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Administrative resumptions example.

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Daniel Hillerström 5 years ago
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  1. 2
      macros.tex
  2. 170
      thesis.tex

2
macros.tex
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@ -460,6 +460,8 @@
\newcommand{\Ready}{\dec{Ready}}
\newcommand{\Blocked}{\dec{Blocked}}
\newcommand{\init}{\dec{init}}
\newcommand{\Reader}{\dec{Reader}}
\newcommand{\Other}{\dec{Other}}
%%
%% Some control operators

170
thesis.tex
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@ -8450,38 +8450,7 @@ follow the notation and style of item \ref{en:ch-cps-HLA20}.
\section{Initial target calculus}
\label{sec:target-cps}
The syntax, semantics, and syntactic sugar for the target calculus
$\UCalc$ is given in Figure~\ref{fig:cps-cbv-target}. The calculus
largely amounts to an untyped variation of $\BCalc$, specifically
we retain the syntactic distinction between values ($V$) and
computations ($M$).
%
The values ($V$) comprise lambda abstractions ($\lambda x.M$),
% recursive functions ($\Rec\,g\,x.M$),
empty tuples ($\Record{}$), pairs ($\Record{V,W}$), and first-class
labels ($\ell$).
%
Computations ($M$) comprise values ($V$), applications ($M~V$), pair
elimination ($\Let\; \Record{x, y} = V \;\In\; N$), label elimination
($\Case\; V \;\{\ell \mapsto M; x \mapsto N\}$), and explicit marking
of unreachable code ($\Absurd$). A key difference from $\BCalcRec$ is
that the function position of an application is allowed to be a
computation (i.e., the application form is $M~W$ rather than
$V~W$). Later, when we refine the initial CPS translation we will be
able to rule out this relaxation.
We give a standard small-step evaluation context-based reduction
semantics. Evaluation contexts comprise the empty context and function
application.
To make the notation more lightweight, we define syntactic sugar for
variant values, record values, list values, let binding, variant
eliminators, and record eliminators. We use pattern matching syntax
for deconstructing variants, records, and lists. For desugaring
records, we assume a failure constant $\ell_\bot$ (e.g. a divergent
term) to cope with the case of pattern matching failure.
\begin{figure}
\flushleft
\textbf{Syntax}
@ -8533,10 +8502,41 @@ term) to cope with the case of pattern matching failure.
\caption{Untyped target calculus for the CPS translations.}
\label{fig:cps-cbv-target}
\end{figure}
%
The syntax, semantics, and syntactic sugar for the target calculus
$\UCalc$ is given in Figure~\ref{fig:cps-cbv-target}. The calculus
largely amounts to an untyped variation of $\BCalc$, specifically
we retain the syntactic distinction between values ($V$) and
computations ($M$).
%
The values ($V$) comprise lambda abstractions ($\lambda x.M$),
% recursive functions ($\Rec\,g\,x.M$),
empty tuples ($\Record{}$), pairs ($\Record{V,W}$), and first-class
labels ($\ell$).
%
Computations ($M$) comprise values ($V$), applications ($M~V$), pair
elimination ($\Let\; \Record{x, y} = V \;\In\; N$), label elimination
($\Case\; V \;\{\ell \mapsto M; x \mapsto N\}$), and explicit marking
of unreachable code ($\Absurd$). A key difference from $\BCalcRec$ is
that the function position of an application is allowed to be a
computation (i.e., the application form is $M~W$ rather than
$V~W$). Later, when we refine the initial CPS translation we will be
able to rule out this relaxation.
The reduction semantics follows the trend of the previous reduction
semantics in the sense that it is a small-step context-based reduction
semantics. Evaluation contexts comprise the empty context and function
application.
\dhil{Most of the primitives are Church encodable. Discuss the value
of treating them as primitive rather than syntactic sugar (target
languages such as JavaScript has similar primitives).}
To make the notation more lightweight, we define syntactic sugar for
variant values, record values, list values, let binding, variant
eliminators, and record eliminators. We use pattern matching syntax
for deconstructing variants, records, and lists. For desugaring
records, we assume a failure constant $\ell_\bot$ (e.g. a divergent
term) to cope with the case of pattern matching failure.
