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      thesis.tex

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@ -4129,7 +4129,7 @@ realisable function in \BCalc{} is effect-free and total.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
% %
% \begin{corollary} % \begin{corollary}
@ -4146,7 +4146,7 @@ some other computation $M'$, then $M'$ is also well typed.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
\section{A primitive effect: recursion} \section{A primitive effect: recursion}
@ -4787,7 +4787,7 @@ getting stuck on an unhandled operation.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
% %
\begin{theorem}[Subject reduction] \begin{theorem}[Subject reduction]
@ -4796,7 +4796,7 @@ getting stuck on an unhandled operation.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
\section{Composing \UNIX{} with effect handlers} \section{Composing \UNIX{} with effect handlers}
@ -7190,7 +7190,7 @@ the basic properties of $\HCalc$.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
% %
\begin{theorem}[Subject reduction] \begin{theorem}[Subject reduction]
@ -7199,7 +7199,7 @@ the basic properties of $\HCalc$.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
\subsection{\UNIX{}-style pipes} \subsection{\UNIX{}-style pipes}
@ -7747,7 +7747,7 @@ The metatheoretic properties of $\HCalc$ carries over to $\HPCalc$.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
% %
\begin{theorem}[Subject reduction] \begin{theorem}[Subject reduction]
@ -7756,7 +7756,7 @@ The metatheoretic properties of $\HCalc$ carries over to $\HPCalc$.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
\subsection{Process synchronisation} \subsection{Process synchronisation}
@ -11367,12 +11367,35 @@ If $\typc{}{M : A}{E}$ and $M \reducesto^+ N \not\reducesto$, then $M
\chapter{Interdefinability of effect handlers} \chapter{Interdefinability of effect handlers}
\label{ch:deep-vs-shallow} \label{ch:deep-vs-shallow}
In this section we show that shallow handlers and general recursion
On the surface, shallow handlers seem to offer more flexibility than
deep handlers as they do not enforce a particular recursion scheme
over effectful computations. An interesting hypothesis worth
investigating is whether this flexibility is a mere programming
convenience or whether it enables shallow handlers to implement
programs that would otherwise be impossible to implement with deep
handlers. Put slightly different, the hypothesis to test is whether
handlers can implement one another. To test this sort of hypothesis we
first need to pin down what it means for `something to be able to
implement something else'.
%
For example in Section~\ref{sec:pipes} I asserted that shallow
handlers provide the natural basis for implementing pipes, suggesting
that an implementation based on deep handlers would be fiddly. It
turns out that deep handlers offer a direct implementation of pipes by
shifting recursion from terms to the level of types (the interested
reader may consult either \citet{KammarLO13} or \citet{HillerstromL18}
for more details).
Through the lens of \emph{typability-preserving macro-expressiveness},
which, in our particular setting, asks whether there exists a local
transformation that can transform one kind of handler into another
kind of handler, whilst preserving typability in the image.
In this chapter we show that shallow handlers and general recursion
can simulate deep handlers up to congruence, and that deep handlers can simulate deep handlers up to congruence, and that deep handlers
can simulate shallow handlers up to administrative reductions. The
latter construction generalises the example of pipes implemented using
deep handlers that we gave in
Section~\ref{sec:pipes}.
can simulate shallow handlers up to administrative reductions. % The
% latter construction generalises the example of pipes implemented
% using deep handlers that we gave in Section~\ref{sec:pipes}.
% %
\paragraph{Relation to prior work} The results in this chapter has \paragraph{Relation to prior work} The results in this chapter has
been published previously in the following papers. been published previously in the following papers.
@ -11428,43 +11451,60 @@ $\Return$-clauses is the identity, and thus ignores the handler
name. However, the translation of operation clauses uses the name to name. However, the translation of operation clauses uses the name to
simulate a deep resumption by guarding invocations of the shallow simulate a deep resumption by guarding invocations of the shallow
resumption $r$ with $h$. resumption $r$ with $h$.
% \paragraph{Example} We illustrate the translation $\dstrans{-}$ on the
% EXAMPLE handler from Section~\ref{sec:strategies} (recall that the
% variable $m$ is bound to the input computation). The example here is
% reproduced in ANF notation.
% \[
% \ba{@{~}l@{~}l@{}l}
% &\mathcal{S}&\left\llbracket
% \ba[m]{@{~}l}
% \Handle\; m~\Unit\; \With\\
% \quad
% \ba{@{}l@{~}c@{~}l}
% \Return~x &\mapsto& \Return\;x\\
% \Move~\Record{\_;n}~resume &\mapsto& \Let\;y \revto ps~n\;\In\; resume~y
% \ea
% \ea\right\rrbracket =\\\\
% &
% &\ba[m]{@{}l}
% \hspace{-1ex}
% \left(
% \ba[m]{@{~}l}
% \Rec~h~f.
