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      macros.tex
  2. 269
      thesis.tex

1
macros.tex
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@ -17,6 +17,7 @@
% \newcommand{\Delim}[1]{\ensuremath{\langle\!\!\mkern-1.5mu\langle#1\rangle\!\!\mkern-1.5mu\rangle}}
\newcommand{\Delim}[1]{\ensuremath{\keyw{del}.#1}}
\newcommand{\sembr}[1]{\ensuremath{\llbracket #1 \rrbracket}}
\newcommand{\BigO}{\ensuremath{\mathcal{O}}}
%%
%% Partiality

269
thesis.tex
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@ -11377,23 +11377,38 @@ 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
that an implementation based on deep handlers would be fiddly. If we
were to consider the wider design space of programming language
features, then 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 the precise details). Thus in some sense
pipes are implementable with deep handlers, however, this particular
implementation strategy is not realisable in the $\HCalc$-calculus
since it has no notion of recursive types, meaning we cannot use this
strategy to argue that deep handlers can implement pipes in our
setting.
%
We will restrict our attention to the calculi $\HCalc$, $\SCalc$, and
$\HPCalc$ and use the notion of \emph{typability-preserving
macro-expressiveness} to determine whether handlers are
interdefinable. In our particular setting, typability-preserving
macro-expressiveness asks whether there exists a \emph{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 shallow handlers up to administrative reductions. % The
kind of handler, whilst preserving typability in the image of the
transformation. By mandating that the transform is local we rule out
the possibility of rewriting the entire program in, say, CPS notation
to implement deep and shallow handlers as in Chapter~\ref{ch:cps}.
%
In this chapter we use the notion of typability-preserving
macro-expressiveness to show that shallow handlers and general
recursion 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}.
%
@ -11523,9 +11538,9 @@ the resumption to wrap the handler around its body.
Nevertheless, the simulation proof makes minimal use of this power,
merely using it to rename a single variable.
%
We write $R_{\Cong}$ for the compatible closure of relation
$R$, that is the smallest relation including $R$ and closed under term
constructors for $\SCalc$.
% We write $R_{\Cong}$ for the compatible closure of relation
% $R$, that is the smallest relation including $R$ and closed under term
% constructors for $\SCalc$.
%% , otherwise known as \emph{reduction up to
%% congruence}.
@ -11535,10 +11550,27 @@ If $M \reducesto N$ then $\dstrans{M} \reducesto_{\Cong}^+
\end{theorem}
\begin{proof}
By induction on $\reducesto$ using a substitution lemma. The
interesting case is $\semlab{Op}$, which is where we apply a single
$\beta$-reduction, renaming a variable, under the lambda abstraction
representing the resumption.
By induction on $\reducesto$ using
Lemma~\ref{lem:dstrans-subst}. The interesting case is
$\semlab{Op}$, which is where we apply a single $\beta$-reduction,
renaming a variable, under the $\lambda$-abstraction representing
the resumption. The proof of this case is as follows.
%
\begin{derivation}
& \dstrans{\Handle\;\EC[\Do\;\ell~V]\;\With\;H}\\
=& \reason{definition of $\dstrans{-}$}\\
& (\Rec\;h\;f.\ShallowHandle\;f\,\Unit\;\With\;\dstrans{H}_h)\,(\lambda\Unit.\dstrans{\EC}[\Do\;\ell~\dstrans{V}])\\
\reducesto^+& \reason{\semlab{Rec}, \semlab{App}, \semlab{Op^\dagger} with $\hell = \{\OpCase{\ell}{p}{r} \mapsto N\}$}\\
& (\Let\;r \revto \Return\;\lambda x.h\,(\lambda\Unit.r~x)\;\In\;\dstrans{N})[\lambda y.\dstrans{\EC}[\Return\;y]/r,\dstrans{V}/p]\\
=& \reason{definition of [-]}\\
&(\Let\;r \revto \Return\;\lambda x.h\,(\lambda\Unit.(\lambda y.\dstrans{\EC}[\Return\;y])~x)\;\In\;\dstrans{N}[\dstrans{V}/p]\\
\reducesto_\Cong & \reason{\semlab{App} reduction under $\lambda x.\cdots$}\\
&(\Let\;r \revto \Return\;\lambda x.h\,(\lambda\Unit.\dstrans{\EC}[\Return\;x])\;\In\;\dstrans{N}[\dstrans{V}/p]\\
\reducesto& \reason{\semlab{Let} and Lemma~\ref{lem:dstrans-subst}}\\
% & \dstrans{N}[\dstrans{V}/p,\lambda x.h\,(\lambda\Unit.\dstrans{\EC}[\Return\;x])/r]\\
% =& \reason{}\\
&\dstrans{N[V/p,\lambda x.h\,(\lambda\Unit.\EC[\Return\;x])/r]}
\end{derivation}
\end{proof}
\section{Shallow as deep}
@ -11592,9 +11624,15 @@ 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.
