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Notes on control and prompt

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Daniel Hillerström
2020-11-23 21:45:45 +00:00
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6
macros.tex
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@@ -437,7 +437,11 @@
\newcommand{\Prompt}{\#}
\newcommand{\Escape}{\keyw{escape}}
\newcommand{\shift}{\ensuremath{\mathcal{S}}}
\newcommand{\reset}[1]{\ensuremath{\langle #1 \rangle}}
\def\sigh#1{%
\pmb{\left\langle\vphantom{#1}\right.}%
#1%
\pmb{\left.\vphantom{#1}\right\rangle}}
\newcommand{\reset}[1]{\pmb{\langle} #1 \pmb{\rangle}}
\newcommand{\fcontrol}{\keyw{fcontrol}}
\newcommand{\fprompt}{\%}
\newcommand{\splitter}{\keyw{splitter}}

12
thesis.bib
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@@ -761,6 +761,18 @@
address = {Cambridge, Massachusetts, USA}
}
@inproceedings{MeyerW85,
author = {Albert R. Meyer and
Mitchell Wand},
title = {Continuation Semantics in Typed Lambda-Calculi (Summary)},
booktitle = {Logic of Programs},
series = {Lecture Notes in Computer Science},
volume = {193},
pages = {219--224},
publisher = {Springer},
year = {1985}
}
@inproceedings{DanvyF90,
author = {Olivier Danvy and
Andrzej Filinski},

310
thesis.tex
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@@ -15,6 +15,7 @@
\usepackage{amssymb} % Provides math fonts
\usepackage{amsthm} % Provides \newtheorem, \theoremstyle, etc.
\usepackage{mathtools}
\usepackage{bm}
\usepackage{pkgs/mathpartir} % Inference rules
\usepackage{pkgs/mathwidth}
\usepackage{stmaryrd} % semantic brackets
@@ -511,7 +512,7 @@ handlers.
\subsection{Monadic reflection: best of both worlds}
\chapter{Controlling continuations}
\chapter{Continuations}
\label{ch:continuations}
% The significance of continuations in the programming languages
% literature is inescapable as continuations have found widespread use .
@@ -567,7 +568,7 @@ the translation preserve typeability.
\citet{Shan04} shows that dynamic delimited control and static
delimited control is macro-expressible in an untyped setting.
\section{Notions of continuations}
\section{Classifying continuations}
% \citeauthor{Reynolds93} has written a historical account of the
% various early discoveries of continuations~\cite{Reynolds93}.
@@ -674,7 +675,7 @@ continuations, composable continuations.
Downward and upward use of continuations.
\section{First-class control operators}
\section{Controlling continuations}
Table~\ref{tbl:classify-ctrl} provides a classification of a
non-exhaustive list of first-class control operators.
@@ -1016,11 +1017,16 @@ aborts the current continuation after capture. It was introduced by
1986~\cite{FelleisenF86,FelleisenFKD86}. The year after
\citet{FelleisenFDM87} introduced the F operator which is a variation
of C, where the captured continuation is composable.
Both operators are values.
%
\[
V,W \in \ValCat ::= \cdots \mid \FelleisenC \mid \FelleisenF
\]
%
The static semantics of $\FelleisenC$ are the same as $\Callcc$,
whilst the static semantics of $\FelleisenF$ are the same as
$\Callcomc$.
\begin{mathpar}
\inferrule*
{~}
@@ -1031,6 +1037,8 @@ of C, where the captured continuation is composable.
{\typ{\Gamma}{\FelleisenF : (\Cont\,\Record{A;A} \to A) \to A}}
\end{mathpar}
%
The dynamic semantics of $\FelleisenC$ and $\FelleisenF$ also differ.
%
\begin{reductions}
\slab{C\textrm{-}Capture} & \EC[\FelleisenC\,V] &\reducesto& V~\cont_{\EC}\\
\slab{C\textrm{-}Resume} & \EC[\Continue~\cont_{\EC'}~V] &\reducesto& \EC'[V] \medskip\\
@@ -1038,18 +1046,25 @@ of C, where the captured continuation is composable.