% \dhil{Most of the primitives are Church encodable. Discuss the value
% of treating them as primitive rather than syntactic sugar (target
% languages such as JavaScript has similar primitives).}
\section{Transforming fine-grain call-by-value}
\label{sec:cps-cbv}
@ -8627,7 +8627,8 @@ operation and its argument. The effect continuation then either
handles the operation, invoking the resumption as appropriate, or
forwards the operation to an outer handler. In the latter case, the
resumption is modified to ensure that the context of the original
operation invocation can be reinstated when the resumption is invoked.
operation invocation can be reinstated upon invocation of the
resumption.
%
\subsection{Curried translation}
@ -8699,11 +8700,13 @@ translation also suffers two displeasing properties which makes it
unviable in practice.
\begin{enumerate}
\item The image of the translation is not \emph{properly
tail-recursive}~\citep{DanvyF92,Steele78}, and as a result the
image is not stackless, meaning it cannot readily be used as the
basis for an implementation. This deficiency is essentially due to
the curried representation of the continuation stack.
\item The image of the translation is not \emph{properly
tail-recursive}~\citep{Danvy06,DanvyF92,Steele78}, meaning not
every function application occur in tail position in the image, and
thus the image is not stackless. Consequently, the translation
cannot readily be used as the basis for an implementation. This
deficiency is essentially due to the curried representation of the
continuation stack.
\item The image of the translation yields static administrative
redexes, i.e. redexes that could be reduced statically. This is a
@ -8806,9 +8809,9 @@ As a first step we may restrict the syntax of the target calculus such
that the term in function position must be a value. With this
restriction the syntax of $\UCalc$ implements the property that any
term constructor features at most one computation term, and this
computation term always appears in tail position. Thus this
restriction suffices to ensure that every function application will be
in tail position.
computation term always appears in tail position. This restriction
suffices to ensure that every function application will be in tail
position.
%
Figure~\ref{fig:refined-cps-cbv-target} contains the adjustments to
syntax and semantics of $\UCalc$. The target calculus now supports
@ -8816,13 +8819,12 @@ both unary and binary application forms. As we shall see shortly,
binary application turns out be convenient when we enrich the notion
of continuation. Both application forms are comprised only of value
terms. As a result the dynamic semantics of $\UCalc$ no longer makes
use of evaluation contexts, hence we remove them altogether. The
reduction rule $\usemlab{App_1}$ applies to unary application and it
is the same as the $\usemlab{App}$-rule in
Figure~\ref{fig:cps-cbv-target}. The new $\usemlab{App_2}$-rule
applies to binary application: it performs a simultaneous substitution
of the arguments $V$ and $W$ for the parameters $x$ and $y$,
respectively, in the function body $M$.
use of evaluation contexts. The reduction rule $\usemlab{App_1}$
applies to unary application and it is the same as the
$\usemlab{App}$-rule in Figure~\ref{fig:cps-cbv-target}. The new
$\usemlab{App_2}$-rule applies to binary application: it performs a
simultaneous substitution of the arguments $V$ and $W$ for the
parameters $x$ and $y$, respectively, in the function body $M$.
%
These changes to $\UCalc$ immediately invalidate the curried
@ -8844,7 +8846,7 @@ representation of the runtime handler stack.
The change of continuation representation means the CPS translation
for effect handlers is no longer a conservative extension. The
translation is adjusted as follows to account for the new
representation of continuations.
representation.
%
\begin{equations}
\cps{-} &:& \CompCat \to \UCompCat\\
@ -8962,7 +8964,7 @@ a more clever implementation of resumptions.
\subsection{Resumptions as explicit reversed stacks}
\label{sec:first-order-explicit-resump}
%
\dhil{Show an example involving administrative redexes produced by resumptions}
% \dhil{Show an example involving administrative redexes produced by resumptions}
%
Thus far resumptions have been represented as functions, and
forwarding has been implemented using function composition. The
@ -8970,10 +8972,68 @@ composition of resumption gives rise to unnecessary dynamic
administrative redexes as function composition necessitates the
introduction of an additional lambda abstraction.
%
As an illustration of how and where these administrative redexes arise
let us consider an example with an operation $\Ask : \Unit \opto \Int$
and two handlers $H_\Reader$ and $H_\Other$ such that
$H_\Reader^\Ask = \{\OpCase{\Ask}{\Unit}{r} \mapsto r~42\}$ whilst
$\Ask \not\in \dom(H_\Other)$. We denote the top-level continuation by
$ks_\top$.
