|
|
@ -946,20 +946,21 @@ macro-expressible. |
|
|
\end{equations} |
|
|
\end{equations} |
|
|
|
|
|
|
|
|
\paragraph{Call-with-composable-continuation} A variation of callcc is |
|
|
\paragraph{Call-with-composable-continuation} A variation of callcc is |
|
|
call-with-composable-continuation, or as I call it \textCallcomc{}. |
|
|
|
|
|
|
|
|
call-with-composable-continuation, abbreviated \textCallcomc{}. |
|
|
% |
|
|
% |
|
|
As the name suggests the captured continuation is composable rather |
|
|
As the name suggests the captured continuation is composable rather |
|
|
than abortive. |
|
|
|
|
|
% |
|
|
|
|
|
According to the history log of Racket it appeared in November 2006 |
|
|
|
|
|
(Racket was then known as MzScheme, version 360)~\cite{Flatt20}. The |
|
|
|
|
|
history log classifies it as a delimited control operator. Truth to be |
|
|
|
|
|
told nowadays in Racket virtually all control operators are delimited, |
|
|
|
|
|
even callcc, because they are parameterised by an optional prompt |
|
|
|
|
|
tag. If the programmer does not supply a prompt tag at invocation time |
|
|
|
|
|
then the optional parameter assume the actual value of the top-level |
|
|
|
|
|
prompt, effectively making the extent of the captured continuation |
|
|
|
|
|
undelimited. |
|
|
|
|
|
|
|
|
than abortive. It was introduced by \citet{FlattYFF07} in 2007, and |
|
|
|
|
|
and implemented in November 2006 according to the history log of |
|
|
|
|
|
Racket (Racket was then known as MzScheme, version |
|
|
|
|
|
360)~\cite{Flatt20}. The history log classifies it as a delimited |
|
|
|
|
|
control operator. |
|
|
|
|
|
% |
|
|
|
|
|
Truth to be told nowadays in Racket virtually all control operators |
|
|
|
|
|
are delimited, even callcc, because they are parameterised by an |
|
|
|
|
|
optional prompt tag. If the programmer does not supply a prompt tag at |
|
|
|
|
|
invocation time then the optional parameter assume the actual value of |
|
|
|
|
|
the top-level prompt, effectively making the extent of the captured |
|
|
|
|
|
continuation undelimited. |
|
|
% |
|
|
% |
|
|
In other words its default mode of operation is undelimited, hence the |
|
|
In other words its default mode of operation is undelimited, hence the |
|
|
justification for categorising it as such. |
|
|
justification for categorising it as such. |
|
|
@ -971,30 +972,31 @@ Like $\Callcc$ this operator is a value. |
|
|
V,W \in \ValCat ::= \cdots \mid \Callcomc |
|
|
V,W \in \ValCat ::= \cdots \mid \Callcomc |
|
|
\] |
|
|
\] |
|
|
% |
|
|
% |
|
|
Unlike $\Callcc$, the continuation returns, which the typing rule for |
|
|
|
|
|
$\Callcomc$ reflects. |
|
|
|
|
|
% |
|
|
|
|
|
\begin{mathpar} |
|
|
|
|
|
\inferrule* |
|
|
|
|
|
{~} |
|
|
|
|
|
{\typ{\Gamma}{\Callcomc : (\Cont\,\Record{A;A} \to A) \to A}} |
|
|
|
|
|
|
|
|
% Unlike $\Callcc$, the continuation returns, which the typing rule for |
|
|
|
|
|
% $\Callcomc$ reflects. |
|
|
|
|
|
% % |
|
|
|
|
|
% \begin{mathpar} |
|
|
|
|
|
% \inferrule* |
|
|
|
|
|
% {~} |
|
|
|
|
|
% {\typ{\Gamma}{\Callcomc : (\Cont\,\Record{A;A} \to A) \to A}} |
|
|
|
|
|
|
|
|
\inferrule* |
|
|
|
|
|
{\typ{\Gamma}{V : A} \\ \typ{\Gamma}{W : \Cont\,\Record{A;B}}} |
|
|
|
|
|
{\typ{\Gamma}{\Continue~W~V : B}} |
|
|
|
|
|
\end{mathpar} |
|
|
|
|
|
% |
|
|
|
|
|
Both the domain and codomain of the continuation are the same as the |
|
|
|
|
|
body type of function argument. The capture rule for $\Callcomc$ is |
|
|
|
|
|
identical to the rule for $\Callcomc$, but the resume rule is |
|
|
|
|
|
different. |
|
|
|
|
|
|
|
|
% \inferrule* |
|
|
|
|
|
% {\typ{\Gamma}{V : A} \\ \typ{\Gamma}{W : \Cont\,\Record{A;B}}} |
|
|
|
|
|
% {\typ{\Gamma}{\Continue~W~V : B}} |
|
|
|
|
|
% \end{mathpar} |
|
|
|
|
|
% % |
|
|
|
|
|
% Both the domain and codomain of the continuation are the same as the |
|
|
|
|
|
% body type of function argument. |
|
|
|
|
|
Unlike $\Callcc$, captured continuations behave as functions. |
|
|
% |
|
|
% |
|
|
\begin{reductions} |
