dhil/phd-dissertation
1
0
Fork 0
mirror of https://github.com/dhil/phd-dissertation synced 2026-03-13 11:08:25 +00:00
Code Issues Releases Wiki Activity

Maintain static continuation invariant in the higher-order CPS translation for deep handlers.

Browse Source
This commit is contained in:
Daniel Hillerström
2020-09-04 19:00:01 +01:00
parent d00a0271ad
commit ebbaa18c39
1 changed files with 97 additions and 38 deletions
Download Patch File Download Diff File Expand all files Collapse all files

135
thesis.tex
View File

@@ -2569,8 +2569,9 @@ resumption stack with the current continuation pair.
\begin{eqs} \begin{eqs}
\cps{-} &:& \ValCat \to \UValCat\\ \cps{-} &:& \ValCat \to \UValCat\\
\cps{x} &\defas& x \\ \cps{x} &\defas& x \\
\cps{\lambda x.M} &\defas& \dlam x\,ks.\cps{M} \sapp \reflect ks \\ \cps{\lambda x.M} &\defas& \dlam x\,ks.\Let\;(k \dcons h \dcons ks') = ks \;\In\;\cps{M} \sapp (\reflect k \scons \reflect h \scons \reflect ks') \\
\cps{\Lambda \alpha.M} &\defas& \dlam z\,ks.\cps{M} \sapp \reflect ks \\ % \cps{\Rec\,g\,x.M} &\defas& \Rec\;f\,x\,ks.\cps{M} \sapp \reflect ks\\
\cps{\Lambda \alpha.M} &\defas& \dlam z\,ks.\Let\;(k \dcons h \dcons ks') = ks \;\In\;\cps{M} \sapp (\reflect k \scons \reflect h \scons \reflect ks') \\
\cps{\Record{}} &\defas& \Record{} \\ \cps{\Record{}} &\defas& \Record{} \\
\cps{\Record{\ell = V; W}} &\defas& \Record{\ell = \cps{V}; \cps{W}} \\ \cps{\Record{\ell = V; W}} &\defas& \Record{\ell = \cps{V}; \cps{W}} \\
\cps{\ell~V} &\defas& \ell~\cps{V} \\ \cps{\ell~V} &\defas& \ell~\cps{V} \\
@@ -2587,10 +2588,15 @@ resumption stack with the current continuation pair.
\cps{\Case~V~\{\ell~x \mapsto M; y \mapsto N\}} &\defas& \cps{\Case~V~\{\ell~x \mapsto M; y \mapsto N\}} &\defas&
\slam \sks.\Case~\cps{V}~\{\ell~x \mapsto \cps{M} \sapp \sks; y \mapsto \cps{N} \sapp \sks\} \\ \slam \sks.\Case~\cps{V}~\{\ell~x \mapsto \cps{M} \sapp \sks; y \mapsto \cps{N} \sapp \sks\} \\
\cps{\Absurd~V} &\defas& \slam \sks.\Absurd~\cps{V} \\ \cps{\Absurd~V} &\defas& \slam \sks.\Absurd~\cps{V} \\
\cps{\Return~V} &\defas& \slam \sk \scons \sks.\sk \dapp \cps{V} \dapp \reify \sks \\ \cps{\Return~V} &\defas& \slam \sk \scons \sks.\reify \sk \dapp \cps{V} \dapp \reify \sks \\
\cps{\Let~x \revto M~\In~N} &\defas& \slam \sk \scons \sks.\cps{M} \sapp ((\dlam x\,ks.\cps{N} \sapp (\sk \scons \reflect ks)) \scons \sks) \\ \cps{\Let~x \revto M~\In~N} &\defas& \slam \sk \scons \sks.\cps{M} \sapp
(\reflect (\dlam x\,ks.