subtyping. Technically we could insert an administrative identity
handler to coerce the effect signature. There are practical reasons
for avoiding administrative handlers, though, as we shall discuss
momentarily the inordinate administrative overhead of this
transformation might conceal the additional overhead incurred by the
introduction of administrative identity handlers. 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 the translation on an instance of
the $\Pipe$ operator from Section~\ref{sec:pipes} using the consumer
@ -11639,6 +11677,32 @@ image contains a redundant pair, because the $\Return$-case of $\Pipe$
is the identity. The translation of the $\Await$-case sets up the
forwarding component and handling component of the pair of thunks.
The distinction between deep and shallow handlers is that the latter
is discharged after handling a single operation, whereas the former is
persistent and apt for continual operation interpretations. The
persistence of deep handlers means that any handler in the image of
the translation remains in place for the duration of the handled
computation after handling a single operation, which has noticeable
asymptotic performance implications. Each activation of a handler in
the image introduces another layer of indirection that any subsequent
operation invocation have to follow. Supposing some source program
contains $n$ handlers and performs $k$ operation invocations, then the
image introduces $k$ additional handlers, meaning the total amount of
handlers in the image is $n+k$. Viewed through the practical lens of
the CPS translation (Chapter~\ref{ch:cps}) or abstract machine
(Chapter~\ref{ch:abstract-machine}) it means that in the worst case
handler lookup takes $\BigO(n+k)$ time. For example, consider the
extreme case where $n = 1$, that is, the handler lookup takes
$\BigO(1)$ time in the source, but in the image it takes $\BigO(k)$
time.
%
Thus this translation is more of theoretical significance than
practical interest. It also demonstrates that typability-preserving
macro-expressiveness is rather coarse-grained notion of
expressiveness, as it blindly considers whether some construct is
computable using another construct without considering the
computational cost.
The translation commutes with substitution and preserves typability.
%
\begin{lemma}\label{lem:sdtrans-subst}
@ -11744,41 +11808,42 @@ For all shallow handlers $H$, the following context is administrative
\end{lemma}
%
\begin{proof}
We need to check both conditions of Definition~\ref{def:admin-eval}.
We have to check both conditions of Definition~\ref{def:admin-eval}.
\begin{enumerate}
\item We need to show that for all $V \in \ValCat$
%
\[
\Let\; z \revto
\Handle\; \Return\;V \;\With\; \sdtrans{H}\;
\In\;
\Let\; \Record{f;\_} = z\; \In\; f\,\Unit \reducesto^\ast \Return\;V.
\]
%
We show this by direct calculation using the assumption $\hret = \{\Return\;x \mapsto N\}$.
\begin{derivation}
&\Let\; z \revto
\Handle\; \Return\;V \;\With\; \sdtrans{H}\;
\In\;
\Let\; \Record{f;\_} = z\; \In\; f\,\Unit\\
\reducesto^+& \reason{\semlab{Lift}, \semlab{Ret}, \semlab{Let}, \semlab{Split}}\\
% &\Let\;\Record{f;\_} = \Record{\lambda\Unit.\Return\;x;\lambda\Unit.\sdtrans{N}}\;\In\;f\,\Unit
& (\lambda\Unit.\Return\;V)\,\Unit \reducesto \Return\;V
\end{derivation}
\item We need to show that for all evaluation contexts
$\EC' \in \EvalCat$, operations
$\ell \in \BL(\EC) \backslash \BL(\EC')$, and values
$V \in \ValCat$
%
\[
\ba{@{~}l@{~}l}
&\Let\; z \revto
\Handle\; \EC'[\Do\;\ell\;V]\;\With\;\sdtrans{H} \;\In\;\Record{f;\_} = z\;\In\;f\,\Unit\\
\reducesto_\Cong^\ast& \Let\; x \revto \Do\;\ell\;V \;\In\; \Let\; z \revto \Handle\; \EC'[\Return\;x]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;\_} = z\;\In\;f\,\Unit
\ea
\]
\item Follows by direct calculation.