\slab{F\textrm{-}Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]
\end{reductions}
%
Their capture rules are identical.
%
Whilst the static semantics of $\FelleisenF$ are similar to that of
$\Callcomc$, their dynamic semantics differ. The former has the power
to discard the current continuation upon capture built in.
Their capture rules are identical. Both operators abort the current
continuation upon capture. This is what set $\FelleisenF$ apart from
the other composable control operator $\Callcomc$.
%
The resume rules of $\FelleisenC$ and $\FelleisenF$ show the
difference between the two operators. The $\FelleisenC$ operator
aborts the current continuation and reinstall the then-current
continuation just like $\Callcc$, whereas the resumption of a
continuation captured by $\FelleisenF$ composes the current
continuation with the then-current continuation.
\citet{FelleisenFDM87} show that $\FelleisenF$ can simulate
$\FelleisenC$.
%
\[
\sembr{\FelleisenC} \defas \lambda m.\FelleisenF\,(\lambda k. m\,(\lambda v.\FelleisenF\,(\lambda\_.\Continue~k~v)))
\]
%
\citet{FelleisenFDM87} also postulate that $\FelleisenC$ cannot express $\FelleisenF$.
\paragraph{Landin's J operator}
%
@@ -1153,6 +1168,8 @@ annotate the evaluation contexts for ordinary applications.
\slab{Resume} & \EC[\Continue~\cont_{\Record{\EC';W}}\,V] &\reducesto& \EC'[W\,V]
\end{reductions}
%
\dhil{The continuation object should have time $\Cont\,\Record{A;\Zero}$}
%
The $\slab{Capture}$ rule only applies if the application of $\J$
takes place inside an annotated evaluation context. The continuation
object produced by a $\J$ application encompasses the caller's
@@ -1187,39 +1204,111 @@ encode a familiar control operator like $\Callcc$~\cite{Thielecke98}.
\citet{Felleisen87b} has shown that the J operator can be
syntactically embedded using callcc.
%
% \[
% \sembr{\lambda x.M} \defas \lambda x.\Callcc\,(\lambda k.\sembr{M}[\J \mapsto \lambda f. k\,f])
% \]
% \[
% \sembr{\J} \defas \lambda k.\Callcc\,(\lambda d.\Continue~k~(\lambda x.\lambda
% \]
\[
\sembr{\lambda x.M} \defas \lambda x.\Callcc\,(\lambda k.\sembr{M}[\J \mapsto \lambda f.\lambda y. \Continue~k~(f\,y)])
\]
% %
% \dhil{The types do not match up.}
\dhil{I am sort of torn between whether to treat continuations as
first-class functions\dots}
\subsection{Delimited operators}
Delimited control: Control delimiters form the basis for delimited
control. \citeauthor{Felleisen88} introduced control delimiters in
1988, although allusions to control delimiters were made a year
earlier by \citet{FelleisenFDM87} and in \citeauthor{Felleisen87}'s
PhD dissertation~\cite{Felleisen87}. The basic idea was teased even
earlier in \citeauthor{Talcott85}'s teased the idea of control
delimiters in her PhD dissertation~\cite{Talcott85}.
%
Common Lisp resumable exceptions (condition system)~\cite{Steele90},
F~\cite{FelleisenFDM87,Felleisen88}, control/prompt~\cite{SitaramF90},
shift/reset~\cite{DanvyF89,DanvyF90}, splitter~\cite{QueinnecS91},
fcontrol~\cite{Sitaram93}, catchcont~\cite{LongleyW08}, effect
handlers~\cite{PlotkinP09}.
Comparison of various delimited control
operators~\cite{Shan04}. Simulation of delimited control using
undelimited control~\cite{Filinski94}
%\paragraph{Felleisen's F and prompt}
\paragraph{Control/prompt}
A delimited control operator captures only a designated segment of the
evaluation stack. Typically, a delimited control operator is a pair
consisting of a \emph{delimiter} and \emph{capture operation}. The
delimiter delimits the extent of the continuation captured by the
capture operation.