\ba[t]{@{}l}
\Let\;(h \dcons ks') = ks\;\In\\
\cps{N} \sapp (\sk \scons \reflect h \scons \reflect ks')) \scons \sks)
\ea\\
\cps{\Do\;\ell\;V} \cps{\Do\;\ell\;V}
&\defas& \slam \sk \scons \sh \scons \sks.\sh \dapp \Record{\ell,\Record{\cps{V}, \sh \dcons \sk \dcons \dnil}} \dapp \reify \sks\\ &\defas& \slam \sk \scons \sh \scons \sks.\reify \sh \dapp \Record{\ell,\Record{\cps{V}, \reify \sh \dcons \reify \sk \dcons \dnil}} \dapp \reify \sks\\
\cps{\Handle \; M \; \With \; H} &\defas& \slam \sks . \cps{M} \sapp (\cps{\hret} \scons \cps{\hops} \scons \sks) \cps{\Handle \; M \; \With \; H} &\defas& \slam \sks . \cps{M} \sapp (\cps{\hret} \scons \cps{\hops} \scons \sks)
% %
\end{equations} \end{equations}
@@ -2599,18 +2605,27 @@ resumption stack with the current continuation pair.
% %
\begin{equations} \begin{equations}
\cps{-} &:& \HandlerCat \to \UCompCat\\ \cps{-} &:& \HandlerCat \to \UCompCat\\
\cps{\{\Return \; x \mapsto N\}} &\defas& \dlam x\, ks.\Let\; (h \dcons ks') = ks \;\In\; \cps{N} \sapp \reflect ks' \cps{\{\Return \; x \mapsto N\}} &\defas& \dlam x\, ks.
\ba[t]{@{~}l}
\Let\; (h \dcons k \dcons h' \dcons ks') = ks \;\In\\
\cps{N} \sapp (\reflect k \scons \reflect h' \scons \reflect ks')
\ea
\\ \\
\cps{\{(\ell \; p \; r \mapsto N_\ell)_{\ell \in \mathcal{L}}\}} \cps{\{(\ell \; p \; r \mapsto N_\ell)_{\ell \in \mathcal{L}}\}}
&\defas& \bl &\defas& \bl
\dlam \Record{z,\Record{p,rs}}\,ks.\Case \;z\; \{ \dlam \Record{z,\Record{p,rs}}\,ks.\Case \;z\; \{
\bl \ba[t]{@{}l@{}c@{~}l}
(\ell \mapsto \Let\;r=\Res\;rs \;\In\; \cps{N_{\ell}} \sapp \reflect ks)_{\ell \in \mathcal{L}};\,\\ &(\ell \mapsto&
y \mapsto \hforward((y,p,rs),ks) \} \\ \ba[t]{@{}l}
\el \\ \Let\;r=\Res\;rs \;\In\\
\Let\;(k \dcons h \dcons ks') = ks \;\In\\
\cps{N_{\ell}} \sapp (\reflect k \scons \reflect h \scons \reflect ks'))_{\ell \in \mathcal{L}};
\ea\\
&y \mapsto& \hforward((y,p,rs),ks) \} \\
\ea \\
\el \\ \el \\
\hforward((y,p,rs),ks) \hforward((y,p,rs),ks)
&\defas&\Let\; (k' \dcons h' \dcons ks') = ks \;\In\; h'\,\Record{y,\Record{p,h' \dcons k' \dcons rs}} \,ks' &\defas&\Let\; (k' \dcons h' \dcons ks') = ks \;\In\; h' \dapp \Record{y,\Record{p,h' \dcons k' \dcons rs}} \dapp ks'
\end{equations} \end{equations}
% %
\textbf{Top level program} \textbf{Top level program}
@@ -2648,10 +2663,10 @@ constructors in the respective calculi we mark static constructors
with a {\color{blue}$\overline{\text{blue overline}}$}, whilst dynamic with a {\color{blue}$\overline{\text{blue overline}}$}, whilst dynamic
constructors are marked with a constructors are marked with a
{\color{red}$\underline{\text{red underline}}$}. The principal idea is {\color{red}$\underline{\text{red underline}}$}. The principal idea is
that redexes marked as static are reduced as part of the translation that redexes marked as static are reduced as part of the translation,
(at compile time), whereas those marked as dynamic are reduced at whereas those marked as dynamic are reduced at runtime. To facilitate
runtime. To facilitate this notation we write application in both this notation we write application explicitly using an infix ``at''
calculi with an infix ``at'' symbol ($@$). symbol ($@$) in both calculi.