% \item We need to show that for all $V \in \ValCat$
% %
% \[
% \Let\; z \revto
% \Handle\; \Return\;V \;\With\; \sdtrans{H}\;
% \In\;
% \Let\; \Record{f;\_} = z\; \In\; f\,\Unit \reducesto^\ast \Return\;V.
% \]
% %
% We show this by direct calculation using the assumption $\hret = \{\Return\;x \mapsto N\}$.
% \begin{derivation}
% &\Let\; z \revto
% \Handle\; \Return\;V \;\With\; \sdtrans{H}\;
% \In\;
% \Let\; \Record{f;\_} = z\; \In\; f\,\Unit\\
% \reducesto^+& \reason{\semlab{Lift}, \semlab{Ret}, \semlab{Let}, \semlab{Split}}\\
% % &\Let\;\Record{f;\_} = \Record{\lambda\Unit.\Return\;x;\lambda\Unit.\sdtrans{N}}\;\In\;f\,\Unit
% & (\lambda\Unit.\Return\;V)\,\Unit \reducesto \Return\;V
% \end{derivation}
\item % We need to show that for all evaluation contexts
% $\EC' \in \EvalCat$, operations
% $\ell \in \BL(\EC) \backslash \BL(\EC')$, and values
% $V \in \ValCat$
% %
% \[
% \ba{@{~}l@{~}l}
% &\Let\; z \revto
% \Handle\; \EC'[\Do\;\ell\;V]\;\With\;\sdtrans{H} \;\In\;\Record{f;\_} = z\;\In\;f\,\Unit\\
% \reducesto_\Cong^\ast& \Let\; x \revto \Do\;\ell\;V \;\In\; \Let\; z \revto \Handle\; \EC'[\Return\;x]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;\_} = z\;\In\;f\,\Unit
% \ea
% \]
%
We show this by direct calculation using the assumption that
Follows by direct calculation using the assumption that
$\ell \notin \BL(\EC')$.
\begin{derivation}
&\Let\; z \revto
@ -12008,60 +12073,74 @@ to a variable to be reduced.
\end{theorem}
%
\begin{proof}
By induction on $M$. There are only two interesting cases to
consider.
\begin{description}
\item[Case]
$M = \ParamHandle\; \Return\;V\;\With\;(q.~H)(W) \reducesto
N[V/x,W/q]$, where $\hret = \{ \Return\; x \mapsto N \}$.
%
\begin{derivation}
&\PD{\ParamHandle\;\Return\;V\;\With\;(q.~H)(W)}\\
=& \reason{definition of $\PD{-}$}\\
&(\Handle\;\Return\;\PD{V}\;\With\;\PD{H}_q)~\PD{W}\\
\reducesto& \reason{$\semlab{Ret}$ with $\PD{\hret}_q = \{\Return~x \mapsto \lambda q. \PD{N}\}$}\\
&(\lambda q. \PD{N}[\PD{V}/x])~\PD{W}\\
\reducesto& \reason{$\semlab{App}$}\\
&\PD{N}[\PD{V}/x,\PD{W}/q]\\
=& \reason{lemma~\ref{lem:pd-subst}}\\
&\PD{N[V/x,W/q]}
\end{derivation}
\item[Case] $M = \ParamHandle\; \EC[\Do\;\ell~V]\;\With\;(q.~H)(W) \reducesto \\
\qquad
N[V/p,W/q,\lambda \Record{x,q'}. \ParamHandle\;\EC[\Return\;x]\;\With\;(q.~H)(q')/r]$, where $\hell = \{ \OpCase{\ell}{p}{r} \mapsto N \}$.
By induction on $M$. The interesting case is \semlab{Op^\param},
which is where we need to reduce under the $\lambda$-abstraction
representing the resumption.
% \begin{description}
% \item[Case]
% $M = \ParamHandle\; \Return\;V\;\With\;(q.~H)(W) \reducesto
% N[V/x,W/q]$, where $\hret = \{ \Return\; x \mapsto N \}$.