%
The operator control is a rebranding of Felleisen's F operator.
The first delimited control operator was introduced by
\citet{Felleisen88} with prompt and control, where prompt is the
delimiter and control is the capture operation. Shortly after
\citet{DanvyF89} introduced their shift and reset as an alternative
capture operation and delimiter, respectively. Since then a whole
variety of delimited control operators have been invented.
% Delimited control: Control delimiters form the basis for delimited
% control. \citeauthor{Felleisen88} introduced control delimiters in
% 1988, although allusions to control delimiters were made a year
% earlier by \citet{FelleisenFDM87} and in \citeauthor{Felleisen87}'s
% PhD dissertation~\cite{Felleisen87}. The basic idea was teased even
% earlier in \citeauthor{Talcott85}'s teased the idea of control
% delimiters in her PhD dissertation~\cite{Talcott85}.
% %
% Common Lisp resumable exceptions (condition system)~\cite{Steele90},
% F~\cite{FelleisenFDM87,Felleisen88}, control/prompt~\cite{SitaramF90},
% shift/reset~\cite{DanvyF89,DanvyF90}, splitter~\cite{QueinnecS91},
% fcontrol~\cite{Sitaram93}, catchcont~\cite{LongleyW08}, effect
% handlers~\cite{PlotkinP09}.
% Comparison of various delimited control
% operators~\cite{Shan04}. Simulation of delimited control using
% undelimited control~\cite{Filinski94}
\paragraph{\citeauthor{Felleisen88}'s control and prompt}
%
Control and prompt were introduced by \citeauthor{Felleisen88} in
1988~\cite{Felleisen88}. The capture operation `control' is a
rebranding of the F operator. Although, the name `control' was first
introduced a little later by \citet{SitaramF90}. A prompt acts as a
control-flow barrier that delimits different parts of a program,
enabling programmers to manipulate and reason about control locally in
different parts of a program. The name `prompt' is intended to draw
connections to shell prompts, and how they act as barriers between the
user and operating system.
%
A prompt is a computation form, whilst control is a value.
%
\begin{syntax}
&V,W \in \ValCat &::=& \cdots \mid \Control\\
&M,W \in \CompCat &::=& \cdots \mid \Prompt~M
\end{syntax}
%
A prompt is written using the sharp ($\Prompt$) symbol. An application
of $\Control$ reifies the current continuation up to the nearest,
dynamically determined, enclosing $\Prompt$.
%
The prompt remains in place after the reification, and thus any
subsequent application of $\Control$ will be delimited by the same
prompt.
%
\citeauthor{Felleisen88}'s original treatment of control and prompt
was untyped. Typing them, particularly using a simple type system,
affect their expressivity, because the type of the continuation object
produced by $\Control$ must be compatible with the type of its nearest
enclosing prompt -- this type is often called the \emph{answer} type
(this terminology is adopted from typed continuation passing style
transforms, where the codomain of every function is transformed to
yield the type of whatever answer the entire program
yields~\cite{MeyerW85}).
%
\dhil{Give intuition for why soundness requires the answer type to be fixed.}
%
In the static semantics we extend the typing judgement relation to
contain an up front fixed answer type $A$.
%
\begin{mathpar}
\inferrule*
{\typ{\Gamma;A}{M : A}}
{\typ{\Gamma;A}{\Prompt~M : A}}
\inferrule*
{~}
{\typ{\Gamma;A}{\Control : (\Cont\,\Record{A;A} \to A) \to A}}
\end{mathpar}
%
A prompt has the same type as its computation constituent, which in
turn must have the same type as fixed answer type.
%
Similarly, the type of $\Control$ is governed by the fixed answer
type. Discarding the answer type reveals that $\Control$ has the same
typing judgement as $\FelleisenF$.
