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CPS for parameterised handlers

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Daniel Hillerström
2021-03-03 22:41:00 +00:00
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thesis.tex
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@@ -10293,28 +10293,28 @@ $M \reducesto^\ast V$ if and only if $\pcps{M} \reducesto^\ast \pcps{V}$.
\end{corollary} \end{corollary}
\section{Transforming parameterised handlers} \section{Transforming parameterised handlers}
\begin{figure}[t]
\textbf{Continuation reductions}
% %
\begin{reductions} \begin{figure}
\usemlab{KAppNil} & % \textbf{Continuation reductions}
\kapp \; (\dRecord{\dnil, \dRecord{q, v, e}} \dcons \dhk) \, V &\reducesto& v \dapp \dRecord{q,V} \dapp \dhk \\ % %
\usemlab{KAppCons} & % \begin{reductions}
\kapp \; (\dRecord{\dlf \dcons \dlk, h} \dcons \dhk) \, V &\reducesto& \dlf \dapp V \dapp (\dRecord{\dlk, h} \dcons \dhk) \\ % \usemlab{KAppNil} &
\end{reductions} % \kapp \; (\dRecord{\dnil, \dRecord{q, v, e}} \dcons \dhk) \, V &\reducesto& v \dapp \dRecord{q,V} \dapp \dhk \\
% % \usemlab{KAppCons} &
\textbf{Resumption reductions} % \kapp \; (\dRecord{\dlf \dcons \dlk, h} \dcons \dhk) \, V &\reducesto& \dlf \dapp V \dapp (\dRecord{\dlk, h} \dcons \dhk) \\
% % \end{reductions}
\[ % %
\ba{@{}l@{\quad}l@{}} % \textbf{Resumption reductions}
\usemlab{Res}^\ddag & % %
\Let\;r=\Res(V_n \dcons \dots \dcons V_1 \dcons \dnil)\;\In\;N \reducesto \\ % \[
&\quad N[\dlam x\,\dhk. \bl\Let\;\dRecord{fs, \dRecord{q,\vhret, \vhops}}\dcons \dhk' = \dhk\;\In\\ % \ba{@{}l@{\quad}l@{}}
\kapp\;(V_1 \dcons \dots \dcons V_n \dcons \dRecord{fs, \dRecord{q,\vhret, \vhops}} \dcons \dhk')\;x/r]\el % \usemlab{Res}^\ddag &
\\ % \Let\;r=\Res(V_n \dcons \dots \dcons V_1 \dcons \dnil)\;\In\;N \reducesto \\
\ea % &\quad N[\dlam x\,\dhk. \bl\Let\;\dRecord{fs, \dRecord{q,\vhret, \vhops}}\dcons \dhk' = \dhk\;\In\\
\] % \kapp\;(V_1 \dcons \dots \dcons V_n \dcons \dRecord{fs, \dRecord{q,\vhret, \vhops}} \dcons \dhk')\;x/r]\el
% \\
% \ea
% \]
% %
\textbf{Computations} \textbf{Computations}
% %
@@ -10374,64 +10374,86 @@ $M \reducesto^\ast V$ if and only if $\pcps{M} \reducesto^\ast \pcps{V}$.
\label{fig:param-cps} \label{fig:param-cps}
\end{figure} \end{figure}
\paragraph{Continuation Passing Style} To accommodate parameterised Generalised continuations provide a versatile implementation strategy
handlers, we generalise the notion of continuations once more. A for effect handlers as exemplified in the previous section. In this
continuation becomes a triple consisting of a pure continuation, section we add further emphasis on the versatility of generalised
effect continuation, and the handler parameter. This effectively continuations by demonstrating how to adapt the continuation structure
amounts to explicit state passing as the parameter value gets threaded to accommodate parameterised handlers. In order to support
through every function application. The pure continuation invocation parameterised handlers, each effect continuation must store the
rule $\usemlab{KAppNil}$ is slightly modified to account for the third current value of the handler parameter. Thus, an effect continuation
component. becomes a triple consisting of the parameter, return clause, and
operation clause(s). Furthermore, the return clause gets transformed
into a binary function, that takes the current value of the handler
parameter as its first argument and the return value of the handled
computation as its second argument. Similarly, the operation clauses
are transformed into a binary function, that takes the handler
parameter first and the operation package second. This strategy
effectively amounts to explicit state passing as the parameter value
gets threaded through every handler continuation
function. Operationally, the pure continuation invocation rule
$\usemlab{KAppNil}$ requires a small adjustment to account for the
handler parameter.