%
% \[
% \bl
% \Reader \defas \{\Ask : \Unit \opto \Int\} \smallskip\\
% H_{\Reader} : \alpha \eff \Reader \Harrow \alpha, \{ \OpCase{\Ask}{\Unit}{r} \mapsto r~42 \} \in H_{\Reader}\\
% H_{\Other} : \alpha \eff \varepsilon \Harrow \alpha, \Ask \not\in \dom(H_{\Reader})
% \el
% \]
%
\begin{derivation}
&\pcps{\Handle\; (\Handle\; \Do\;\Ask\,\Unit\;\With\;H_{\Other})\;\With\;H_{\Reader}}\\
% =& \reason{definition of $\cps{-}$}\\
% % &\lambda ks.\cps{\Handle\; \Do\;\Ask\,\Unit\;\With\;H_{\Other}}(\cps{H_{\Reader}^\mret} \cons H_{\Reader}^\mops \cons ks)\\
% % =& \reason{}\\
% &(\lambda ks.(\lambda ks'.\cps{\Do\;\Ask\,\Unit}(\cps{H_{\Other}^\mret} \cons \cps{H_{\Other}^\mops} \cons ks'))(\cps{H_{\Reader}^\mret} \cons H_{\Reader}^\mops \cons ks))\,ks_\top\\
=& \reason{definition of $\pcps{-}$}\\
&(\lambda ks.
\bl
(\lambda ks'.
\bl
(\lambda (k \cons h \cons ks'').h\,\Record{\Ask,\Record{\Unit,\lambda x.\lambda ks'''.k~x~(h \cons ks''')}}\,ks'')\\
(\cps{H_{\Other}^\mret} \cons \cps{H_{\Other}^\mops} \cons ks'))
\el\\
(\cps{H_{\Reader}^\mret} \cons H_{\Reader}^\mops \cons ks))\,ks_\top
\el\\
% \reducesto^\ast& \reason{apply continuation}\\
% & (\lambda (k \cons h \cons ks'').h\,\Record{\Ask,\Record{\Unit,\lambda x.\lambda ks'''.k~x~(h \cons ks''')}})(\cps{H_{\Other}^\mret} \cons \cps{H_{\Other}^\mops} \cons \cps{H_{\Reader}^\mret} \cons H_{\Reader}^\mops \cons ks_\top)\\
\reducesto^\ast & \reason{multiple applications of \usemlab{App}, activation of $H_\Other$}\\
& \cps{H_{\Other}^\mops}\,\Record{\Ask,\Record{\Unit,\lambda x.\lambda ks'''.\cps{H_{\Other}^\mret}~x~(\cps{H_{\Other}^\mops} \cons ks''')}}\,(\cps{H_{\Reader}^\mret} \cons H_{\Reader}^\mops \cons ks_\top)\\
\reducesto^\ast & \reason{effect forwarding to $H_\Reader$}\\
& \bl
H_{\Reader}^\mops\,\Record{\Ask,\Record{\Unit,\lambda x.\lambda ks''. r_\dec{admin}~x~(H_{\Reader}^\mret \cons H_{\Reader}^\mops \cons ks'')}}\,ks_\top\\
\where~r_\dec{admin} \defas \lambda x.\lambda
ks'''.\cps{H_{\Other}^\mret}~x~(\cps{H_{\Other}^\mops} \cons ks''')
\el\\
\reducesto^\ast & \reason{invocation of the administrative resumption} \\
& r_\dec{admin}~42~(H_{\Reader}^\mret \cons H_{\Reader}^\mops \cons ks_\top)\\
\reducesto^\ast & \reason{invocation of the resumption of the operation invocation site}\\
& \cps{H_{\Other}^\mret}~42~(\cps{H_{\Other}^\mops} \cons
H_{\Reader}^\mret \cons H_{\Reader}^\mops \cons ks_\top)
\end{derivation}
%
Effect forwarding introduces the administrative abstraction
$r_{\dec{admin}}$, whose sole purpose is to forward the interpretation
of the operation to the operation invocation site. In a certain sense
$r_{\dec{admin}}$ is a sort of identity frame. The insertion of
identities ought to always trigger the alarm bells as an identity
computation is typically extraneous.
%
The amount of identity frames being generated scales linearly with the
number of handlers the operation needs to pass through before reaching
a suitable handler.
We can avoid generating these administrative redexes by applying a
variation of the technique that we used in the previous section to
uncurry the curried CPS translation.
We can avoid generating these administrative resumption redexes by
applying a variation of the technique that we used in the previous
section to uncurry the curried CPS translation.
%
Rather than representing resumptions as functions, we move to an
explicit representation of resumptions as \emph{reversed} stacks of

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