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Initial stab at higher-order CPS translation.

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Daniel Hillerström 5 years ago
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  1. 12
      macros.tex
  2. 182
      thesis.tex

12
macros.tex
View File

@ -212,7 +212,6 @@
\newcommand{\sRecord}[1]{\static{\langle}#1\static{\rangle}}
\newcommand{\dRecord}[1]{\dynamic{\langle}#1\dynamic{\rangle}}
%\newcommand{\sP}{\mathcal{P}}
\newcommand{\sQ}{\mathcal{Q}}
\newcommand{\sV}{\mathcal{V}}
\newcommand{\sW}{\mathcal{W}}
@ -237,12 +236,23 @@
\newcommand{\dhf}{q} % handler frames
\newcommand{\dhk}{k} % handler continuations
\newcommand{\dhkr}{rk} % reverse handler continuations
\newcommand{\dLet}{\dynamic{\Let}}
\newcommand{\dIn}{\dynamic{\In}}
% static
\newcommand{\slf}{\phi} % let frames
\newcommand{\slk}{\sigma} % let continuations
\newcommand{\shf}{\theta} % handler frames
\newcommand{\shk}{\kappa} % handler continuations
\newcommand{\sLet}{\static{\Let}}
\newcommand{\sIn}{\static{\In}}
\newcommand{\sk}{\kappa}
\newcommand{\sks}{\mathit{\kappa s}}
\newcommand{\sh}{\eta}
\newcommand{\sx}{\varepsilon}
\newcommand{\sP}{\mathcal{P}}
\newcommand{\VS}{VS}
\newcommand{\Vmap}{\keyw{vmap}}
\newcommand{\Vmapsnd}{\keyw{vmapsnd}}
\newcommand{\Fun}{\keyw{fun}}

