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Composable continuations. Controlling continuations intro.

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Daniel Hillerström
2020-12-05 15:23:47 +00:00
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@@ -827,7 +827,7 @@ itself have no effect.
% %
The innermost application erases the outermost application term, The innermost application erases the outermost application term,
consequently only the first application of $\cont$ occurs during consequently only the first application of $\cont$ occurs during
runtime. runtime. It is as if the first application occurred in tail position.
The continuations introduced by the early control operators were all The continuations introduced by the early control operators were all
abortive, since they were motivated by modelling unrestricted jumps abortive, since they were motivated by modelling unrestricted jumps
@@ -837,11 +837,33 @@ An abortive continuation is also known as an `escape' continuation in
the literature. the literature.
\paragraph{Composable continuation} A composable continuation splices \paragraph{Composable continuation} A composable continuation splices
its captured context with the its invocation context. its captured context with the its invocation context, i.e.
% %
\[ \[
\EC[\Continue~\cont_{\EC'}~V] \reducesto \EC[\EC'[V]] \Continue~\cont_{\EC}~V \reducesto \EC[V]
\] \]
%
The application of a composable continuation can be understood
locally, because it has no effect on its invocation context. A
composable continuation behaves like a function in the sense that it
returns to its caller, and thus composition is well-defined, e.g.
%
\[
\Continue~\cont_{\EC}~(\Continue~\cont_{\EC}~V) \reducesto \Continue~\cont_{\EC}~\EC[V]
\]
%
The innermost application composes the captured context with the
outermost application. Thus, the outermost application occurs when
$\EC[V]$ has been reduced to a value.
In the literature, virtually every delimited control operator provides
composable continuations. However, the notion of composable
continuation is not intimately connected to delimited control. It is
perfect possible to conceive of a undelimited composable continuation,
just as a delimited abortive continuation is conceivable.
A composable continuation is also known as a `functional' continuation
in the literature.
% % \citeauthor{Reynolds93} has written a historical account of the % % \citeauthor{Reynolds93} has written a historical account of the
% % various early discoveries of continuations~\cite{Reynolds93}. % % various early discoveries of continuations~\cite{Reynolds93}.
@@ -950,8 +972,17 @@ its captured context with the its invocation context.
\section{Controlling continuations} \section{Controlling continuations}
\label{sec:controlling-continuations} \label{sec:controlling-continuations}
As suggested in the previous section, the design space for
continuation is rich. This richness is to an extent reflected by the
considerable amount of control operators that appear in the literature
and in practice.
%
Table~\ref{tbl:classify-ctrl} provides a classification of a Table~\ref{tbl:classify-ctrl} provides a classification of a
non-exhaustive list of first-class control operators. non-exhaustive list of first-class control operators.
It is worth remarking that a \emph{first-class} control operator is
typically not itself a first-class citizen, rather, it means that the
reified continuation is a first-class citizen.
% %
\begin{table} \begin{table}
\centering \centering
@@ -973,7 +1004,7 @@ non-exhaustive list of first-class control operators.
\hline \hline
control/prompt & Delimited & Composable & \citet{Felleisen88}\\ control/prompt & Delimited & Composable & \citet{Felleisen88}\\
\hline \hline
effect handlers & Delimited & Composable & \citet{PlotkinP09,PlotkinP13} \\ effect handlers & Delimited & Composable & \citet{PlotkinP13} \\
\hline \hline
escape & Undelimited & Abortive & \citet{Reynolds98a}\\ escape & Undelimited & Abortive & \citet{Reynolds98a}\\
\hline \hline
@@ -985,19 +1016,22 @@ non-exhaustive list of first-class control operators.
