L3-internship-report/report.tex

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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

\begin{abstract}
	During my internship, I have worked on a \emph{deterministic}
	game semantics model for a minimalistic \emph{concurrent} language,
	described using the \emph{games as event structures} formalism. I have
	proved the \emph{adequacy} of this model with the operational semantics of
	the language modelled; and I have implemented this model, allowing one to
	input an expression of the language and getting its representation
	as Dot graph. This implementation also supports basic operations on event
	structures, which could be useful to other people working in this domain.
\end{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}

\begin{itemize}
	\item Simple channel usage: $T_1 \eqdef \newch{a} (a \cdot 1 \parallel
		\bar{a} \cdot 1)$. This term converges: $\left(a \cdot 1 \parallel
		\bar{a} \cdot 1	\right) \redarrow{\tau_a} 1 \parallel 1$, thus $T_1
		\redarrow{\tau} \left(1 \parallel 1\right) \redarrow{\tau} 1$.

	\item Deadlock: $T_2 \eqdef \newch{a} (a \cdot \bar{a} \cdot 1)$. This term
		does not reduce at all: no reduction is possible under the $\nu$.

	\item \todo{More examples?}
\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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}\label{sssec:es_operations}

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 $\sigma \cap \tau$.
\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 $\leq_{A^\perp \parallel A}
	\cup \set{ (1-i, e) \leq (i, e) \vert e \in A~\&~\rho((i,e)) = \oplus}$.
\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}\label{ssec:llccs_interp}

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{align*}
		\seman{x^A} &\eqdef \cc_{\seman{A}} &
		\seman{A \linarrow B} &\eqdef \seman{A}^\perp \parallel \seman{B} \\
		\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} \\
		\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}

A lot of these interpretations is what was expected: $\proc$ and $\chan$ have
the same interpretation as presented before, $1$ is interpreted as the game
playing \textit{done} while $0$ does not, the arrow is the type introduced for
functions.

A variable is represented by a \emph{copycat} of its type, to keep a version of
the variable in the environment, reflecting the judgement $x:A \vdash x:A$.
Same goes for the abstraction: the typing rule states that given a term
verifying $\Gamma, x:A \vdash t:B$, we should obtain a term verifying $\Gamma
\vdash \lambda x^A \cdot t : A \linarrow B$. The only thing we have to do is to
``move'' $x$ from the environment to the term, which is transcribed by the
associativity of $\parallel$. In the application, one can notice that the
copycat wires up perfectly $\seman{t} \parallel \seman{u}$: the version of $x$
from the environment of $t$ is connected to $u$, and the output of $t$ is
connected to the output of the term.

In the representation above, the drawn strategies are organized in columns: for
instance, the strategy involved in $\seman{P \parallel Q}$ has type $P^\perp
\parallel P^\perp \parallel P$. Each column of events stands for one of those
games, in this order.


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

We will now prove the major result of this study, the \emph{adequacy} of the
game semantics.

\begin{theorem}[Adequacy]
	The previous interpretation is \emph{adequate} with the operational
	semantics, that is
	\[ \forall P \textit{ st. } \vdash P : \proc,~
	\left(P \Downarrow\right) \iff \left(\seman{P} = \seman{1}\right) \]
\end{theorem}

In order to prove the theorem, a few intermediary definitions and results are
required.

\begin{definition}[Evaluation in a context]
	For $l$ a list of channels and $P$ a term, the evaluation of $P$ in the
	context $l$, $\seman{P}_l$, is defined by induction by $\seman{P}_{[]}
	\eqdef \seman{P}$ and $\seman{P}_{h::t} \eqdef \seman{(\nu h)P}_t$.
\end{definition}

\begin{definition}[valid contexts]
	The \emph{valid contexts} for a reduction $P \redarrow{x} Q$, $\validctx_{P
	\redarrow{x} Q}$, is the set of (ordered) lists of channels $l$ such that
	\begin{enumerate}[(i)]
		\item $\forall a:\chan \in \freevars(P)~\st~\bar{a} \in \freevars(P),~
			a \in l \text{ or } \bar{a} \in l$;
		\item if $x = a : \chan$, $a \not\in l$;
		\item $\forall a : \chan,~a \in l \implies \bar{a} \not\in l$.
	\end{enumerate}

\end{definition}

\begin{lemma}
	For all $a: \chan$, $P, Q, R : \proc$, the following properties hold:
	\begin{enumerate}[(i)]
		\item $\newch{a} (P \parallel Q) = \left(\newch{a} P\right) \parallel
			Q$ when $a,\bar{a} \not\in \freevars{Q}$;
		\item $\newch{a} (P \parallel Q) = P \parallel \left(\newch{a}
			Q\right)$ when $a,\bar{a} \not\in \freevars{P}$;
		\item $\newch{a} \left(P\cdot Q\right) = \left(\newch{a} P\right) \cdot
			Q$ when $a,\bar{a} \not\in \freevars{Q}$;
		\item $\seman{\parallel}_l$ is associative, that is, for all $l$,
			$\seman{(A \parallel B) \parallel C}_l = \seman{A \parallel (B
			\parallel C)}_l$.
	\end{enumerate}
\end{lemma}

The theorem is mostly a consequence of the following lemma:

\begin{lemma}
	$\forall P \redarrow{x} Q, \forall l \in \validctx_{P \redarrow{x} Q}$,
	\begin{enumerate}[(i)]
		\item if $x = \tau$, then $\seman{P}_l = \seman{Q}_l$;
		\item if $x = a: \chan$, then $\seman{P}_l = \seman{a \cdot Q}_l$;
		\item if $x = \tau_a$ ($a : \chan$), then $\seman{P}_{a::l} =
			\seman{Q}_l$.
	\end{enumerate}
\end{lemma}

\begin{proof}
	We prove this by induction over the rules of the operational semantics of
	\llccs. Most of the cases are straightforward, thus, we will only sketch
	the proof for a few of those cases.