% \ba[t]{@{~}l}
% \ShallowHandle\; f~\Unit\; \With\\
% \quad
% \ba{@{}l@{~}c@{~}l}
% \Return~x &\mapsto& \Return\; x\\
% \Move~\Record{\_;n}~resume &\mapsto&\\
% \quad\ba[t]{@{~}l@{}}
% \Let\; r \revto \Return\; \lambda x. h (\lambda\Unit.resume~x)\;\In\\
% \Let\; y \revto ps~n\; \In\; r~y
% \ea
% \ea
% \ea
% \ea\right)~(\lambda \Unit. m~\Unit)
% \ea
% \ea
% \]\\
In order to exemplify the translation, let us consider a variation of
the $\environment$ handler from Section~\ref{sec:tiny-unix-env}, which
handles an operation $\Ask : \UnitType \opto \Int$.
%
\[
\ba{@{~}l@{~}l}
&\mathcal{D}\left\llbracket
\ba[m]{@{}l}
\Handle\;\Do\;\Ask\,\Unit + \Do\;\Ask\,\Unit\;\With\\
\quad\ba[m]{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& \Return\;x\\
\OpCase{\Ask}{\Unit}{r} &\mapsto& r~42
\ea
\ea
\right\rrbracket \medskip\\
=& \bl
(\Rec\;env~f.
\bl
\ShallowHandle\;f\,\Unit\;\With\\
\quad\ba[t]{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& \Return\;x\\
\OpCase{\Ask}{\Unit}{r} &\mapsto&
\bl
\Let\;r \revto \Return\;\lambda x.env~(\lambda\Unit.r~x)\;\In\\
r~42)~(\lambda\Unit.\Do\;\Ask\,\Unit + \Do\;\Ask\,\Unit)
\el
\ea
\el
\el
\ea
\]
%
The deep semantics are simulated by generating the name $env$ for the
shallow handlers and recursively apply the handler under the modified
resumption.
The translation commutes with substitution and preserves typability.
%
\begin{lemma}\label{lem:dstrans-subst}
Let $\sigma$ denote a substitution. The translation $\dstrans{-}$
commutes with substitution, i.e.
%
\[
\dstrans{V}\dstrans{\sigma} = \dstrans{V\sigma},\quad
\dstrans{M}\dstrans{\sigma} = \dstrans{M\sigma},\quad
\dstrans{H}\dstrans{\sigma} = \dstrans{H\sigma}.
\]
%
\end{lemma}
%
\begin{proof}
By induction on the structures of $V$, $M$, and $H$.
\end{proof}
\begin{theorem} \begin{theorem}
If $\Delta; \Gamma \vdash M : C$ then $\dstrans{\Delta}; \dstrans{\Gamma} \vdash If $\Delta; \Gamma \vdash M : C$ then $\dstrans{\Delta}; \dstrans{\Gamma} \vdash
@ -11472,7 +11512,7 @@ If $\Delta; \Gamma \vdash M : C$ then $\dstrans{\Delta}; \dstrans{\Gamma} \vdash
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
In order to obtain a simulation result, we allow reduction in the In order to obtain a simulation result, we allow reduction in the
@ -11547,22 +11587,26 @@ environments.
\el \el
\] \]
% %
Each shallow handler is encoded as a deep handler that returns a pair
of thunks. The first forwards all operations, acting as the identity
on computations. The second interprets a single operation before
reverting to forwarding.
As evident from the translation of handler types, each shallow handler
is encoded as a deep handler that returns a pair of thunks. It is
worth noting that the handler construction is actually pure, yet we
need to annotate the pair with the translated effect signature
$\sdtrans{E_2}$, because the calculus has no notion of effect
subtyping. The first component of the pair forwards all operations,
acting as the identity on computations. The second component
interprets a single operation before reverting to forwarding.
% %
The following example illustrates an application of the translation on
the $\Pipe$ operator from Section~\ref{sec:pipes} applied to the
producer and consumer pair
$\Record{\Rec\;ones\,\Unit.\Do\;\Yield~1;ones\,\Unit;\lambda\Unit.\Do\;\Await\,\Unit
+ \Do\;\Await}$
The following example illustrates the translation on an instance of
the $\Pipe$ operator from Section~\ref{sec:pipes} using the consumer
computation $\Do\;\Await\,\Unit + \Do\;\Await\,\Unit$ and the
suspended producer computation
$\Rec\;ones\,\Unit.\Do\;\Yield~1;ones\,\Unit$.
