Uncurried CPS translation

thesis.bib

@@ -169,7 +169,7 @@
                Matija Pretnar},
   title     = {On the Expressive Power of User-Defined Effects: Effect Handlers,
                Monadic Reflection, Delimited Control},
-  journal   = {Proc. ACM Program. Lang.},
+  journal   = {{PACMPL}},
   volume    = {1},
   number    = {ICFP},
   articleno = {13},
@@ -178,6 +178,19 @@
   year      = {2017}
 }
 
+@article{ForsterKLP19,
+  author    = {Yannick Forster and
+               Ohad Kammar and
+               Sam Lindley and
+               Matija Pretnar},
+  title     = {On the expressive power of user-defined effects: Effect handlers,
+               monadic reflection, delimited control},
+  journal   = {J. Funct. Program.},
+  volume    = {29},
+  pages     = {e15},
+  year      = {2019}
+}
+
 # Eff
 @article{BauerP15,
   author    = {Andrej Bauer and
@@ -538,6 +551,14 @@
   year      = {1975}
 }
 
+@techreport{Steele78,
+  title = {{RABBIT}: A Compiler for {SCHEME}},
+  author = {Guy Steele},
+  number = {AITR-474},
+  institution = {{INRIA}},
+  year = {1978},
+}
+
 @inproceedings{DanvyF90,
   author    = {Olivier Danvy and
                Andrzej Filinski},
@@ -547,6 +568,17 @@
   year      = {1990}
 }
 
+@article{DanvyF92,
+  author    = {Olivier Danvy and
+               Andrzej Filinski},
+  title     = {Representing Control: {A} Study of the {CPS} Transformation},
+  journal   = {Mathematical Structures in Computer Science},
+  volume    = {2},
+  number    = {4},
+  pages     = {361--391},
+  year      = {1992}
+}
+
 @book{Appel92,
   author    = {Andrew W. Appel},
   title     = {Compiling with Continuations},

thesis.tex

@@ -2121,7 +2121,7 @@ translate to value terms in the target.
 \label{fig:cps-cbv}
 \end{figure}
 
-\subsection{First-order CPS translations of effect handlers}
+\section{First-order CPS translations of effect handlers}
 \label{sec:fo-cps}
 
 The translation of a computation term by the basic CPS translation in
@@ -2157,7 +2157,7 @@ resumption is modified to ensure that the context of the original
 operation invocation can be reinstated when the resumption is invoked.
 %
 
-\subsubsection{Curried translation}
+\subsection{Curried translation}
 \label{sec:first-order-curried-cps}
 
 We first consider a curried CPS translation that builds the
@@ -2173,19 +2173,14 @@ have been $\eta$-reduced.  The translation of operations and handlers
 is as follows.
 %
 \begin{equations}
-\cps{\Do\;\ell\;V} &=& \lambda k.\lambda h.h~\Record{\ell,\Record{\cps{V}, \lambda x.k~x~h}} \\
+\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} &=& \cps{M}~\cps{\hret}~\cps{\hops}, ~\text{where} \smallskip\\
+\cps{\Handle \; M \; \With \; H} &\defas& \cps{M}~\cps{\hret}~\cps{\hops}, ~\text{where} \smallskip\\
-\multicolumn{3}{@{}l@{}}{
+\cps{\{ \Return \; x \mapsto N \}} &\defas& \lambda x . \lambda h . \cps{N} \\
-\begin{eqs}
-\qquad
-\cps{\{ \Return \; x \mapsto N \}} &=& \lambda x . \lambda h . \cps{N} \\
 \cps{\{ \ell~p~r \mapsto N_\ell \}_{\ell \in \mathcal{L}}}
-&=&
+&\defas&
 \lambda \Record{z,\Record{p,r}}. \Case~z~
            \{ (\ell \mapsto \cps{N_\ell})_{\ell \in \mathcal{L}}; y \mapsto \hforward(y,p,r) \} \\
-           \hforward(y,p,r) &=& \lambda k. \lambda h. h\,\Record{y,\Record{p, \lambda x.\,r\,x\,k\,h}}
+\hforward(y,p,r) &\defas& \lambda k. \lambda h. h\,\Record{y,\Record{p, \lambda x.\,r\,x\,k\,h}}
-\end{eqs}
-}
 \end{equations}
 %
 The translation of $\Do\;\ell\;V$ abstracts over the current pure
@@ -2207,9 +2202,8 @@ dispatches on encoded operations, and in the default case forwards to
 an outer handler.
 %
 In the forwarding case, the resumption is extended by the parent
-continuation pair in order to reinstate the handler stack, thereby
+continuation pair to ensure that an eventual invocation of the
-ensuring subsequent invocations of the same operation are handled
+resumption reinstates the handler stack.
-uniformly.
 