%
The dynamic semantics for control and prompt consist of three rules:
1) to handle return through a prompt, 2) continuation capture, and 3)
continuation invocation.
%
\begin{reductions}
\slab{Value} &
@@ -1228,7 +1317,23 @@ The operator control is a rebranding of Felleisen's F operator.
\Prompt~\EC[\Control~V] &\reducesto& \Prompt~(V~\cont_{\EC}), \text{ where $\EC$ contains no \Prompt}\\
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]
\end{reductions}
%
The \slab{Value} rule accounts for the case when the computation
constituent of $\Prompt$ has been reduced to a value, in which case
the prompt is removed and the value is returned.
%
The \slab{Capture} rule states that an application of $\Control$
captures the current continuation up to the nearest enclosing
prompt. The current continuation (up to the nearest prompt) is also
aborted. If we erase $\Prompt$ from the rule, then it is clear that
$\Control$ has the same dynamic behaviour as $\FelleisenF$.
%
It is evident from the \slab{Resume} rule that control and prompt are
an instance of a dynamic control operator, because resuming the
continuation object produced by $\Control$ does not insert a new
prompt.
\dhil{Multi-prompts: more liberal typing, no interference}
\paragraph{Cupto}
%
@@ -1242,50 +1347,7 @@ The operator control is a rebranding of Felleisen's F operator.
\slab{Resume} & \Continue~\cont_{\EC}~V, \rho &\reducesto& \EC[V], \rho
\end{reductions}
\paragraph{Effect handlers}
%
\begin{reductions}
\slab{Value} & \Handle\; V \;\With\;H &\reducesto& M[V/x], \text{ where } \{\Return\;x \mapsto M\} \in H\\
\slab{Capture} & \Handle\;\EC[\Do\;\ell~V] \;\With\; H &\reducesto& M[V/p,\cont_{\Record{\EC;H}}/k],\\
\multicolumn{4}{l}{\hfill\bl\text{where $\ell$ is not handled in $\EC$}\\\text{and }\{\ell~p~k \mapsto M\} \in H\el}\\
\slab{Resume} & \Continue~\cont_{\Record{\EC;H}}~V &\reducesto& \Handle\;\EC[V]\;\With\;H\\
\end{reductions}
%
\begin{reductions}
\slab{Capture} & \Handle\;\EC[\Do\;\ell~V] \;\With\; H &\reducesto& M[V/p,\cont_{\EC}/k],\\
\multicolumn{4}{l}{\hfill\bl\text{where $\ell$ is not handled in $\EC$}\\\text{and }\{\ell~p~k \mapsto M\} \in H\el}\\
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]\\
\end{reductions}
%
\paragraph{Sitaram's fcontrol}
%
\begin{reductions}
\slab{Value} &
\fprompt~V.W &\reducesto& W\\
\slab{Capture} &
\fprompt~V.\EC[\fcontrol~W] &\reducesto& V~W~\cont_{\EC}, \text{ where $\EC$ contains no \fprompt}\\
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]
\end{reductions}
%
\paragraph{Longley's catch-with-continue}
%
\begin{mathpar}
\inferrule*
{\typ{\Gamma, f : A \to B}{M : C \times D} \\ \Ground\;C}
{\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}
%
\begin{reductions}
\slab{Value} &
\Catchcont \; f . \Record{V;W} &\reducesto& \Inl\; \Record{V;\lambda\,f. W}\\
\slab{Capture} &
\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]
\end{reductions}
\paragraph{Shift/reset}
\paragraph{\citeauthor{DanvyF90}'s shift and reset}
%
\begin{reductions}
\slab{Value} & \reset{V} &\reducesto& V\\
@@ -1303,16 +1365,96 @@ The operator control is a rebranding of Felleisen's F operator.
