented in
figure~\ref{fig:llam:typing}. The $(\textit{Abs})$ rule is the usual one.

\begin{figure}[h]
	\begin{align*}
		\frac{~}{x : A \vdash x : A} & (\textit{Ax}) &
		\frac{\Gamma \vdash t : A \linarrow B \quad \Delta \vdash u : A}
			{\Gamma,\Delta \vdash t~u : B} & (\textit{App}) &
		\frac{\Gamma, x : A \vdash t : B}
			{\Gamma \vdash \lambda x^{A} \cdot t : A \linarrow B} & (Abs)
	\end{align*}
	\caption{Linear \lcalc{} typing rules}\label{fig:llam:typing}
\end{figure}

The linearity is here guaranteed: in the (\textit{Ax}) rule, the environment
must be $x:A$ instead of the usual $\Gamma, x:A$, ensuring that each variable
is used \emph{at least once}; while the environment split in the binary
operators' rules ensures that each variable is used \emph{at most once}
(implicitly, $\Gamma \cap \Delta = \emptyset$).

To lift the operational semantics to \llccs, we only need to add one rule:
\[
	\frac{P \longrightarrow_\beta P'}{P \redarrow{\tau} P'}
\]


%%%%%%%%%%%%%%%%%%%%%
\subsection{Examples}

\todo{Examples}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{A games model}

Our goal is now to give a games model for the above language. For that purpose,
we will use \emph{event structures}, providing an alternative formalism to
the often-used tree-like games.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{The event structures framework}

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 found along a
path from the root of the tree.

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 goes even worse with less structure: since there is $n!$ % chktex 40
permutations for $n$ elements, the tree can grow way bigger.

This problem motivated the use of \emph{event structures} as a formalism to
describe such games~\cite{rideau2011concurrent}. Informally, an event structure
is a partial order $\leq$ on \emph{events} (here, the game's moves), alongside
with a \emph{consistency} relation.

The purpose of the consistency relation is to describe non-determinism, in
which we are not interested here, since we seek a deterministic model: in all
the following constructions, I will omit the consistency set. The original
constructions including it can be found for instance
in~\cite{castellan2016concurrent,winskel1986event}.

The partial order $e_1 \leq e_2$ means that $e_1$ must have been played before
$e_2$ can be played. For instance, the Hasse diagram of the previous game would
look like
\[
	\begin{tikzpicture}
		\node (1) {A};
		\node (2) [right of=1] {B};
		\node (3) [below left of=1, below right of=2] {C};
		\path[->]
			(1) edge (3)
			(2) edge (3);
	\end{tikzpicture}
\]

%%%%%
\subsubsection{Event structures}

\begin{definition}[event structure]
	An \emph{event structure}~\cite{winskel1986event} is a poset $(E, \leq_E)$,
	where $E$ is a set of \emph{events} and $\leq_E$ is a partial order on $E$
	such that for all $e \in E$, $\downclose{e} \eqdef \set{e' \in E~\vert~e'
	\leq_E e}$ is finite.

	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.

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 $\leq_E$.

\begin{definition}[event structure with polarities]
	An \emph{event structure with polarities} (\textit{ESP}) is an event
	structure $(E, \leq_E, \rho)$, where $\rho : E \to \set{+,-}$ is a
	function associating a \emph{polarity} to each event.
\end{definition}

In order to model games, this is used to represent whether a move is to be
played by Player or Opponent. To represent polarities, we will often use colors
instead of $+$ and $-$ signs: a red-circled event will have a negative
polarity, \ie{} will be played by Opponent, while a green-circled one will have
a positive polarity.

The ESP of the previous example would then be
\[
	\begin{tikzpicture}
		\node (1) [draw=red,ellipse] {A};
		\node (3) [draw=green,ellipse,right of=1] {C};
		\node (2) [draw=red,ellipse,right of=3] {B};
		\path[->]
			(1) edge (3)
			(2) edge (3);
	\end{tikzpicture}
\]


\begin{definition}[configuration]
	A \emph{configuration} of an ESP $A$ is a finite subset $X \subseteq A$
	that is \emph{down-closed}, \ie{}
	\vspace{-0.5em}
	\[ {\forall x \in X}, {\forall e \in A}, {e \leq_A x} \implies {e \in X}.\]

	$\config(A)$ is the set of configurations of $A$.
\end{definition}

A configuration can thus be seen as a valid state of the game. $\config(A)$
plays a major role in definitions and proofs on games and strategies.