|
|
\begin{reductions} |
|
|
\slab{Capture} & \EC[\Callcomc~V] &\reducesto& \EC[V~\qq{\cont_{\EC}}]\\ |
|
|
\slab{Capture} & \EC[\Callcomc~V] &\reducesto& \EC[V~\qq{\cont_{\EC}}]\\ |
|
|
% \slab{Resume} & \EC[\Continue~\cont_{\EC'}~V] &\reducesto& \EC[\EC'[V]] |
|
|
|
|
|
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V] |
|
|
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V] |
|
|
\end{reductions} |
|
|
\end{reductions} |
|
|
% |
|
|
% |
|
|
|
|
|
The capture rule for $\Callcomc$ is identical to the rule for |
|
|
|
|
|
$\Callcomc$, but the resume rule is different. |
|
|
|
|
|
% |
|
|
The effect of continuation invocation can be understood locally as it |
|
|
The effect of continuation invocation can be understood locally as it |
|
|
does not erase the global evaluation context, but rather composes with |
|
|
does not erase the global evaluation context, but rather composes with |
|
|
it. |
|
|
it. |
|
|
@ -2132,16 +2134,31 @@ computation. |
|
|
|
|
|
|
|
|
If the reader ever find themselves in a quiz show asked to single out |
|
|
If the reader ever find themselves in a quiz show asked to single out |
|
|
a canonical example of continuation use, then implementation of |
|
|
a canonical example of continuation use, then implementation of |
|
|
cooperative concurrency would be a qualified guess. Various forms of |
|
|
|
|
|
coroutines as continuations occur so frequently in the literature and |
|
|
|
|
|
in the wild that they have become routine. |
|
|
|
|
|
|
|
|
concurrency would be a qualified guess. Cooperative concurrency in |
|
|
|
|
|
terms of various forms of coroutines as continuations occur so |
|
|
|
|
|
frequently in the literature and in the wild that they have become |
|
|
|
|
|
routine. |
|
|
% |
|
|
% |
|
|
\citet{HaynesFW86} published one of the first implementations of |
|
|
\citet{HaynesFW86} published one of the first implementations of |
|
|
coroutines using first-class control. \citet{HiebD90} demonstrated how |
|
|
|
|
|
to implement engines. An engine is a kind of scheduler for |
|
|
|
|
|
computations running on a time budget~\cite{HaynesF84}. |
|
|
|
|
|
|
|
|
coroutines using first-class control. |
|
|
|
|
|
% |
|
|
|
|
|
Preemptive concurrency in the form of engines were implemented by |
|
|
|
|
|
\citet{DybvigH89}. An engine is a control abstraction that runs |
|
|
|
|
|
computations with an allotted time budget~\cite{HaynesF84}. They used |
|
|
|
|
|
continuations to represent strands of computation and timer interrupts |
|
|
|
|
|
to suspend continuations. |
|
|
|
|
|
% |
|
|
|
|
|
\citet{KiselyovS07a} used delimited continuations to explain various |
|
|
|
|
|
phenomena of operating systems, including multi-tasking. |
|
|
|
|
|
% |
|
|
|
|
|
On the web, \citet{Queinnec04} used continuations to model the |
|
|
|
|
|
client-server interactions. This model was adapted by |
|
|
|
|
|
\citet{CooperLWY06} in Links with support for an Erlang-style |
|
|
|
|
|
concurrency model~\cite{ArmstrongVW93}. |
|
|
% |
|
|
% |
|
|
Various other forms of schedulers were developed by \citet{GanzFW99}. |
|
|
|
|
|
|
|
|
\citet{Leijen17a} and \citet{DolanEHMSW17} gave two different ways of |
|
|
|
|
|
implementing the asynchronous programming operator async/await as a |
|
|
|
|
|
user-definable library. |
|
|
|
|
|
|
|
|
Continuations have also been used in meta-programming to speed up |
|
|
Continuations have also been used in meta-programming to speed up |
|
|
partial evaluation and |
|
|
partial evaluation and |
|
|
@ -2149,35 +2166,73 @@ multi-staging~\cite{LawallD94,KameyamaKS11,OishiK17,Yallop17}. Let |
|
|
insertion is a canonical example of use of continuations in |
|
|
insertion is a canonical example of use of continuations in |
|
|
multi-staging~\cite{Yallop17}. |
|
|
multi-staging~\cite{Yallop17}. |
|
|
|
|
|
|
|
|
The aforementioned applications of continuations is by no means |
|
|
|
|
|
exhaustive, however, the diverse application spectrum underlines the |
|
|
|
|
|
|
|
|