\hline \hline
shift/reset & Delimited & Composable & \citet{DanvyF90}\\ shift/reset & Delimited & Composable & \citet{DanvyF90}\\
\hline \hline
spawn & Delimited & Composable & \citet{HiebDA94}\\ spawn & Delimited & Composable & \citet{HiebD90}\\
\hline \hline
splitter & Delimited & Abortive, composable & \citet{QueinnecS91}\\ splitter & Delimited & Abortive, composable & \citet{QueinnecS91}\\
\hline \hline
\end{tabular} \end{tabular}
\caption{Classification of first-class sequential control operators.}\label{tbl:classify-ctrl} \caption{Classification of first-class sequential control operators.}\label{tbl:classify-ctrl}
\dhil{TODO: Possibly split into two tables: undelimited and delimited. Change the table to display the behaviour of control reifiers.}
\end{table} \end{table}
% %
\paragraph{An optical device for control} \paragraph{A small calculus for control}
% %
To look at control we will a simply typed fine-grain call-by-value To look at control we will a simply typed fine-grain call-by-value
calculus. The calculus is essentially the same as the one used in calculus. Although, we will sometimes have to discard the types, as
many of the control operators were invented and studied in a untyped
setting. The calculus is essentially the same as the one used in
Chapter~\ref{ch:handlers-efficiency}, except that here we will have an Chapter~\ref{ch:handlers-efficiency}, except that here we will have an
explicit invocation form for continuations. Although, in practice most explicit invocation form for continuations. Although, in practice most
systems disguise continuations as first-class functions, but for a systems disguise continuations as first-class functions, but for a
@@ -1009,20 +1043,20 @@ depicts the syntax of types and terms in the calculus.
\begin{figure} \begin{figure}
\centering \centering
\begin{syntax} \begin{syntax}
\slab{Types} & A,B &::=& \UnitType \mid \Zero \mid A \to B \mid A + B \mid A \times B \mid \Cont\,\Record{A;B} \smallskip\\ \slab{Types} & A,B &::=& \UnitType \mid A \to B \mid A \times B \mid \Cont\,\Record{A;B} \mid A + B \smallskip\\
\slab{Values} & V,W &::=& x \mid \lambda x^A.M \mid V + W \mid \Record{V;W} \mid \Unit \mid \cont_\EC\\ \slab{Values} & V,W &::=& \Unit \mid \lambda x^A.M \mid \Record{V;W} \mid \cont_\EC \mid \Inl~V \mid \Inr~W \mid x\\
\slab{Computations} & M,N &::=& \Return\;V \mid \Let\;x \revto M \;\In\;N \mid \Let \Record{x;y} = V \;\In\; N \\ \slab{Computations} & M,N &::=& \Return\;V \mid \Let\;x \revto M \;\In\;N \mid \Let\;\Record{x;y} = V \;\In\; N \\
& &\mid& \Absurd^A\;V \mid V\,W \mid \Continue~V~W \smallskip\\ & &\mid& V\,W \mid \Continue~V~W \smallskip\\
\slab{Evaluation\textrm{ }contexts} & \EC &::=& [\,] \mid \Let\;x \revto \EC \;\In\;N \slab{Evaluation\textrm{ }contexts} & \EC &::=& [\,] \mid \Let\;x \revto \EC \;\In\;N
\end{syntax} \end{syntax}
\caption{Types and term syntax}\label{fig:pcf-lang-control} \caption{Types and term syntax}\label{fig:pcf-lang-control}
\end{figure} \end{figure}
% %
The types are the standard simple types with the addition of the empty The types are the standard simple types with the addition of the
type $\Zero$ and the continuation object type $\Cont\,\Record{A;B}$, continuation object type $\Cont\,\Record{A;B}$, which is parameterised
which is parameterised by an argument type and a result type, by an argument type and a result type, respectively. The static
respectively. The static semantics is standard as well, except for the semantics is standard as well, except for the continuation invocation
continuation invocation primitive $\Continue$. primitive $\Continue$.
% %
\begin{mathpar} \begin{mathpar}
\inferrule* \inferrule*
@@ -1096,7 +1130,7 @@ $\Escape$, however, it is worth noting that this idiom require
recursive types to type check. Even in a language without recursive recursive types to type check. Even in a language without recursive
types, the continuation may propagate outside its binding types, the continuation may propagate outside its binding
$\Escape$-expression if the language provides an escape hatch such as $\Escape$-expression if the language provides an escape hatch such as
mutable reference cells. mutable references.
% In our simply-typed setting it is not possible for the continuation to % In our simply-typed setting it is not possible for the continuation to
% propagate outside its binding $\Escape$-expression as it would require % propagate outside its binding $\Escape$-expression as it would require
% the addition of either recursive types or some other escape hatch like % the addition of either recursive types or some other escape hatch like