 The translation of complete programs feeds the translated term the
 identity pure continuation (which discards its handler argument), and
@@ -2223,52 +2217,72 @@ Conceptually, this translation encloses the translated program in a
 top-level handler with an empty collection of operation clauses and an
 identity return clause.
 
-% There are three shortcomings of this initial translation that we
+We may regard this particular CPS translation as being simple, because
-% address below.
+it requires virtually no extension to the CPS translation for
+$\BCalc$. However, this translation suffers from two deficiencies
+which makes it unviable in practice.
 
-% \begin{itemize}
+\begin{enumerate}
-% \item First, it is not \emph{properly tail-recursive}~\citep{DanvyF92,
+  \item The image of the translation is not \emph{properly
-%   Steele78} due to the curried representation of the continuation
+    tail-recursive}~\citep{DanvyF92,Steele78}, and as a result the
-%   stack, as a result the image of the translation is not stackless,
+    image is not stackless, meaning it cannot readily be used as the
-%   which makes it problematic to implement using a trampoline in, say,
+    basis for an implementation. This deficiency is essentially due to
-%   JavaScript. (Properly tail-recursion CPS translations ensure that
+    the curried representation of the continuation stack.
-%   all calls to functions and continuations are in tail position, hence
-%   there is no need to maintain a stack.) We rectify this issue with an
-%   explicit list representation in the next subsection.
 
-% \item Second, it yields administrative redexes (redexes that could be
+  \item The image of the translation yields context independent
-%   reduced statically). We will rectify this with a higher-order
+    administrative redexes, i.e. redexes that could be reduced
-%   one-pass translation in
+    statically. This is a classic problem with CPS translation, which
-%   Section~\ref{sec:higher-order-uncurried-cps}.
+    is typically dealt with by introducing a second pass to clean up
+    the image~\cite{DanvyF92}.
+\end{enumerate}
 
-% \item Third, this translation cannot cope with shallow handlers. The
+The following example readily illustrates both issues.
-%   pure continuations $k$ are abstract and include the return clause of
+%
-%   the corresponding handler. Shallow handlers require that the return
+\begin{equations}
-%   clause of a handler is discarded when one of its operations is
+\pcps{\Return\;\Record{}}
-%   invoked. We will fix this in Section~\ref{sec:pure-as-stack}, where we
+              &=& (\lambda k.k\,\Record{})\,(\lambda x.\lambda h.x)\,(\lambda \Record{z,\_}.\Absurd\,z)  \\
-%   will represent pure continuations as explicit stacks.
+     &\reducesto& ((\lambda x.\lambda h.x)\,\Record{})\,(\lambda \Record{z,\_}.\Absurd\,z)\\
-% \end{itemize}
+     &\reducesto& (\lambda h.\Record{})\,(\lambda \Record{z,\_}.\Absurd\,z)\\
+     &\reducesto& \Record{} \\
+\end{equations}
+%
+The first reduction is administrative: it has nothing to do with the
+dynamic semantics of the original term and there is no reason not to
+eliminate it statically. We will address this issue in
+Section~\ref{sec:higher-order-cps}. The second and third reductions
+simulate handling $\Return\;\Record{}$ at the top level. The second
+reduction partially applies $\lambda x.\lambda h.x$ to $\Record{}$,
+which must return a value so that the third reduction can be applied:
+evaluation is not tail-recursive. We will address this issue in
+Section~\ref{sec:first-order-uncurried-cps}.
+%
+The lack of tail-recursion is also apparent in our relaxation of
+fine-grain call-by-value in Figure~\ref{fig:cps-cbv-target}: the
+function position of an application can be a computation, and the
+calculus makes use of evaluation contexts.
 
