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Final example

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Daniel Hillerström 5 years ago
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  1. 14
      thesis.bib
  2. 92
      thesis.tex

14
thesis.bib
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@ -2005,6 +2005,20 @@
year = {1994} year = {1994}
} }
# marker and callpc
@inproceedings{MoreauQ94,
author = {Luc Moreau and
Christian Queinnec},
title = {Partial Continuations as the Difference of Continuations - {A} Duumvirate
of Control Operators},
booktitle = {{PLILP}},
series = {{LNCS}},
volume = {844},
pages = {182--197},
publisher = {Springer},
year = {1994}
}
# Comparison of (some) delimited control operators # Comparison of (some) delimited control operators
@misc{Shan04, @misc{Shan04,
author = {Chung{-}chieh Shan}, author = {Chung{-}chieh Shan},

92
thesis.tex
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@ -4361,6 +4361,11 @@ $k$ restores the context with the prompt. The $\abort$ application
erases the evaluation context up to this prompt, however, the body of erases the evaluation context up to this prompt, however, the body of
the functional argument to $\abort$ reinvokes the continuation $k$ the functional argument to $\abort$ reinvokes the continuation $k$
which restores the prompt context once again. which restores the prompt context once again.
\citet{MoreauQ94} proposed a variation of splitter called
\emph{marker}, which is also built on top of multi-prompt
semantics. The key difference is that the control reifier strips the
reified context of all prompts.
% \begin{derivation} % \begin{derivation}
% &1 + \splitter\,(\lambda p.2 + \splitter\,(\lambda p'.3 + \calldc\,\Record{p';\lambda k. k\,0 + \calldc\,\Record{p';\lambda k'. k\,(k'\,1)}}))), \emptyset\\ % &1 + \splitter\,(\lambda p.2 + \splitter\,(\lambda p'.3 + \calldc\,\Record{p';\lambda k. k\,0 + \calldc\,\Record{p';\lambda k'. k\,(k'\,1)}}))), \emptyset\\
% \reducesto& \reason{\slab{AppSplitter}}\\ % \reducesto& \reason{\slab{AppSplitter}}\\
@ -4437,9 +4442,44 @@ continuation. This continuation gets passed to the functional value of
the control expression. The captured continuation contains the label the control expression. The captured continuation contains the label
$\ell$, and as specified by the $\slab{Resume}$ rule an invocation of $\ell$, and as specified by the $\slab{Resume}$ rule an invocation of
the continuation causes this label to be reinstalled. the continuation causes this label to be reinstalled.
% concas structured
% alternative for using undelimited continuations, such as provided by
% callcc, in concurrent programming
The following example usage of $\Spawn$ is a slight variation on an
example due to \citet{HiebDA94}.
%
\begin{derivation}
& 1 \cons (\Spawn\,(\lambda c. 2 \cons (c\,(\lambda k. 3 \cons k\,(k\,\nil))))), \emptyset\\
\reducesto& \reason{\slab{AppSpawn}}\\
&1 \cons (\ell : (\lambda c. 2 \cons (c\,(\lambda k. 3 \cons k\,(k\,\nil))))\,(\lambda f.f \reflect \ell)), \{\ell\}\\
\reducesto& \reason{$\beta$-reduction}\\
&1 \cons (\ell : 2 \cons ((\lambda f.f \reflect \ell)\,(\lambda k. 3 \cons k\,(k\,\nil)))), \{\ell\}\\
\end{derivation}
%
\begin{derivation}
\reducesto& \reason{$\beta$-reduction}\\
&1 \cons (\ell : 2 \cons ((\lambda k. 3 \cons k\,(k\,\nil)) \reflect \ell)), \{\ell\}\\
\reducesto& \reason{\slab{Capture} $\EC = 2 \cons [\,]$}\\
& 1 \cons 3 \cons \qq{\cont_{\EC}}\,(\qq{\cont_{\EC}}\,\nil), \{\ell\}\\
\reducesto& \reason{\slab{Resume} $\EC$ with $\nil$}\\
&1 \cons 3 \cons \qq{\cont_{\EC}}\,(\ell : 2 \cons \nil), \{\ell\}\\
\reducesto^+& \reason{\slab{Value}}\\
&1 \cons 3 \cons \qq{\cont_{\EC}}\,[2], \{\ell\}\\
\reducesto^+& \reason{\slab{Resume} $\EC$ with $[2]$}\\
&1 \cons 3 \cons (\ell : 2 \cons [2]), \{\ell\}\\
\reducesto^+& \reason{\slab{Value}}\\
&1 \cons 3 \cons [2,2], \{\ell\} \reducesto^+ [1,3,2,2], \{\ell\}
\end{derivation}
%
When the controller $c$ is invoked the current continuation is
$1 \cons (\ell : 2 \cons [\,])$. The control expression reifies the
$\ell : 2 \cons [\,]$ portion of the continuation and binds it to
$k$. The first invocation of $k$ reinstates the reified portion and
computes the singleton list $[2]$ which is used as argument to the
second invocation of $k$.