	\begin{itemize}
		\item The basic rules (for $\parallel$, $\cdot$, \ldots) are working
			thanks to the previous lemma.
		\item $\dfrac{P \redarrow{a} P'\quad Q \redarrow{\bar{a}} Q'}
				{P \parallel Q \redarrow{\tau_a} P' \parallel Q'}$:
			for all $l$, by (ii), $\seman{P}_l = \seman{a \cdot P'}_l$ and
			$\seman{Q} = \seman{\bar{a} \cdot Q'}$, thus $\seman{(\nu a)(P
			\parallel Q)}_l = \seman{\newch{a}(a \cdot P' \parallel \bar{a}
			\cdot Q')}$ (inline replacement of terms, permitted by the previous
			lemma), thus $\seman{\newch{a} (P \parallel Q)}_l = \seman{P'
			\parallel Q'}_l$.
		\item $\dfrac{P \redarrow{\tau_c} Q}{\newch{a} P \redarrow{\tau} Q} (c
			\in \set{a,\bar{a}})$ clearly works thanks to (iii).
	\end{itemize}
\end{proof}

\begin{proof}[Proof: theorem]
	\emph{Forwards implication}: $\left(P \Downarrow\right) \implies
	\left(\seman{P} = \seman{1}\right)$.
	Proof by induction over the derivation of $P \redarrow{\tau}^\ast 1$, by
	iterating the previous lemma.

	\emph{Backwards implication}: $\left(\seman{P} = \seman{1}\right) \implies
	\left(P \Downarrow\right)$.
	We prove its contrapositive judgement, $P \neq 1\,\&\,\seman{P} =
	\seman{1} \implies \exists Q\,:\,P \redarrow{\tau} Q$ by induction over the
	syntax of \linccs{} (\wlogen, we can assume that $P
	\not\longrightarrow_\beta$, thus the terms are in \linccs). \todo{?}
\end{proof}

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

\hfill\href{https://tobast.fr/l3/demo.html}
	{\includegraphics[height=2em]{tryme32.png}~\raisebox{0.5em}{Try online}}
\qquad \href{https://github.com/tobast/cam-strategies/}
	{\includegraphics[height=2em]{github32.png}~\raisebox{0.5em}{Github
	repository}}

\vspace{1em}

One of the objectives of my internship was also to implement the operations on
games and strategies described in §\ref{sssec:es_operations}, and to use them
to provide a convenient Dot representation of the operational semantics of
\llccs{} described in §\ref{ssec:llccs_interp}.

\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.

\subsection{Modelling \llccs}

The modelling of \llccs{} required to implement a lexer/parser for the
language and a function transforming \llccs{} terms into strategies, as well as
a rendering backend, displaying a strategy as a Dot graph. This could then just
be plugged into an HTML/Javascript frontend using \texttt{js\_of\_ocaml}; this
frontend is linked above in the document.

The major difficulty came from the necessity to massively reorder the sub-terms
of the strategies on the go: indeed, in order to know how to compose two
strategies $\sigma$ and $\tau$, the implementation keeps track of the parallel
compositions that were taken to get both strategies. For instance, if $\sigma$
was obtained by putting in parallel strategies so that the game is $(A
\parallel B^\perp) \parallel C^\perp$, and $\tau$ was obtained the same way,
reaching a game $(B \parallel C) \parallel D$, the implementation would refuse
to compute $\tau \strComp \sigma$, because it would try to match games
$C^\perp$ and $(B \parallel C)$.

The theoretical construction extensively uses the associativity (up to
isomorphism) of $\parallel$, but in the code, each use of the associativity
must be explicit, leading to a large amount of code.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{Conclusion}

During this internship, I have established deterministic game semantics for a
simple concurrent language. Although the language is fairly simple, it should
not be too hard to lift it to a language closer to real-world programming
languages, through the inclusion of the imperative primitives found in the
literature.

I also explored the possibility to reach a full-abstraction result, but
abandoned this path by lack of time. Indeed, this would have required either to
modify \llccs{} or to restrict the authorized strategies, because of the
following legal strategy, which cannot be expressed as the semantics of a term
in \llccs{}:
\[ \begin{tikzpicture}
	\node (0P) at (2,2) {$\proc$};
	\node (0C) at (0,2) {$\chan$};
	\node (1) at (0,1) [draw=red, ellipse] {call};
	\node (2) at (0,0) [draw=green, ellipse] {done};
	\node (3) at (2,1) [draw=red, ellipse] {call};
	\node (4) at (2,0) [draw=green, ellipse] {done};
	\path[->]
		(1) edge (2)
		(3) edge (4)
			edge (1);
\end{tikzpicture}
\]

This strategy behaves like a ``forget'' strategy: its effect is to call one end
of a channel, and then to resume the execution of the term without waiting for
the other end to be called. If we integrate this operator to the language as
$(f\,a)$ for any channel $a$, this construction can discriminate terms that
would not have been discriminated before. For instance $\lambda x^{\chan
\linarrow \proc} \cdot \newch{a} ((x a) \cdot \bar{a} \cdot 1)$ and $\lambda
x^{\chan \linarrow \proc} \cdot \newch{a} ((x a) \cdot \bar{a} \cdot 0)$ can
be discriminated by the context $\calC[X] = X (\lambda a^\chan \cdot (f a))$.

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

\bibliography{biblio}
\bibliographystyle{alpha}

\end{document}