182
thesis.tex
View File

@ -2193,8 +2193,10 @@ have been $\eta$-reduced. The translation of operations and handlers
is as follows.
%
\begin{equations}
\cps{-} &:& \CompCat \to \UCompCat\\
\cps{\Do\;\ell\;V} &\defas& \lambda k.\lambda h.h~\Record{\ell,\Record{\cps{V}, \lambda x.k~x~h}} \\
\cps{\Handle \; M \; \With \; H} &\defas& \cps{M}~\cps{\hret}~\cps{\hops} \smallskip\\
\cps{\Handle \; M \; \With \; H} &\defas& \cps{M}~\cps{\hret}~\cps{\hops} \medskip\\
\cps{-} &:& \HandlerCat \to \UCompCat\\
\cps{\{ \Return \; x \mapsto N \}} &\defas& \lambda x . \lambda h . \cps{N} \\
\cps{\{ \ell~p~r \mapsto N_\ell \}_{\ell \in \mathcal{L}}}
&\defas&
@ -2230,7 +2232,8 @@ identity pure continuation (which discards its handler argument), and
an effect continuation that is never intended to be called.
%
\begin{equations}
\pcps{M} &=& \cps{M}~(\lambda x.\lambda h.x)~(\lambda \Record{z,\_}.\Absurd~z) \\
\pcps{-} &:& \CompCat \to \UCompCat\\
\pcps{M} &\defas& \cps{M}~(\lambda x.\lambda h.x)~(\lambda \Record{z,\_}.\Absurd~z) \\
\end{equations}
%
Conceptually, this translation encloses the translated program in a
@ -2391,12 +2394,14 @@ translation is adjusted as follows to account for the new
representation of continuations.
%
\begin{equations}
\cps{-} &:& \CompCat \to \UCompCat\\
\cps{\Return~V} &\defas& \lambda (k \cons ks).k\,\cps{V}\,ks \\
\cps{\Let~x \revto M~\In~N} &\defas& \lambda (k \cons ks).\cps{M}((\lambda x.\lambda ks'.\cps{N}(k \cons ks')) \cons ks)
\smallskip \\
\cps{\Do\;\ell\;V} &\defas& \lambda (k \cons h \cons ks).h\,\Record{\ell,\Record{\cps{V}, \lambda x.\lambda ks'.k\,x\,(h \cons ks')}}\,ks
\smallskip \\
\cps{\Handle \; M \; \With \; H} &\defas& \lambda ks . \cps{M} (\cps{\hret} \cons \cps{\hops} \cons ks) \smallskip\\
\cps{\Handle \; M \; \With \; H} &\defas& \lambda ks . \cps{M} (\cps{\hret} \cons \cps{\hops} \cons ks) \medskip\\
\cps{-} &:& \HandlerCat \to \UCompCat\\
\cps{\{\Return \; x \mapsto N\}} &\defas& \lambda x.\lambda ks.\Let\; (h \cons ks') = ks \;\In\; \cps{N}\,ks'
\\
\cps{\{\ell \; p \; r \mapsto N_\ell\}_{\ell \in \mathcal{L}}}
@ -2409,7 +2414,8 @@ representation of continuations.
\hforward((y,p,r),ks) &\defas& \bl
\Let\; (k' \cons h' \cons ks') = ks \;\In\; \\
h'\,\Record{y, \Record{p, \lambda x.\lambda ks''.\,r\,x\,(k' \cons h' \cons ks'')}}\,ks'\\
\el \smallskip\\
\el \medskip\\
\pcps{-} &:& \CompCat \to \UCompCat\\
\pcps{M} &\defas& \cps{M}~((\lambda x.\lambda ks.x) \cons (\lambda \Record{z,\Record{p,r}}. \lambda ks.\,\Absurd~z) \cons \nil)
\end{equations}
%
@ -2479,9 +2485,10 @@ Rather than representing resumptions as functions, we move to an
explicit representation of resumptions as \emph{reversed} stacks of
pure and effect continuations. By choosing to reverse the order of
pure and effect continuations, we can construct resumptions
efficiently using regular cons-lists. We convert these reversed stacks
to actual functions on demand using a special
$\Let\;r=\Res\,V\;\In\;N$ computation term that reduces as follows.
efficiently using regular cons-lists. We augment the syntax and
semantics of $\UCalc$ with a computation term
$\Let\;r=\Res\,V\;\In\;N$ which allow us to convert these reversed
stacks to actual functions on demand.
%
\begin{reductions}
\usemlab{Res}
@ -2502,9 +2509,12 @@ modified to account for the change in representation of
resumptions.
%
\begin{equations}
\cps{-} &:& \CompCat \to \UCompCat\\
\cps{\Do\;\ell\;V}
&\defas& \lambda k \cons h \cons ks.\,h\, \Record{\ell,\Record{\cps{V}, h \cons k \cons \nil}}\, ks
\\
\medskip\\
%
\cps{-} &:& \HandlerCat \to \UCompCat\\
\cps{\{(\ell \; p \; r \mapsto N_\ell)_{\ell \in \mathcal{L}}\}}
&\defas& \bl
\lambda \Record{z,\Record{p,rs}}.\lambda ks.\Case \;z\; \{
@ -2525,6 +2535,162 @@ resumption stack with the current continuation pair.
% Since we have only changed the representation of resumptions, the
% translation of top-level programs remains the same.
\subsection{Higher-order translation for deep effect handlers}
\label{sec:higher-order-uncurried-deep-handlers-cps}
%
\begin{figure}
%
\textbf{Static patterns and static lists}
%
\begin{syntax}
\slab{Static patterns} &\sP& ::= & \sk \scons \sP \mid \sks \\
\slab{Static lists} &\VS& ::= & V \scons \VS \mid \reflect V \\
\end{syntax}
%
\textbf{Reification}
%
\begin{equations}
\reify (V \scons \VS) &\defas& V \dcons \reify \VS \\
\reify \reflect V &\defas& V \\
\end{equations}
%
\textbf{Values}
%
\begin{displaymath}
\begin{eqs}
\cps{-} &:& \ValCat \to \UValCat\\
\cps{x} &\defas& x \\
\cps{\lambda x.M} &\defas& \dlam x\,ks.\cps{M} \sapp \reflect ks \\
\cps{\Lambda \alpha.M} &\defas& \dlam z\,ks.\cps{M} \sapp \reflect ks \\
\cps{\Record{}} &\defas& \Record{} \\
\cps{\Record{\ell = V; W}} &\defas& \Record{\ell = \cps{V}; \cps{W}} \\
\cps{\ell~V} &\defas& \ell~\cps{V} \\
\end{eqs}
\end{displaymath}
%
\textbf{Computations}
%
\begin{equations}