\paragraph{Static terms} \paragraph{Static terms}
% %
@@ -2663,23 +2678,23 @@ the following productions.
% %
\begin{syntax} \begin{syntax}
\slab{Static patterns} &\sP \in \SPatCat &::=& \sks \mid \sk \scons \sP\\ \slab{Static patterns} &\sP \in \SPatCat &::=& \sks \mid \sk \scons \sP\\
\slab{Static values} & \sV, \sW \in \SValCat &::=& V \mid \reflect V \mid \sV \scons \sW \mid \slam \sP. \sM\\ \slab{Static values} & \sV, \sW \in \SValCat &::=& \reflect V \mid \sV \scons \sW \mid \slam \sP. \sM\\
\slab{Static computations} & \sM \in \SCompCat &::=& \sV \sapp \sW \mid \sV \dapp V \dapp W \slab{Static computations} & \sM \in \SCompCat &::=& \sV \mid \sV \sapp \sW \mid \sV \dapp V \dapp W
\end{syntax} \end{syntax}
% %
The patterns comprise only static list deconstructing. We let $\sP$ The patterns comprise only static list deconstructing. We let $\sP$
range over static patterns. range over static patterns.
% %
The static values comprise dynamic values, reflected dynamic values, The static values comprise reflected dynamic values, static lists, and
static lists, and static lambda abstractions. We let $\sV, \sW$ range static lambda abstractions. We let $\sV, \sW$ range over meta language
over meta language values; by convention we shall use variables $\sk$ values; by convention we shall use variables $\sk$ to denote
to denote statically known pure continuations, $\sh$ to denote statically known pure continuations, $\sh$ to denote statically known
statically known effect continuations, and $\sks$ to denote statically effect continuations, and $\sks$ to denote statically known
known continuations. continuations.
% %
We shall use $\sM$ to range over static computations, which comprise We shall use $\sM$ to range over static computations, which comprise
static application and binary dynamic application of a static value to static values, static application and binary dynamic application of a
two dynamic values. static value to two dynamic values.
% %
Static computations are subject to the following equational axioms. Static computations are subject to the following equational axioms.
% %
@@ -2705,7 +2720,7 @@ $\reify \sV$ by induction on their structure.
\ea \ea
\] \]
% %
\dhil{Need to spell out that static pattern matching may induce dynamic (administrative) reductions} %\dhil{Need to spell out that static pattern matching may induce dynamic (administrative) reductions}
% %
\paragraph{Higher-order translation} \paragraph{Higher-order translation}
% %
@@ -2715,17 +2730,56 @@ same as the refined first-order uncurried CPS translation, although
the notation is slightly more involved due to the separation of static the notation is slightly more involved due to the separation of static
and dynamic parts. and dynamic parts.
% %
The translation on values is mostly homomorphic on the syntax The translation function for computation terms is staged: for any
constructors, except for $\lambda$- and $\Lambda$-abstractions which given computation term the translation yields a static function, which
are translated in the exact same way, because the target calculus has can be applied to a static continuation to ultimately yield a
no notion of types. As per usual continuation passing style practice, $\UCalc$-computation term. Staging is key to eliminating static
the translation of a lambda abstraction yields a dynamic binary lambda administrative redexes as any static function may perform static
abstraction, where the second parameter is intended to be the computation by manipulating its provided static continuation
continuation parameter. However, the translation slightly diverge from
the usual practice in the translation of the body as the result of
$\cps{M}$ is applied statically, rather than dynamically, to the a binary higher-order
(reflected) continuation parameter. function; it is higher-order because its second parameter is a list of
static continuation functions.