Both \citet{HiebD90} and \citet{HiebDA94} give several concurrent
programming examples with spawn. They show how
parallel-or~\cite{Plotkin77} can be codified as a macro using spawn
(and a parallel invocation primitive \emph{pcall}).
\paragraph{\citeauthor{Sitaram93}'s fcontrol} The control operator \paragraph{\citeauthor{Sitaram93}'s fcontrol} The control operator
`fcontrol' was introduced by \citet{Sitaram93} in 1993. It is a `fcontrol' was introduced by \citet{Sitaram93} in 1993. It is a
@ -4817,7 +4857,7 @@ gets stuck as either $\Choose$ or $\Fail$, or both, would be
unhandled. Thus, we have to run the computation in the context of both unhandled. Thus, we have to run the computation in the context of both
handlers. However, we have a choice to make as we can compose the handlers. However, we have a choice to make as we can compose the
handlers in either order. Let us first explore the composition, where handlers in either order. Let us first explore the composition, where
$H_c$ is the outermost handler. Thus instantiate $H_c$ at type
$H_c$ is the outermost handler. Thus we instantiate $H_c$ at type
$\Option~\Bool$ and $H_f$ at type $\Bool$. $\Option~\Bool$ and $H_f$ at type $\Bool$.
% %
\begin{derivation} \begin{derivation}
@ -4830,6 +4870,8 @@ $\Option~\Bool$ and $H_f$ at type $\Bool$.
& (\Handle\;(\Handle\;\EC[\True] \;\With\;H_f)\;\With\;H_c) \concat k~\False\\ & (\Handle\;(\Handle\;\EC[\True] \;\With\;H_f)\;\With\;H_c) \concat k~\False\\
\reducesto & \reason{$\beta$-reduction}\\ \reducesto & \reason{$\beta$-reduction}\\
& (\Handle\;(\Handle\; \Do\;\Choose~\Unit \;\With\; H_f)\;\With\;H_c) \concat k~\False\\ & (\Handle\;(\Handle\; \Do\;\Choose~\Unit \;\With\; H_f)\;\With\;H_c) \concat k~\False\\
\end{derivation}
\begin{derivation}
\reducesto & \reason{\slab{Capture}, $\{\OpCase{\Choose}{\Unit}{k'} \mapsto \cdots\} \in H_c$, $\EC'' = (\Handle\;[\,]\;\cdots)$}\\ \reducesto & \reason{\slab{Capture}, $\{\OpCase{\Choose}{\Unit}{k'} \mapsto \cdots\} \in H_c$, $\EC'' = (\Handle\;[\,]\;\cdots)$}\\
& (k'~\True \concat k'~\False) \concat k~\False, \qquad \text{where $k' = \qq{\cont_{\Record{\EC'';H_c}}}$}\\ & (k'~\True \concat k'~\False) \concat k~\False, \qquad \text{where $k' = \qq{\cont_{\Record{\EC'';H_c}}}$}\\
\reducesto& \reason{\slab{Resume} with $\True$}\\ \reducesto& \reason{\slab{Resume} with $\True$}\\
@ -5031,7 +5073,7 @@ first-order parameter.
The function $\dec{odd}$ expects its environment to provide it with an The function $\dec{odd}$ expects its environment to provide it with an
implementation of a single operation of type $\Int \to \Bool$. The implementation of a single operation of type $\Int \to \Bool$. The
body of $\dec{odd}$ invokes, or queries, this operation twice with body of $\dec{odd}$ invokes, or queries, this operation twice with
arguments $0$ and $1$, respectively. The results are then tested using
arguments $0$ and $1$, respectively. The results are tested using
exclusive-or. exclusive-or.