Probabilistic programming is yet another application domain of |
|
|
|
|
|
continuations. \citet{KiselyovS09} used delimited continuations to |
|
|
|
|
|
speed up probabilistic programs. \citet{GorinovaMH20} used |
|
|
|
|
|
continuations to achieve modularise probabilistic programs and to |
|
|
|
|
|
provide a simple and efficient mechanism for reparameterisation of |
|
|
|
|
|
inference algorithms. |
|
|
|
|
|
% |
|
|
|
|
|
In the subject of differentiable programming \citet{WangZDWER19} |
|
|
|
|
|
explained reverse-mode automatic differentiation operators in terms of |
|
|
|
|
|
delimited continuations. |
|
|
|
|
|
|
|
|
|
|
|
The aforementioned applications of continuations are by no means |
|
|
|
|
|
exhaustive, though, the diverse application spectrum underlines the |
|
|
versatility of continuations. |
|
|
versatility of continuations. |
|
|
|
|
|
|
|
|
\section{Constraining continuations} |
|
|
\section{Constraining continuations} |
|
|
|
|
|
|
|
|
\citet{FriedmanH85} argued in favour of constraining the power of |
|
|
|
|
|
continuations~\cite{HaynesF87}. Although, they were concerned with |
|
|
|
|
|
callcc and undelimited continuations many of their arguments are |
|
|
|
|
|
applicable to other control operators and delimited continuations. |
|
|
|
|
|
|
|
|
|
|
|
For example callec is a variation of callcc where the continuation |
|
|
|
|
|
only can be invoked during the dynamic extent of |
|
|
|
|
|
callec~\cite{Flatt20}. |
|
|
|
|
|
|
|
|
|
|
|
Dynamic-wind\dots |
|
|
|
|
|
|
|
|
|
|
|
To avoid resource leakage the implementation of effect handlers in |
|
|
|
|
|
Multicore OCaml provides both a \emph{continue} primitive for |
|
|
|
|
|
continuing a given continuation and a \emph{discontinue} primitive for |
|
|
|
|
|
aborting a given continuation~\cite{DolanWSYM15,DolanEHMSW17}. The |
|
|
|
|
|
latter throws an exception at the operation invocation site to clean |
|
|
|
|
|
up resources. |
|
|
|
|
|
% |
|
|
|
|
|
A similar approached is used by \citet{Fowler19}, who uses a |
|
|
|
|
|
|
|
|
\citet{FriedmanH85} advocated for constraining the power of |
|
|
|
|
|
(undelimited) continuations~\cite{HaynesF87}. |
|
|
|
|
|
% |
|
|
|
|
|
Even though, they were concerned with callcc and undelimited |
|
|
|
|
|
continuations some of their arguments are applicable to other control |
|
|
|
|
|
operators and delimited continuations. |
|
|
|
|
|
% |
|
|
|
|
|
For example, they argued in favour of restricting continuations to be |
|
|
|
|
|
one-shot, which means continuations may only be invoked once. Firstly, |
|
|
|
|
|
because one-shot continuations admit particularly efficient |
|
|
|
|
|
implementations. Secondly, many applications involve only single use |
|
|
|
|
|
of continuations. Thirdly, one-shot continuations interact more |
|
|
|
|
|
robustly with resources, such as file handles, than general multi-shot |
|
|
|
|
|
continuations, because multiple use of a continuation may accidentally |
|
|
|
|
|
interact with a resource after it has been released. |
|
|
|
|
|
|
|
|
|
|
|
One-shot continuations by themselves are no saving grace for avoiding |
|
|
|
|
|
resource leakage as they may be dropped or used to perform premature |
|
|
|
|
|
exits from a block with resources. For example, Racket provides the |
|
|
|
|
|
programmer with a facility known as \emph{dynamic-wind} to protect a |
|
|
|
|
|
context with resources such that non-local exits properly release |
|
|
|
|
|
whatever resources the context has acquired~\cite{Flatt20}. |
|
|
|
|
|
% |
|
|
|
|
|
An alternative approach is taken by Multicore OCaml, whose |
|
|
|
|
|
implementation of effect handlers with one-shot continuations provides |
|
|
|
|
|
both a \emph{continue} primitive for continuing a given continuation |
|
|
|
|
|
and a \emph{discontinue} primitive for aborting a given |
|
|
|
|
|
continuation~\cite{DolanWSYM15,DolanEHMSW17}. The latter throws an |
|
|
|
|
|
exception at the operation invocation site to which can be caught by |
|
|
|
|
|