% A major difference that has a large cosmetic effect on the
% presentation of the translation is that we maintain the invariant that
% the statically known stack ($\sk$) always contains at least one frame,
% consisting of a triple
% $\sRecord{\reflect V_{fs}, \sRecord{\reflect V_{ret}, \reflect
% V_{ops}}}$ of reflected dynamic pure frame stacks, return
% handlers, and operation handlers. Maintaining this invariant ensures
% that all translations are uniform in whether they appear statically
% within the scope of a handler or not, and this simplifies our
% correctness proof. To maintain the invariant, any place where a
% dynamically known stack is passed in (as a continuation parameter
% $k$), it is immediately decomposed using a dynamic language $\Let$ and
% repackaged as a static value with reflected variable
% names. Unfortunately, this does add some clutter to the translation
% definition, as compared to the translations above. However, there is a
% payoff in the removal of administrative reductions at run time. The
% translations presented by \citet{HillerstromLAS17} and
% \citet{HillerstromL18} did not do this decomposition and repackaging
% step, which resulted in additional administrative reductions in the
% translation due to the translations of $\Let$ and $\Do$ being passed
% dynamic continuations when they were expecting statically known ones.
% %
The translation on values is mostly homomorphic on the syntax
constructors, except for $\lambda$-abstractions and
$\Lambda$-abstractions which are translated in the exact same way,
because the target calculus has no notion of types. As per usual
continuation passing style practice, the translation of a lambda
abstraction yields a dynamic binary lambda abstraction, where the
second parameter is intended to be the continuation
parameter.
%
However, the translation slightly diverge from the usual
practice in the translation of the body as the result of $\cps{M}$ is
applied statically, rather than dynamically, to the (reflected)
continuation parameter.
%
The reason is that the translation on computations is staged: for any The reason is that the translation on computations is staged: for any
given computation term the translation yields a static function, which given computation term the translation yields a static function, which
can be applied to a static continuation to ultimately produce a can be applied to a static continuation to ultimately produce a
@@ -2736,7 +2790,7 @@ for instance done in the translation of $\Let$.
% %
\dhil{Spell out why this translation qualifies as `higher-order'.} \dhil{Spell out why this translation qualifies as `higher-order'.}
Let us again revisit the example from Let us revisit the example from
Section~\ref{sec:first-order-curried-cps} to see that the higher-order Section~\ref{sec:first-order-curried-cps} to see that the higher-order
translation eliminates the static redex at translation time. translation eliminates the static redex at translation time.
% %
@@ -2812,7 +2866,12 @@ translation to evaluation contexts.
\begin{equations} \begin{equations}
\cps{-} &:& \EvalCat \to \SValCat\\ \cps{-} &:& \EvalCat \to \SValCat\\
\cps{[~]} &\defas& \slam \sks.\sks \\ \cps{[~]} &\defas& \slam \sks.\sks \\
\cps{\Let\; x \revto \EC \;\In\; N} &\defas& \slam \sk \scons \sks.\cps{\EC} \sapp ((\dlam x\,ks.\cps{N} \sapp (k \scons \reflect ks)) \scons \sks) \\ \cps{\Let\; x \revto \EC \;\In\; N} &\defas& \slam \sk \scons \sks.\cps{\EC} \sapp
(\reflect(\dlam x\,ks.
\ba[t]{@{}l}
\Let\;(h \dcons ks') = ks\;\In\;\\
\cps{N} \sapp (\sk \scons \reflect h \scons \reflect ks')) \scons \sks)
\ea\\
\cps{\Handle\; \EC \;\With\; H} &\defas& \slam \sks. \cps{\EC} \sapp (\cps{\hret} \scons \cps{\hops} \scons \sks) \cps{\Handle\; \EC \;\With\; H} &\defas& \slam \sks. \cps{\EC} \sapp (\cps{\hret} \scons \cps{\hops} \scons \sks)
\end{equations} \end{equations}
% %