-% To illustrate the first two issues, consider the following example:
+Nevertheless, we can show that the image of this CPS translation
-% \begin{equations}
+simulates the preimage. Due to the presence of administrative
-% \pcps{\Return\;\Record{}}
+reductions, the simulation is not on the nose, but instead up to
-%               &=& (\lambda k.k\,\Record{})\,(\lambda x.\lambda h.x)\,(\lambda \Record{z,\_}.\Absurd\,z)  \\
+congruence.
-%      &\reducesto& ((\lambda x.\lambda h.x)\,\Record{})\,(\lambda \Record{z,\_}.\Absurd\,z)\\
+%
-%      &\reducesto& (\lambda h.\Record{})\,(\lambda \Record{z,\_}.\Absurd\,z)\\
+For reduction in the untyped target calculus we write
-%      &\reducesto& \Record{} \\
+$\reducesto_{\textrm{cong}}$ for the smallest relation containing
-% \end{equations}
+$\reducesto$ that is closed under the term formation constructs.
-% The first reduction is administrative: it has nothing to do with the
+%
-% dynamic semantics of the original term and there is no reason not to
+\begin{theorem}[Simulation]
-% eliminate it statically. The second and third reductions simulate
+  \label{thm:fo-simulation}
-% handling $\Return\;\Record{}$ at the top level. The second reduction
+  If $M \reducesto N$ then $\pcps{M} \reducesto_{\textrm{cong}}^+
-% partially applies $\lambda x.\lambda h.x$ to $\Record{}$, which must
+  \pcps{N}$.
-% return a value so that the third reduction can be applied: evaluation
+\end{theorem}
-% is not tail-recursive.
+
-% %
+\begin{proof}
-% The lack of tail-recursion is also apparent in our relaxation of
+The result follows by composing a call-by-value variant of
-% fine-grain call-by-value in Figure~\ref{fig:cps-cbv-target}: the
+\citeauthor{ForsterKLP19}'s translation from effect handlers to
-% function position of an application can be a computation, and the
+delimited continuations~\citeyearpar{ForsterKLP19} with
-% calculus makes use of evaluation contexts.
+\citeauthor{MaterzokB12}'s CPS translation for delimited
+continuations~\citeyearpar{MaterzokB12}.
+\end{proof}
 
 % \paragraph*{Remark}
 % We originally derived this curried CPS translation for effect handlers
@@ -2277,31 +2291,6 @@ identity return clause.
 % \citeauthor{MaterzokB12}'s CPS translation for delimited
 % continuations~\citeyearpar{MaterzokB12}.
 
-% \medskip
-
-\dhil{Discuss deficiencies of the curried translation}
-\dhil{State simulation result}
-% Because of the administrative reductions, simulation is not on the
-% nose, but instead up to congruence.
-% %
-% For reduction in the untyped target calculus we write
-% $\reducesto_{\textrm{cong}}$ for the smallest relation containing
-% $\reducesto$ that is closed under the term formation constructs.
-% %
-% \begin{theorem}[Simulation]
-%   \label{thm:fo-simulation}
-%   If $M \reducesto N$ then $\pcps{M} \reducesto_{\textrm{cong}}^+
-%   \pcps{N}$.
-% \end{theorem}
-
-% \begin{proof}
-% The result follows by composing a call-by-value variant of
-% \citeauthor{ForsterKLP17}'s translation from effect handlers to
-% delimited continuations~\citeyearpar{ForsterKLP17} with
-% \citeauthor{MaterzokB12}'s CPS translation for delimited
-% continuations~\citeyearpar{MaterzokB12}.
-% \end{proof}
-
 \chapter{Abstract machine semantics}
 
 \part{Expressiveness}