local exception handlers to release resources properly. |
|
|
|
|
|
% |
|
|
|
|
|
This approach is also used by \citet{Fowler19}, who uses a |
|
|
substructural type system to statically enforce the use of |
|
|
substructural type system to statically enforce the use of |
|
|
continuations, either by means of a continue or a discontinue. |
|
|
continuations, either by means of a continue or a discontinue. |
|
|
|
|
|
|
|
|
|
|
|
% For example callec is a variation of callcc where continuation |
|
|
|
|
|
% invocation is only defined during the dynamic extent of |
|
|
|
|
|
% callec~\cite{Flatt20}. |
|
|
|
|
|
|
|
|
\section{Implementing continuations} |
|
|
\section{Implementing continuations} |
|
|
|
|
|
|
|
|
|
|
|
There are numerous strategies for implementing continuations. There is |
|
|
|
|
|
no best implementation strategy. Each strategy has different |
|
|
|
|
|
trade-offs, and as such, there is no ``best'' strategy. In this |
|
|
|
|
|
section, I will briefly outline the gist of some implementation |
|
|
|
|
|
strategies and their trade-offs. For an in depth analysis the |
|
|
|
|
|
interested reader may consult the respective work of |
|
|
|
|
|
\citet{ClingerHO88} and \citet{FarvardinR20}, which contain thorough |
|
|
|
|
|
studies of implementation strategies for first-class continuations. |
|
|
|
|
|
|
|
|
Table~\ref{tbl:ctrl-operators-impls} lists some programming languages |
|
|
Table~\ref{tbl:ctrl-operators-impls} lists some programming languages |
|
|
with support for first-class control operators and their |
|
|
with support for first-class control operators and their |
|
|
implementation strategies. |
|
|
implementation strategies. |
|
|
@ -2208,7 +2263,7 @@ implementation strategies. |
|
|
\hline |
|
|
\hline |
|
|
OchaCaml & shift/reset & Virtual machine\\ |
|
|
OchaCaml & shift/reset & Virtual machine\\ |
|
|
\hline |
|
|
\hline |
|
|
Racket & callcc, callcc$^{\ast}$, cupto, fcontrol, control/prompt, shift/reset, splitter, spawn & Continuation marks\\ |
|
|
|
|
|
|
|
|
Racket & callcc, \textCallcomc{}, cupto, fcontrol, control/prompt, shift/reset, splitter, spawn & Segmented stacks\\ |
|
|
\hline |
|
|
\hline |
|
|
Rhino JavaScript & JI & Interpreter \\ |
|
|
Rhino JavaScript & JI & Interpreter \\ |
|
|
\hline |
|
|
\hline |
|
|
@ -2216,21 +2271,29 @@ implementation strategies. |
|
|
\hline |
|
|
\hline |
|
|
SML/NJ & callcc & CPS\\ |
|
|
SML/NJ & callcc & CPS\\ |
|
|
\hline |
|
|
\hline |
|
|
Wasm/k & control/prompt & ?? \\ |
|
|
|
|
|
|
|
|
Wasm/k & control/prompt & Virtual machine \\ |
|
|
\hline |
|
|
\hline |
|
|
\end{tabular} |
|
|
\end{tabular} |
|
|
\caption{Some languages and their implementation strategies for first-class control.}\label{tbl:ctrl-operators-impls} |
|
|
|
|
|
|
|
|
\caption{Some languages and their implementation strategies for continuations.}\label{tbl:ctrl-operators-impls} |
|
|
\end{table} |
|
|
\end{table} |
|
|
|
|
|
% |
|
|
|
|
|
At runtime the call stack represents the current continuation. |
|
|
|
|
|
% |
|
|
|
|
|
Thus continuation capture can be implemented by copying the current |
|
|
|
|
|
stack, and continuation invocation as reinstatement of the stack. This |
|
|
|
|
|
implementation strategy works well if continuations are captured |
|
|
|
|
|
infrequently. A slight this implementation strategy is to defer |
|
|
|
|
|
copying the stack until the continuation is invoked |
|
|
|
|
|
|
|
|
\paragraph{Continuation marks} |
|
|
|
|
|
|
|
|
% \paragraph{Continuation marks} |
|
|
|
|
|
|
|
|
\paragraph{Continuation passing style} |
|
|
|
|
|
|
|
|
% \paragraph{Continuation passing style} |
|
|
|
|
|
|
|
|
\paragraph{Abstract and virtual machines} |
|
|
|
|
|
|
|
|
% \paragraph{Abstract and virtual machines} |
|
|
|
|
|
|
|
|
\paragraph{Segmented stacks} |
|
|
|
|
|
|
|
|
% \paragraph{Segmented stacks} |
|
|
|
|
|
|
|
|
\paragraph{Stack copying} |
|
|
|
|
|
|
|
|
% \paragraph{Stack copying} |
|
|
|
|
|
|
|
|
\part{Design} |
|
|
\part{Design} |
|
|
|
|
|
|
|
|
|