\cps{-} &:& (\CompCat \times (\UValCat \to \UCompCat)^\ast) \to \UCompCat\\
\cps{V\,W} &\defas& \slam \sks.\cps{V} \dapp \cps{W} \dapp \reify \sks \\
\cps{V\,T} &\defas& \slam \sks.\cps{V} \dapp \Record{} \dapp \reify \sks \\
\cps{\Let\; \Record{\ell=x;y} = V \; \In \; N} &\defas& \slam \sks.\Let\; \Record{\ell=x;y} = \cps{V} \; \In \; \cps{N} \sapp \sks \\
\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\} \\
\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{\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{\Do\;\ell\;V} &\defas& \slam \sk \scons \sh \scons \sks.\sh \dapp
(\ell~\Record{\cps{V}, \sh \dcons \sk \dcons \dnil}) \dapp \reify \sks \\
\cps{\Handle \; M \; \With \; H} &\defas& \slam \sks . \cps{M} \sapp (\cps{\hret} \scons \cps{\hops} \scons \sks),~\text{where}
\smallskip \\
%
\end{equations}
%
\textbf{Handler definitions}
%
\begin{equations}
\cps{-} &:& \HandlerCat \to \UCompCat\\
\cps{\{\Return \; x \mapsto N\}} &\defas& \dlam x\, ks.\dLet\; (h \dcons ks') = ks \;\dIn\; \cps{N} \sapp \reflect ks'
\\
\cps{\{(\ell \; p \; r \mapsto N_\ell)_{\ell \in \mathcal{L}}\}}
&\defas&
\bl
\dlam z \, ks.\Case \;z\; \{
\bl
(\ell~\Record{p, s} \mapsto \dLet\;r=\Fun\,s \;\dIn\; \cps{N_{\ell}} \sapp \reflect ks)_{\ell \in \mathcal{L}};\,
y \mapsto \hforward \} \\
\el \\
\el \\
\hforward &\defas& \bl
\dLet\; (k' \dcons h' \dcons ks') = ks \;\dIn\; \\
\Vmap\,(\dlam\Record{p, s}\,(k \dcons ks).k\,\Record{p, h' \dcons k' \dcons s}\,ks)\,y\,(h' \dcons ks') \\
\el
\end{equations}
\textbf{Top level program}
%
\begin{equations}
\pcps{M} &=& \cps{M} \sapp ((\dlam x\,ks.x) \scons (\dlam z\,ks.\Absurd~z) \scons \snil) \\
\end{equations}
\vspace{-1em}
\caption{Higher-order uncurried CPS translation of $\HCalc$}
\label{fig:cps-higher-order-uncurried}
\end{figure}
%
We now adapt our uncurried CPS translation to a higher-order one-pass
CPS translation~\cite{DanvyF90} that partially evaluates
administrative redexes at translation time.
%
Following Danvy and Nielsen~\cite{DanvyN03}, we adopt a two-level
lambda calculus notation to distinguish between \emph{static} lambda
abstraction and application in the meta language and \emph{dynamic}
lambda abstraction and application in the target language.
%
The idea is that redexes marked as static are reduced as part of the
translation (at compile time), whereas those marked as dynamic are
reduced at runtime.
%
The CPS translation is given in
Figure~\ref{fig:cps-higher-order-uncurried}.
An overline denotes a static syntax constructor and an underline
denotes a dynamic syntax constructor. In order to facilitate this
notation we write application explicitly as an infix ``at'' symbol
($@$). We assume the meta language is pure and hence respects the
usual $\beta$ and $\eta$ equivalences. We extend the overline and
underline notation to distinguish between static and dynamic let
bindings.
The reify operator $\reify$ maps static lists to dynamic ones and the
reflect constructor $\reflect$ allows dynamic lists to be treated as
static.
%
We use list pattern matching in the meta language.
%
\begin{equations}
(\slam (\sk \scons \sP).\sM) \sapp (V \scons \VS)
&=& \sLet\; \sk = V \;\sIn\; (\slam \sP.\sM) \sapp \VS \\
(\slam (\sk \scons \sP).\sM) \sapp \reflect V
&=& \dLet\; (k \dcons ks) = V \;\dIn\;
\sLet\; \sk = k \;\sIn\; (\slam \sP.\sM) \sapp \reflect ks \\
\end{equations}
Here we let $\sM$ range over meta language expressions.
Now the target calculus is refined so that all lambda abstractions and
applications take two arguments, the $\usemlab{Lift}$ rule is removed,
and the $\usemlab{App}$ rule is replaced by the following reduction
rule.
%
\begin{reductions}
\usemlab{AppTwo} & (\dlam x\,ks . \, M) \dapp V \dapp W &\reducesto& M[V/x, W/ks] \\
\end{reductions}
%
We add an extra dummy argument to the translation of type lambda
abstractions and applications in order to ensure that all dynamic
functions take exactly two arguments.
%
The single argument lambdas and applications from the first-order
uncurried translation are still present, but now they are all static.
In order to reason about the behaviour of the \semlab{Handle-op} rule,
which is defined in terms of an evaluation context, we extend the CPS
translation to evaluation contexts.
%
\begin{displaymath}
\ba{@{}r@{~}c@{~}r@{~}l@{}}
\cps{[~]} &=& \slam \sks. &\sks \\
\cps{\Let\; x \revto \EC \;\In\; N} &=& \slam \sk \scons \sks.&\cps{\EC} \sapp ((\dlam x\,ks.\cps{N} \sapp (k \scons \reflect ks)) \scons \sks) \\
\cps{\Handle\; \EC \;\With\; H} &=& \slam \sks. &\cps{\EC} \sapp (\cps{\hret} \scons \cps{\hops} \scons \sks) \\
\ea
\end{displaymath}
The following lemma is the characteristic property of the CPS
translation on evaluation contexts.
%
This allows us to focus on the computation contained within
an evaluation context.
\chapter{Abstract machine semantics}
\part{Expressiveness}

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