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11
thesis.bib
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@@ -3739,3 +3739,14 @@
pages = {1--494}, pages = {1--494},
year = 1894 year = 1894
} }
# Game semantics
@inbook{Hyland97,
author = {Martin Hyland},
title = {Game Semantics},
booktitle = {Semantics and Logics of Computation},
publisher = {Cambridge University Press},
editor = {Andrew M. Pitts and Peter Dybjer},
pages = {131--184},
year = 1997
}

124
thesis.tex
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@@ -4719,49 +4719,75 @@ continuation without the handler.
Chapter~\ref{ch:unary-handlers} contains further examples of deep and Chapter~\ref{ch:unary-handlers} contains further examples of deep and
shallow handlers in action. shallow handlers in action.
% %
\dhil{Consider whether to present the below encodings\dots} % \dhil{Consider whether to present the below encodings\dots}
% % %
Deep handlers can be used to simulate shift0 and % Deep handlers can be used to simulate shift0 and
reset0~\cite{KammarLO13}. % reset0~\cite{KammarLO13}.
% % %
\begin{equations} % \begin{equations}
\sembr{\shiftz~k.M} &\defas& \Do\;\dec{Shift0}~(\lambda k.M)\\ % \sembr{\shiftz~k.M} &\defas& \Do\;\dec{Shift0}~(\lambda k.M)\\
\sembr{\resetz{M}} &\defas& % \sembr{\resetz{M}} &\defas&
\ba[t]{@{~}l}\Handle\;m\,\Unit\;\With\\ % \ba[t]{@{~}l}\Handle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l} % ~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& x\\ % \Return\;x &\mapsto& x\\
\OpCase{\dec{Shift0}}{f}{k} &\mapsto& f\,k % \OpCase{\dec{Shift0}}{f}{k} &\mapsto& f\,k
\ea % \ea
\ea % \ea
\end{equations} % \end{equations}
% % %
Shallow handlers can be used to simulate control0 and % Shallow handlers can be used to simulate control0 and
prompt0~\cite{KammarLO13}. % prompt0~\cite{KammarLO13}.
% % %
\begin{equations} % \begin{equations}
\sembr{\Controlz~k.M} &\defas& \Do\;\dec{Control0}~(\lambda k.M)\\ % \sembr{\Controlz~k.M} &\defas& \Do\;\dec{Control0}~(\lambda k.M)\\
\sembr{\Promptz~M} &\defas& % \sembr{\Promptz~M} &\defas&
\bl % \bl
prompt0\,(\lambda\Unit.M)\\ % prompt0\,(\lambda\Unit.M)\\
\textbf{where}\; % \textbf{where}\;
\bl % \bl
prompt0~m \defas % prompt0~m \defas
\ba[t]{@{~}l}\ShallowHandle\;m\,\Unit\;\With\\ % \ba[t]{@{~}l}\ShallowHandle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l} % ~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& x\\ % \Return\;x &\mapsto& x\\
\OpCase{\dec{Control0}}{f}{k} &\mapsto& prompt0\,(\lambda\Unit.f\,k) % \OpCase{\dec{Control0}}{f}{k} &\mapsto& prompt0\,(\lambda\Unit.f\,k)
\ea % \ea
\ea % \ea
\el % \el
\el % \el
\end{equations} % \end{equations}
% %
%Recursive types are required to type the image of this translation %Recursive types are required to type the image of this translation
\paragraph{\citeauthor{Longley09}'s catch-with-continue} \paragraph{\citeauthor{Longley09}'s catch-with-continue}
% %
\dhil{TODO} The control operator \emph{catch-with-continue} (abbreviated
catchcont) is a delimited extension of the $\Catch$ operator. It was
designed by \citet{Longley09} in 2008~\cite{LongleyW08}. Its origin is
in game semantics, in which program evaluation is viewed as an
interactive dialogue with the ambient environment~\cite{Hyland97} ---
this view aligns neatly with the view of effect handler oriented
programming. Curiously, \citeauthor{Longley09} and
\citeauthor{PlotkinP09} developed catchcont and effect handlers,
respectively, during the same time, in the same country, in the same
city, in the same building. Despite all of this the two operators have
not yet been related formally.
The catchcont operator appears as a computation form in our calculus.
%
\begin{syntax}
&M,N \in \CompCat &::=& \cdots \mid \Catchcont~f.M
\end{syntax}
%
Unlike other delimited control operators, $\Catchcont$ does not
introduce separate explicit syntactic constructs for the control
delimiter and control reifier. Instead it leverages the higher-order
facilities of $\lambda$-calculus: the syntactic construct $\Catchcont$
play the role of control delimiter and the (generative) function name
$f$ is the name of the control reifier. \citet{LongleyW08} describe
$f$ as a `dummy variable'.
The typing rule for $\Catchcont$ is as follows.
% %
\begin{mathpar} \begin{mathpar}
\inferrule* \inferrule*
@@ -4769,6 +4795,20 @@ prompt0~\cite{KammarLO13}.
{\typ{\Gamma}{\Catchcont\;f.M : C \times ((A \to B) \to D) + (A \times (B \to (A \to B) \to C \times D))}} {\typ{\Gamma}{\Catchcont\;f.M : C \times ((A \to B) \to D) + (A \times (B \to (A \to B) \to C \times D))}}
\end{mathpar} \end{mathpar}
% %
The computation handled by $\Catchcont$ must return a pair, where the
first component must be a ground value. This restriction ensures that
the value is not a $\lambda$-abstraction, which means that the value
cannot contain any further occurrence of the control reifier $f$. The
second component is unrestricted, and thus, it may contain further
occurrences of $f$. If $M$ fully reduces then $\Catchcont$ returns a
pair consisting of a ground value (i.e. an answer from $M$) and a
continuation function which allow $M$ to yield further
`answers'. Alternatively, if $M$ invokes the control reifier $f$, then
$\Catchcont$ returns a pair consisting of the argument supplied to $f$
and the current continuation of the invocation of $f$.
The operational rules for $\Catchcont$ are as follows.
%
\begin{reductions} \begin{reductions}
\slab{Value} & \slab{Value} &
\Catchcont \; f . \Record{V;W} &\reducesto& \Inl\; \Record{V;\lambda\,f. W}\\ \Catchcont \; f . \Record{V;W} &\reducesto& \Inl\; \Record{V;\lambda\,f. W}\\
@@ -4776,7 +4816,15 @@ prompt0~\cite{KammarLO13}.
\Catchcont \; f .\EC[\,f\,V] &\reducesto& \Inr\; \Record{V; \lambda x. \lambda f. \Continue~\cont_{\EC}~x}\\ \Catchcont \; f .\EC[\,f\,V] &\reducesto& \Inr\; \Record{V; \lambda x. \lambda f. \Continue~\cont_{\EC}~x}\\
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V] \slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]
\end{reductions} \end{reductions}
%
The \slab{Value} makes sure to bind any lingering instances of $f$ in
$W$ before escaping the delimiter. The \slab{Capture} rule reifies and
aborts the current evaluation up to, but no including, the delimiter,
which gets uninstalled. The reified evaluation context gets stored in
the second component of the returned pair. Importantly, the second
$\lambda$-abstraction makes sure to bind any instances of $f$ in the
captured evaluation context once it has been reinstated by the
\slab{Resume} rule.
% \subsection{Second-class control operators} % \subsection{Second-class control operators}
% Coroutines, async/await, generators/iterators, amb. % Coroutines, async/await, generators/iterators, amb.