\end{reductions}
%
\paragraph{Splitter}
\paragraph{\citeauthor{PlotkinP09}'s effect handlers}
%
\begin{reductions}
\slab{Throw} & \splitter~throw~callpc.\EC[\,throw~V] &\reducesto& V~\Unit\\
\slab{Value} & \Handle\; V \;\With\;H &\reducesto& M[V/x], \text{ where } \{\Return\;x \mapsto M\} \in H\\
\slab{Capture} & \Handle\;\EC[\Do\;\ell~V] \;\With\; H &\reducesto& M[V/p,\cont_{\Record{\EC;H}}/k],\\
\multicolumn{4}{l}{\hfill\bl\text{where $\ell$ is not handled in $\EC$}\\\text{and }\{\ell~p~k \mapsto M\} \in H\el}\\
\slab{Resume} & \Continue~\cont_{\Record{\EC;H}}~V &\reducesto& \Handle\;\EC[V]\;\With\;H\\
\end{reductions}
%
\begin{reductions}
\slab{Capture} & \Handle\;\EC[\Do\;\ell~V] \;\With\; H &\reducesto& M[V/p,\cont_{\EC}/k],\\
\multicolumn{4}{l}{\hfill\bl\text{where $\ell$ is not handled in $\EC$}\\\text{and }\{\ell~p~k \mapsto M\} \in H\el}\\
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]\\
\end{reductions}
%
Deep handlers can be used to simulate shift and reset.
%
\begin{equations}
\sembr{\shift} &\defas& \lambda f. \Do\;\shift~f\\
\sembr{\reset{M}} &\defas&
\ba[t]{@{~}l}\Handle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& x\\
\OpCase{\dec{Shift}}{f}{k} &\mapsto& f\,(\lambda x. \Continue~k~x)
\ea
\ea
\end{equations}
%
Shallow handlers can be used to simulate prompt and control.
%
\begin{equations}
\sembr{\Control} &\defas& \lambda f. \Do\;\dec{Control}~f\\
\sembr{\Prompt~M} &\defas&
\bl
prompt\,(\lambda\Unit.M)\\
\textbf{where}\;
\bl
prompt~m \defas
\ba[t]{@{~}l}\ShallowHandle\;m\,\Unit\;\With\\
~\ba{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& x\\
\OpCase{\dec{Control}}{f}{k} &\mapsto& prompt\,(\lambda\Unit.f\,(\lambda x. \Continue~k~x))
\ea
\ea
\el
\el
\end{equations}
%
%Recursive types are required to type the image of this translation
\paragraph{\citeauthor{Sitaram93}'s fcontrol}
%
\begin{reductions}
\slab{Value} &
\fprompt~V.W &\reducesto& W\\
\slab{Capture} &
\splitter~throw~callpc.\EC[callpc~V] &\reducesto& V~\cont_{\EC} \\
\fprompt~V.\EC[\fcontrol~W] &\reducesto& V~W~\cont_{\EC}, \text{ where $\EC$ contains no \fprompt}\\
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]
\end{reductions}
%
\paragraph{\citeauthor{Longley09}'s catch-with-continue}
%
\begin{mathpar}
\inferrule*
{\typ{\Gamma, f : A \to B}{M : C \times D} \\ \Ground\;C}
{\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}
%
\begin{reductions}
\slab{Value} &
\Catchcont \; f . \Record{V;W} &\reducesto& \Inl\; \Record{V;\lambda\,f. W}\\
\slab{Capture} &
\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]
\end{reductions}
\paragraph{\citeauthor{QueinnecS91}'s splitter}
\begin{reductions}
\slab{Throw} & \splitter~throw~calldc.\EC[\,throw~V] &\reducesto& V~\Unit\\
\slab{Capture} &
\splitter~throw~calldc.\EC[calldc~V] &\reducesto& V~\cont_{\EC} \\
\slab{Resume} & \Continue~\cont_{\EC}~V &\reducesto& \EC[V]
\end{reductions}
\paragraph{Spawn/controller}
\paragraph{Spawn}
\dhil{TODO}
\subsection{Second-class control operators}
Coroutines, async/await, generators/iterators, amb.