% %
\[ \[
\kapp \; (\dRecord{\dnil, \dRecord{\vhret, \vhops, q}} \dcons \dhk) \, V \reducesto \vhret \dapp \Record{V,q} \dapp \dhk \kapp \; (\dRecord{\dnil, \dRecord{q, \vhret, \vhops}} \dcons \dhk) \, V \reducesto \vhret \dapp \Record{q,V} \dapp \dhk
\] \]
% %
The pure continuation $v$ is now applied to a pair consisting of the The pure continuation $v$ is now applied to a pair consisting of the
return value $V$ and the current value of the handler parameter current value of the handler parameter $q$ and the return value
$q$. The resumption rule $\usemlab{Res}$ is adapted to update the $V$. Similarly, the resumption rule $\usemlab{Res}$ must also be
value of the handler parameter. adapted to update the value of the handler parameter.
% %
\[ \[
\bl \bl
\Let\;r=\Res\,(\dRecord{\vhret,\vhops,\_} \dcons \dots \dcons \dhf_1 \dcons \dnil)\;\In\;N\reducesto\\ \Let\;r=\Res^\ddag\,(\dRecord{q,\vhret,\vhops} \dcons \dots \dcons V_1 \dcons \dnil)\;\In\;N\reducesto\\
\qquad N[\lambda \Record{x,p}\,\dhk.\kapp\;(\dhf_1 \dcons \dots \dcons \dRecord{\vhret,\vhops,p} \dcons \dhk)\;x/r] \qquad N[\dlam \dRecord{q',x}\,\dhk.\kapp\;(V_1 \dcons \dots \dcons \dRecord{q',\vhret,\vhops} \dcons \dhk)\;x/r]
\el \el
\] \]
% %
Thus, the parameter of the top-most handler in the resumption stack is The rule is not much different from the original $\usemlab{Res}$
ignored and replaced by a new value $p$. The translation is updated rule. The difference is that this rule unpacks the current handler
accordingly to account for the triple structure. This involves parameter $q$ along with the return clause, $\vhret$, and operation
updating all the parts that previously dynamically decomposed and clauses, $\vhops$. The reduction constructs a resumption function,
statically recomposed frames to now include the additional state whose first parameter $q'$ binds the updated value of the handler
parameter. The key updated translation clauses are shown in parameter. The $q'$ is packaged with the original $\vhret$ and
Figure~\ref{fig:param-cps}. $\vhops$ such that the next activation of the handler gets the
parameter value $q'$ rather than $q$.
The CPS translation is updated accordingly to account for the triple
effect continuation structure. This involves updating all the parts
that previously dynamically decomposed and statically recomposed
effect continuation frames to now include the additional state
parameter. The cases that need to be updated are shown in
Figure~\ref{fig:param-cps}. We write $\xi$ to denote static handler
parameters.
% %
The translation of $\Do$ invokes the effect continuation The translation of $\Let$ unpacks and repacks the effect continuation
$\reify \svhops$ with a pair consisting of the operation and the value to maintain the continuation length invariant. The translation of
of the handler parameter. The parameter is also pushed onto the $\Do$ invokes the effect continuation $\reify \svhops$ with a triple
reversed resumption stack. This is necessary to account for the case consisting of the value of the handler parameter, the operation, and
where the effect continuation $\reify \svhops$ does not handle the operation payload. The parameter is also pushed onto the reversed
operation $\ell$. resumption stack. This is necessary to account for the case where the
effect continuation $\reify \svhops$ does not handle operation $\ell$.
% An alternative option is to push the parameter back % An alternative option is to push the parameter back
% on the resumption stack during effect forwarding. However that means % on the resumption stack during effect forwarding. However that means
% the resumption stack will be nonuniform as the top element sometimes % the resumption stack will be nonuniform as the top element sometimes
% will be a pair. % will be a pair.
% %
The translation of the return and operation clauses yields functions The translation of the return and operation clauses are parameterised
that take a pair as input in addition to the current continuation. The by the name of the binder for the handler parameter. Each translation
forwarding case is adjusted in much the same way as the translation yields functions that take a pair as input in addition to the current
for $\Do$. The current continuation $k$ is destructed in order to continuation. The forwarding case is adjusted in the same way as the
identify the next effect continuation $\vhops$ and its parameter translation for $\Do$. The current continuation $k$ is deconstructed
$q$. Then $\vhops$ is invoked with the updated resumption stack and in order to identify the next effect continuation $\vhops$ and its
the value of its parameter $q$. parameter $q$. Then $\vhops$ is invoked with the updated resumption
stack and the value of its parameter $q$.
The amended CPS translation for parameterised handlers is not a zero %
cost translation for shallow and ordinary deep handlers as they will The top-level translation adds a `dummy' unit value, which is ignored
have to thread a ``dummy'' parameter value through. In contrast, the by both the pure continuation and effect continuation.% The amended
abstract machine implementation of parameterised handlers does not % CPS translation for parameterised handlers is not a zero cost
impose an overhead on shallow and deep handlers. % translation for shallow and ordinary deep handlers as they will have
% to thread a ``dummy'' parameter value through.
\section{Related work} \section{Related work}
\label{sec:cps-related-work} \label{sec:cps-related-work}