\begin{notation}
	For $x,y \in \config(A)$, $x \forkover{e} y$ states that $y =
	x \sqcup \set{e}$ (and that both are valid configurations), where $\sqcup$
	denotes the disjoint union. It is also possible to write $x \forkover{e}$,
	stating that $x \sqcup \set{e} \in \config(A)$, or $x \fork y$.
\end{notation}

%%%%%
\subsubsection{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}

For instance, one could imagine a game modeling the user interface of a coffee
machine: Player is the coffee machine, while Opponent is a user coming to buy a
drink.

\begin{example}[Process game]
	We can represent a process by the following game:
	\[
	\begin{tikzpicture}
		\node (1) at (0,0) [draw=red,ellipse] {call};
		\node (2) at (2,0) [draw=green,ellipse] {done};
		\path[->]
			(1) edge (2);
	\end{tikzpicture}
	\]
	The ``call'' event will be triggered by Opponent (the system) when the
	process is started, and Player will play ``done'' when the process has
	finished, if it ever does. The relation $\text{call} \leq \text{done}$
	means that a process cannot finish \emph{before} it is called.
\end{example}

\begin{definition}[pre-strategy]
	A \emph{pre-strategy} on the game $A$, $\sigma: A$, is an ESP such that
	\begin{enumerate}[(i)]
		\item $\sigma \subseteq A$;
		\item $\config(\sigma) \subseteq \config(A)$;
		\item $\forall s \in \sigma, \rho_A(s) = \rho_\sigma(s)$
	\end{enumerate}
\end{definition}

\begin{example}[processes, cont.]
	A possible \emph{pre-strategy} for the game consisting in two processes put
	side by side (in which the game's events are annotated with a number to
	distinguish the elements of the two processes) would be

	\[
	\begin{tikzpicture}
		\node (1) at (0,1.2) [draw=red,ellipse] {call$_0$};
		\node (2) at (0,0) [draw=green,ellipse] {done$_0$};
		\node (3) at (2,1.2) [draw=red,ellipse] {call$_1$};

		\path[->]
			(1) edge (2)
			(3) edge (2);
	\end{tikzpicture}
	\]

	This pre-strategy is valid: it is a subset of the game that does not
	include $\text{call}_1$, but it does include both $\text{call}_0$ and
	$\text{done}_0$ and inherits the game's partial order.

	This would describe two processes working in parallel. The process $0$
	waits before the process $1$ is called to terminate, and the process $1$
	never returns.
\end{example}

But as it is defined, a pre-strategy does not exactly capture what we expect of
a \emph{strategy}: it is too expressive. For instance, the relation
$\text{call}_0 \leq \text{call}_1$ on the above strategy is allowed, stating
that the operating system cannot decide to start the process $1$ before the
process $0$. It is not up to the program to decide that, this strategy is thus
unrealistic. We then have to restrict pre-strategies to \emph{strategies}:

\begin{definition}[strategy]
	A \emph{strategy} is a pre-strategy $\sigma : A$ that
	``behaves well'', \ie{} that is
	\begin{enumerate}[(i)]
		\item\label{def:receptive}
			\textit{receptive}: for all $x \in \config(A)$ \st{}
			$x \forkover{e^-}$, $e \in \sigma$;

		\item\label{def:courteous}
			\textit{courteous}: $\forall x \edgeArrow_\sigma x' \in \sigma$,
			$(\rho(x),\rho(x')) \neq (-,+) \implies
			x \edgeArrow_A x'$.
	\end{enumerate}
\end{definition}

(\ref{def:receptive}) captures the idea that we cannot prevent Opponent from
playing one of its moves. Indeed, not including an event in a strategy means
that this event \emph{will not} be played. It is unreasonable to consider that
a strategy could forbid Opponent to play a given move.

(\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, \ie{} every direct arrow in the partial order that is not inherited from
the game should be ${\ominus \edgeArrow \oplus}$. Clearly, it is unreasonable
to consider an arrow ${\ostar \edgeArrow \ominus}$, which would mean forcing
Opponent to wait for a move (either from Player or Opponent) before playing
their move; but ${\oplus \edgeArrow \oplus}$ is also unreasonable, since we're
working in a concurrent context. Intuitively, one could think that when playing
$x$ then $y$, it is undefined whether Opponent will receive $x$ then $y$ or $y$
then $x$.