Now, let us implement the environment for $\dec{odd}$. Now, let us implement the environment for $\dec{odd}$.
@ -5051,18 +5093,16 @@ Now, let us implement the environment for $\dec{odd}$.
\el \el
\] \]
% %
We allow ourselves to use recursive data types to make the example
concise. Type $\dec{Dialogue}$ represents the dialogue between
$\dec{odd}$ and its parameter. The data structure is a standard binary
tree with two constructors: $!$ constructs a leaf holding a value of
type $\Int$ and $?$ constructs an interior node holding a value of
type $\Bool$ and two subtrees. The function $\dec{env}$ implements the
environment that $\dec{odd}$ will be run in. This function evaluates
its parameter $m$ under $\Catchcont$ which injects the operation
$f$. If $m$ returns, then the left component gets tagged with $!$,
otherwise the argument to the operation $q$ gets tagged with a $?$
along with the subtrees constructed by the two recursive applications
of $\dec{env}$.
Type $\dec{Dialogue}$ represents the dialogue between $\dec{odd}$ and
its parameter. The data structure is a standard binary tree with two
constructors: $!$ constructs a leaf holding a value of type $\Int$ and
$?$ constructs an interior node holding a value of type $\Bool$ and
two subtrees. The function $\dec{env}$ implements the environment that
$\dec{odd}$ will be run in. This function evaluates its parameter $m$
under $\Catchcont$ which injects the operation $f$. If $m$ returns,
then the left component gets tagged with $!$, otherwise the argument
to the operation $q$ gets tagged with a $?$ along with the subtrees
constructed by the two recursive applications of $\dec{env}$.
% %
% The primitive structure of catchcont makes it somewhat fiddly to % The primitive structure of catchcont makes it somewhat fiddly to
% program with it compared to other control operators as we have to % program with it compared to other control operators as we have to
@ -5071,6 +5111,14 @@ of $\dec{env}$.
The following derivation gives the high-level details of how The following derivation gives the high-level details of how
evaluation proceeds. evaluation proceeds.
% %
\begin{figure}
\begin{center}
\scalebox{1.3}{\SXORTwoModel}
\end{center}
\caption{Visualisation of the result obtained by
$\dec{env}~\dec{odd}$.}\label{fig:decision-tree-cc}
\end{figure}
%
\begin{derivation} \begin{derivation}
&\dec{env}~\dec{odd}\\ &\dec{env}~\dec{odd}\\
\reducesto^+ & \reason{$\beta$-reduction}\\ \reducesto^+ & \reason{$\beta$-reduction}\\
@ -5107,14 +5155,6 @@ Figure~\ref{fig:decision-tree-cc} visualises this result as a binary
tree. The example here does not make use of the `continuation tree. The example here does not make use of the `continuation
component', the interested reader may consult \citet{LongleyW08} for component', the interested reader may consult \citet{LongleyW08} for
an example usage. an example usage.
%
\begin{figure}
\begin{center}
\scalebox{1.3}{\SXORTwoModel}
\end{center}
\caption{Visualisation of the result obtained by
$\dec{env}~\dec{odd}$.}\label{fig:decision-tree-cc}
\end{figure}
% \subsection{Second-class control operators} % \subsection{Second-class control operators}
% Coroutines, async/await, generators/iterators, amb. % Coroutines, async/await, generators/iterators, amb.
@ -19209,7 +19249,7 @@ language~\cite{Plotkin77}. It is also well known that the presence of
control features or local state enables observational distinctions control features or local state enables observational distinctions
that cannot be made in a purely functional setting: for instance, that cannot be made in a purely functional setting: for instance,
there are programs involving call/cc that detect the order in which a there are programs involving call/cc that detect the order in which a
(call-by-name) `+' operation evaluates its arguments
(call-by-name) `$+$' operation evaluates its arguments
\citep{CartwrightF92}. Such operations are `non-functional' in the \citep{CartwrightF92}. Such operations are `non-functional' in the
sense that their output is not determined solely by the extension of sense that their output is not determined solely by the extension of
their input (seen as a mathematical function their input (seen as a mathematical function

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