% %
\[ \[
\ba{@{~}l@{~}l} \ba{@{~}l@{~}l}
&\mathcal{D}\left\llbracket &\mathcal{D}\left\llbracket
\ba[m]{@{}l} \ba[m]{@{}l}
\ShallowHandle\;(\lambda\Unit.\Do\;\Await\,\Unit + \Do\;\Await\,\Unit)\,\Unit\;\With\\
\ShallowHandle\;\Do\;\Await\,\Unit + \Do\;\Await\,\Unit\;\With\\
\quad\ba[m]{@{~}l@{~}c@{~}l} \quad\ba[m]{@{~}l@{~}c@{~}l}
\Return\;x &\mapsto& \Return\;x\\ \Return\;x &\mapsto& \Return\;x\\
\OpCase{\Await}{\Unit}{r} &\mapsto& \Copipe\,\Record{r;\Rec\;ones\,\Unit.\Do\;\Yield~1;ones\,\Unit} \OpCase{\Await}{\Unit}{r} &\mapsto& \Copipe\,\Record{r;\Rec\;ones\,\Unit.\Do\;\Yield~1;ones\,\Unit}
@ -11612,45 +11656,6 @@ The translation commutes with substitution and preserves typability.
\begin{proof} \begin{proof}
By induction on the structures of $V$, $M$, and $H$. By induction on the structures of $V$, $M$, and $H$.
\end{proof} \end{proof}
% \paragraph{Example} We demonstrate the translation $\sdtrans{-}$ on
% the $\Pipe$ handler from Section~\ref{sec:shallow-handlers-tutorial}
% (recall that the variables $c$ and $p$ are bound to the consumer and
% producer functions respectively). The example is reproduced in ANF
% notation.
% %
% \[
% \ba{@{~}l@{~}l@{}l}
% &\mathcal{D}&\left\llbracket
% \ba[m]{@{~}l}
% \lambda\Record{p;c}.\ShallowHandle\; c\,\Unit \;\With\; \\
% \quad
% \ba[m]{@{}l@{~}c@{~}l@{}}
% \Return~x &\mapsto& \Return\; x \\
% \OpCase{\Await}{\Unit}{r} &\mapsto& \Copipe\; \Record{r;p} \\
% \ea \\
% \ea\right\rrbracket = \\
% &
% &\ba[t]{@{~}l}
% \Let\; z \revto
% \ba[t]{@{~}l}
% \Handle\; c~\Unit \; \With\\
% \quad \ba[m]{@{~}l}
% \ba[m]{@{~}l@{~}l}
% \Return~x &\mapsto \Return\; \Record{\lambda\Unit. \Return\; x; \lambda\Unit. \Return\;x}\\
% \Await~\Unit~resume &\mapsto
% \ea\\
% \qquad\ba[t]{@{~}l}
% \Let\;r \revto \lambda x.\Let\; z \revto resume~x\;
% \In\; \Let\; \Record{f; g} = z \;\In\; f~\Unit\; \In\\
% \Return\;\Record{\lambda \Unit. \Let\; x \revto \Do\; \ell~p \;\In\; r~x;
% \lambda\Unit.\sdtrans{\Copipe}\Record{r; p}}
% \ea
% \ea
% \ea\\
% \In\;\Let\;\Record{f; g} = z\; \In\; g~\Unit
% \ea
% \ea
% \]
% %
\begin{theorem} \begin{theorem}
If $\Delta; \Gamma \vdash M : C$ then $\sdtrans{\Delta}; If $\Delta; \Gamma \vdash M : C$ then $\sdtrans{\Delta};
@ -11658,7 +11663,7 @@ If $\Delta; \Gamma \vdash M : C$ then $\sdtrans{\Delta};
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
\newcommand{\admin}{admin} \newcommand{\admin}{admin}
@ -11873,10 +11878,20 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on $\reducesto$ using Lemma~\ref{lem:sdtrans-subst} and
Lemma~\ref{lem:sdtrans-admin}. The interesting case is
$\semlab{Op^\dagger}$, which uses Lemma~\ref{lem:sdtrans-admin} to
approximate the body of the resumption up to administrative reduction.
By induction on $\reducesto$ using Lemma~\ref{lem:sdtrans-subst} and
Lemma~\ref{lem:sdtrans-admin}. We show only the interesting case
$\semlab{Op^\dagger}$, which uses Lemma~\ref{lem:sdtrans-admin} to
approximate the body of the resumption up to administrative
reduction.
% $\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H \reducesto
% N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ where
% $\ell \notin \BL(\EC)$ and
% $H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$.
% %
% \begin{derivation}
% \end{derivation}
%
\end{proof} \end{proof}
\section{Parameterised handlers as ordinary deep handlers} \section{Parameterised handlers as ordinary deep handlers}
@ -11978,7 +11993,7 @@ If $\Delta; \Gamma \vdash M : C$ then $\PD{\Delta};
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on typing derivations.
By induction on the typing derivations.
\end{proof} \end{proof}
% %
This translation of parameterised handlers simulates the native This translation of parameterised handlers simulates the native

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