%%%%%
\subsubsection{Operations on games and strategies}

\todo{Better progression in this part.}

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, 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 $\left(\set{0} \times E\right) \sqcup \left(\set{1} \times
	F\right)$ (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 $F$.

	One can then naturally expand this definition to games (by preserving
	polarities) and to strategies.
\end{definition}

In the example before, when talking of ``two processes side by side'', we
actually referred formally to the parallel composition of two processes.

\smallskip

Given two strategies on dual games $A$ and $A^\perp$, it is natural and
interesting to compute their \emph{interaction}, that is, ``what will happen if
one strategy plays against the other''.

\begin{definition}[Closed interaction]
	Given two strategies $\sigma : A$ and $\tau : A^\perp$, their
	\emph{interaction} $\sigma \wedge \tau$ is the ESP
	$\sigma \cap \tau \subseteq A$ from which causal loops have been removed.

	More precisely, $\sigma \cap \tau$ is a set adjoined with a \emph{preorder}
	${(\leq_\sigma \cup \leq_\tau)}^\ast$ (transitive closure) that may not
	respect antisymmetry, that is, may have causal loops. The event structure
	$\sigma \wedge \tau$ is then obtained by removing all the elements
	contained in such loops from $\sigma \cup \tau$.

\end{definition}
\textit{This construction is a simplified version of the analogous one
from~\cite{castellan2016concurrent} (the pullback), taking advantage of the
fact that our event structures are deterministic --- that is, without a
consistency set.}

This indeed captures what we wanted: $\sigma \wedge \tau$ contains the moves
that both $\sigma$ and $\tau$ are ready to play, including both orders, except
for the events that can never be played because of a ``deadlock'' (\ie{} a
causal loop).

\smallskip

We might now try to generalize that to an \emph{open} case, where both
strategies don't play on the same games, but only have a common part. Our
objective here is to \emph{compose} strategies: indeed, a strategy on $A^\perp
\parallel B$ can be seen as a strategy \emph{from $A$ to $B$}, playing as
Opponent on a board $A$ and as Player on a board $B$. This somehow looks like a
function, that could be composed with another strategy on $B^\perp \parallel
C$.

\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 an event structure defined as $(\sigma \parallel C) \wedge
	(A^\perp \parallel \tau)$, where $A^\perp$ and $C$ are seen as strategies.
\end{definition}

The idea is to put in correspondence the ``middle'' states (those of $B$) while
adding ``neutral'' states for $A$ and $C$.

$\tau \strInteract \sigma$ is an \emph{event structure} (\ie, without
polarities): indeed, the two strategies disagree on the polarities of the
middle part. Alternatively, it can be seen as an ESP with a polarity function
over $\set{+,-,?}$.

From this point, the notion of composition we sought is only a matter of
``hiding'' the middle part:

\begin{definition}[Strategies composition]
	Given two strategies $\sigma : A^\perp \parallel B$ and $\tau : B^\perp
	\parallel C$, their \emph{composition} $\tau \strComp \sigma$ is the ESP
	$(\tau \strInteract \sigma) \cap (A^\perp \parallel C)$, on which the
	partial order is the restriction of $\leq_{\tau \strInteract \sigma}$ and
	the polarities those of $\sigma$ and $\tau$.
\end{definition}

It is then useful to consider an identity strategy \wrt{} the composition
operator. This identity is called the \emph{copycat} strategy:

\begin{definition}[Copycat]
	The \emph{copycat strategy} of a game $A$, $\cc_A$, is the strategy on the
	game $A^\perp \parallel A$ whose events are $A^\perp \parallel A$ wholly,
	on which the order is the transitive closure of $\leq_{A^\perp \parallel A}
	\cup \set{ (1-i, x) \leq (i, x) \vert x \in A~\&~\rho((i,x)) = \oplus}$.
\end{definition}

The copycat strategy of a game is indeed an identity for the composition of
\emph{strategies}. In fact, it even holds that for a \emph{pre-}strategy
$\sigma : A$, $\sigma$ is a strategy $\iff$ $\cc_A \strComp \sigma = \sigma$.

\begin{example}[copycat]
	If we consider the following game $A$
	\[
	\begin{tikzpicture}
		\node (1) [draw=red,ellipse] {A};
		\node (3) [draw=green,ellipse,right of=1] {C};
		\node (2) [draw=red,ellipse,right of=3] {B};
		\path[->]
			(3) edge (1);
	\end{tikzpicture}
	\]
	its copycat strategy $\cc_A$ is
	\[
	\begin{tikzpicture}
		\node (01) {($A^\perp$)};
		\node (02) [below of=01] {($A$)};
		\node (11) [draw=green,ellipse,right of=01] {A};
		\node (31) [draw=red,ellipse,right of=11] {C};
		\node (21) [draw=green,ellipse,right of=31] {B};

		\node (12) [draw=red,ellipse,right of=02] {A};
		\node (32) [draw=green,ellipse,right of=12] {C};
		\node (22) [draw=red,ellipse,right of=32] {B};
		\path[->]
			(12) edge (11)
			(22) edge (21)
			(31) edge (32)
			(31) edge (11)
			(32) edge (12);
	\end{tikzpicture}
	\]
\end{example}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Interpretation of \llccs}

We can now equip \llccs{} with denotational semantics, interpreting the
language as strategies as defined in figure~\ref{fig:llccs:interp}.

\begin{figure}[h]
	\begin{minipage}[t]{0.45\textwidth} \begin{align*}
		\seman{x^A} &\eqdef \cc_{\seman{A}} \\
		\seman{t^{A \linarrow B}~u^{A}} &\eqdef
			\cc_{A \linarrow B} \strComp \left( \seman{t} \parallel \seman{u}
			\right) \\
		\seman{\lambda x^A \cdot t} &\eqdef \seman{t}
	\end{align*} \end{minipage} \hfill
	\begin{minipage}[t]{0.45\textwidth} \begin{align*}
		\seman{\alpha} &\eqdef \ominus \\
		\seman{A \linarrow B} &\eqdef \seman{A}^\perp \parallel \seman{B} \\
	\end{align*}\end{minipage}
	\begin{align*}
		\seman{P \parallel Q} &\eqdef \left(
			\begin{tikzpicture}[baseline, scale=0.8]
				\node (4) at (0,0.65) [draw=green,ellipse] {call $P$};
				\node (5) at (0,-0.65) [draw=red,ellipse] {done $P$};
				\node (2) at (2.5,0.65) [draw=green,ellipse] {call $Q$};
				\node (3) at (2.5,-0.65) [draw=red,ellipse] {done $Q$};
				\node (0) at (5,0.65) [draw=red,ellipse] {call};
				\node (1) at (5,-0.65) [draw=green,ellipse] {done};
				\path[->]
					(0) edge (1)
						edge [bend right] (2)
						edge [bend right] (4)
					(2) edge (3)
					(4) edge (5)
					(3) edge [bend right] (1)
					(5) edge [bend right] (1);
			\end{tikzpicture}
			\right) \strComp \left(\seman{P} \parallel \seman{Q}\right) &
		\seman{\proc} = \seman{\chan} &\eqdef \begin{tikzpicture}[baseline]
			\node (1) at (0,0.5) [draw=red,ellipse] {call};
			\node (2) at (0,-0.5) [draw=green,ellipse] {done};
			\draw [->] (1) -- (2);
			\end{tikzpicture}
		\\ %%%%%%%%%%%%%%%%%%%%%%%%%
		\seman{P \cdot Q} &\eqdef \left(
			\begin{tikzpicture}[baseline,scale=0.8]
				\node (4) at (0,0.65) [draw=green,ellipse] {call $P$};
				\node (5) at (0,-0.65) [draw=red,ellipse] {done $P$};
				\node (2) at (2.5,0.65) [draw=green,ellipse] {call $Q$};
				\node (3) at (2.5,-0.65) [draw=red,ellipse] {done $Q$};
				\node (0) at (5,0.65) [draw=red,ellipse] {call};
				\node (1) at (5,-0.65) [draw=green,ellipse] {done};
				\path[->]
					(0) edge (1)
						edge [bend right] (4)
					(2) edge (3)
					(4) edge (5)
					(3) edge [bend right] (1)
					(5) edge (2);
			\end{tikzpicture}
			\right) \strComp \left(\seman{P} \parallel \seman{Q}\right) &
		\seman{1} &\eqdef \begin{tikzpicture}[baseline]
			\node (1) at (0,0.5) [draw=red,ellipse] {call};
			\node (2) at (0,-0.5) [draw=green,ellipse] {done};
			\draw [->] (1) -- (2);
			\end{tikzpicture}
		\\ %%%%%%%%%%%%%%%%%%%%%%%%%
		\seman{(a : \chan) \cdot P} &\eqdef \left(
			\begin{tikzpicture}[baseline,scale=0.8]
				\node (4) at (0,0.65) [draw=green,ellipse] {call $P$};
				\node (5) at (0,-0.65) [draw=red,ellipse] {done $P$};
				\node (2) at (2.5,0.65) [draw=green,ellipse] {call $a$};
				\node (3) at (2.5,-0.65) [draw=red,ellipse] {done $a$};
				\node (0) at (5,0.65) [draw=red,ellipse] {call};
				\node (1) at (5,-0.65) [draw=green,ellipse] {done};
				\path[->]
					(0) edge (1)
						edge [bend right] (2)
					(2) edge (3)
					(4) edge (5)
					(3) edge (4)
					(5) edge [bend right] (1);
			\end{tikzpicture}
			\right) \strComp \left(\seman{P} \parallel \seman{a}\right) &
		\seman{0} &\eqdef \begin{tikzpicture}[baseline]
			\node (1) at (0,0.2) [draw=red,ellipse] {call};
			\end{tikzpicture}
		\\ %%%%%%%%%%%%%%%%%%%%%%%%%
		\seman{(\nu a) P} &\eqdef \left(
			\begin{tikzpicture}[baseline,scale=0.8]
				\node (6) at (0,0.65) [draw=green,ellipse] {call $a$};
				\node (7) at (0,-0.65) [draw=red,ellipse] {done $a$};
				\node (4) at (2.5,0.65) [draw=green,ellipse] {call $\bar{a}$};
				\node (5) at (2.5,-0.65) [draw=red,ellipse] {done $\bar{a}$};
				\node (2) at (5,0.65) [draw=green,ellipse] {call $P$};
				\node (3) at (5,-0.65) [draw=red,ellipse] {done $P$};
				\node (0) at (7.5,0.65) [draw=red,ellipse] {call};
				\node (1) at (7.5,-0.65) [draw=green,ellipse] {done};
				\path[->]
					(0) edge (1)
						edge [bend right] (2)
					(2) edge (3)
					(3) edge [bend right] (1)
					(4) edge (5)
						edge (7)
					(6) edge (7)
						edge (5);
			\end{tikzpicture}
			\right) \strComp \seman{P} &
	\end{align*}
	\caption{\llccs{} interpretation as strategies}\label{fig:llccs:interp}
\end{figure}


%%%%%%%%%%%%%%%%%%%%%
\subsection{Adequacy}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Implementation of deterministic concurrent games}

\hfill\href{https://github.com/tobast/cam-strategies/}
	{\includegraphics[height=2em]{github32.png}~\raisebox{0.5em}{Github
	repository}}

\vspace{1em}

The first part of my internship mostly consisted --- apart from understanding
the bibliography and the underlying concepts --- in the implementation of
operations on \emph{deterministic} concurrent games, that is, concurrent games
as event structures without conflicts. The work had to be done from scratch, as
no one implemented this before.

This implementation aims to provide
\begin{enumerate}[(i)]
	\item a --- more or less --- convenient way to input games/strategies;
	\item basic operations over those games and strategies: parallel
		composition, pullback, interaction, composition, copycat, \ldots;
	\item a clean display as a Dot graph.
\end{enumerate}

\subsection{Structures}

The implementation aims to stay as close as possible to the mathematical model,
while still providing quite efficient operations.

As we do not handle non-determinism, an event structure can be easily
represented as a DAG in memory. The actual representation that was chosen is a
set of nodes, each containing (as well as a few other information) a list of
incoming and outgoing edges.

A \emph{game} is, in the literature, a simple ESP\@. However, to provide
interaction and composition operations, we have to somehow keep track of the
parallel compositions that were used to reach this game: if the user wants to
compose strategies on $A \strParallel B$ and $B \strParallel C$, we have to
remember that those games were indeed parallel compositions of the right games,
and not just a set where the events from, \eg, $A$ and $B$ are mixed. \\
This information is kept in a tree, whose leaves are the base games that were
put in parallel, and whose nodes represent a parallel composition operation.

Finally, a \emph{strategy} consists in a game and an ESP (the strategy itself),
plus a map from the nodes of the strategy to the nodes of the game.  This
structure is really close to the mathematical definition of a strategy, and yet
is only a lesser loss in efficiency.

\subsection{Operations}

The usual operations on games and strategies, namely \emph{parallel
composition}, \emph{pullback}, \emph{interaction} and \emph{composition} are
implemented in a very modular way: each operation is implemented in a functor,
whose arguments are the other operations it makes use of, each coming with its
signature. Thus, one can simply \lstocaml{open Operations.Canonical} to use the
canonical implementation, or define its own implementation, build it into an
\lstocaml{Operations} module (which has only a few lines of code) and then
open it. This is totally transparent to the user, who can use the same infix
operators.

\subsubsection{Parallel composition}

While the usual construction (\cite{castellan2016concurrent}) involves defining
the events of $A \strParallel B$ as ${\set{0} \times A} \cup {\set{1}
\times B}$, the parallel composition of two strategies is here simply
represented as the union of both event structures, while altering the
composition tree.

\subsubsection{Pullback}

Given two strategies on the same game, the pullback operation attempts to
extract a ``common part'' of those two strategies. Intuitively, the pullback of
two strategies is ``what happens'' if those two strategies play together.

The approach that was implemented (and that is used as
\lstocaml{Pullback.Canonical}) is a \emph{bottom-up} approach: iteratively, the
algorithm looks for an event that has no dependencies in both strategies, adds
it and removes the satisfied dependencies.\\
One could also imagine a \emph{top-bottom} approach, where the algorithm starts
working on the merged events of both strategies, then looks for causal loops
and removes every event involved.

\subsubsection{Interaction}

Once the previous operations are implemented, \emph{interaction} is easily
defined as in the literature (\cite{castellan2016concurrent}) and nearly is a
one-liner.

\subsubsection{Composition}

Composition is also quite easy to implement, given the previous operations. The
only difficulty is that hiding the central part means computing the new
$\edgeArrow$ relation (that is the transitive reduction of $\leq$), which means
computing the transitive closure of the interaction, hiding the central part
and then computing the transitive reduction of the DAG\@.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linear lambda-calculus}

Concurrent games can be used as a model of lambda-calculus.  To keep the
strategies finite and to avoid non-determinism, and to have a somehow easier
approach, one can use concurrent games as a model of \emph{linear}
lambda-calculus, that is, a variant of the simply-typed lambda-calculus where
each variable in the environment can and must be used exactly once.

\subsection{Definition}

The linear lambda calculus we use has the same syntax as the usual simply-typed
lambda calculus with type annotations:

\medskip

Only the following typing rules differ from the usual rules and are worth
noting:


Note that in~(\ref{typ:llam:ax}), the left part is $x : A$ and not (as usual)
$\Gamma, x:A$. This ensures that each defined variable present in the
environment will be used. The implicit condition $\Gamma \cap \Delta =
\emptyset$ in~(\ref{typ:llam:app}) ensures that each defined variable will be
used at most once.

The terms can then be interpreted as strategies through the $\seman{\cdot}$
operator defined as in figure~\ref{fig:llam:interp}. The $\ominus$ stands for a
game whose only event is negative. The interpretation operator maps a type to a
game and a term to a strategy playing on the game associated to its type, put
in parallel with its environment's dual. For instance, if $x:A \vdash t:B$, the
strategy $\seman{t}$ will play on $\seman{A}^\perp \parallel \seman{B}$.

This explains the definition of $\seman{\lambda x^A \cdot t}$: $\seman{t}$
plays on $\seman{A}^\perp \parallel \seman{B}$, same as $\seman{\lambda x^A
\cdot t}$.

\subsection{Implementation}

The implementation, which was supposed to be fairly simple, turned out to be
not as straightforward as expected due to technical details: while, in the
theory, the parallel composition is obviously associative and commutative (up
to isomorphism), and thus used as such when dealing with environment and typing
rules, things get a bit harder in practice when one is supposed to provide the
exact strategy.

For instance, the above rule~(\ref{typ:llam:app}) states that the resulting
environment is $\Gamma,\Delta$, while doing so in the actual implementation
(that is, simply considering $\seman{\Gamma} \strParallel \seman{\Delta}$)
turns out to be a nightmare: it is better to keep the environment ordered by
the variables introduction order, thus intertwining $\Gamma$ and $\Delta$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Linear \lccs}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\bibliography{biblio}
\bibliographystyle{ieeetr}

\end{document}