% %
% \begin{derivation}
% &\PD{\ParamHandle\;\Return\;V\;\With\;(q.~H)(W)}\\
% =& \reason{definition of $\PD{-}$}\\
% &(\Handle\;\Return\;\PD{V}\;\With\;\PD{H}_q)~\PD{W}\\
% \reducesto& \reason{$\semlab{Ret}$ with $\PD{\hret}_q = \{\Return~x \mapsto \lambda q. \PD{N}\}$}\\
% &(\lambda q. \PD{N}[\PD{V}/x])~\PD{W}\\
% \reducesto& \reason{$\semlab{App}$}\\
% &\PD{N}[\PD{V}/x,\PD{W}/q]\\
% =& \reason{lemma~\ref{lem:pd-subst}}\\
% &\PD{N[V/x,W/q]}
% \end{derivation}
% \item[Case] $M = \ParamHandle\; \EC[\Do\;\ell~V]\;\With\;(q.~H)(W) \reducesto \\
% \qquad
% N[V/p,W/q,\lambda \Record{x,q'}. \ParamHandle\;\EC[\Return\;x]\;\With\;(q.~H)(q')/r]$, where $\hell = \{ \OpCase{\ell}{p}{r} \mapsto N \}$.
\begin{derivation}
&\PD{\ParamHandle\;\EC[\Do\;\ell~V]\;\With\;(q.~H)(W)}\\
=& \reason{definition of $\PD{-}$}\\
&(\Handle\;\EC[\Do\;\ell~\PD{V}]\;\With\;\PD{H}_q)~\PD{W}\\
\reducesto& \reason{$\semlab{Op}$ with $\PD{\hell}_q = \{\ell~p~r \mapsto \lambda q. \Let\,r' \revto \lambda \Record{x,q'}. r~x~q\,\In\, \PD{N}[r'/r]\}$}\\
&(\Handle\;\PD{\EC}[\Do\;\ell~\PD{V}]\;\With\;\PD{H}_q)~\PD{W}\\
\reducesto& \reason{$\semlab{Op}$ with $\hell = \{\OpCase{\ell}{p}{r} \mapsto N\}$}\\
&((\lambda q. \bl
\Let\;r' \revto \lambda \Record{x,q'}. r~x~q\; \In \\
\PD{N}[r'/r])[\PD{V}/p,\lambda x.\Handle\;\EC[\Return\;x]\;\With\;\PD{H}_q)~\PD{W}\\
\Let\;r \revto \lambda \Record{x;q'}. r~x~q\;\In \\
\PD{N})[\PD{V}/p,\lambda x.\Handle\;\PD{\EC}[\Return\;x]\;\With\;\PD{H}_q/r])~\PD{W}\\
\el \\
=& \reason{definition of $[-]$}\\
&(\lambda q. \bl
\Let\; r' \revto \lambda \Record{x,q'}. (\lambda x. \Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~x~q'\;\In \\
\PD{N}[\PD{V}/p,r'/r])\\
\Let\; r \revto \lambda \Record{x,q'}. (\lambda x. \Handle\;\PD{\EC}[\Return\;x]\;\With\; \PD{H}_q)~x~q'\;\In \\
\PD{N}[\PD{V}/p])\,\PD{W}\\
\el \\
\reducesto& \reason{$\semlab{App}$}\\
&\Let\;r' \revto \lambda \Record{x,q'}. r~x~q'\; \In\; \PD{N}[r'/r,\PD{V}/p,\PD{V}/q]\\
\reducesto& \reason{$\semlab{Let}$}\\
\reducesto& \reason{\semlab{App}}\\
&\bl
\Let\; r \revto \lambda \Record{x;q'}. (\lambda x. \Handle\;\PD{\EC}[\Return\;x]\;\With\; \PD{H}_q)~x~q'\;\In\\
\PD{N}[\PD{V}/p,\PD{W}/q]
\el\\
\reducesto_\Cong& \reason{\semlab{App} under $\lambda\Record{x;q'}.\cdots$}\\
&\bl
\Let\; r \revto \lambda \Record{x;q'}.(\Handle\;\PD{\EC}[\Return\;x]\;\With\; \PD{H}_q)~q'\;\In\\
\PD{N}[\PD{V}/p,\PD{W}/q]
\el\\
\reducesto& \reason{\semlab{Let}}\\
&\PD{N}[\bl
\PD{V}/p,\PD{W}/q, \\
\lambda \Record{x,q'}. (\lambda x. \Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~x~q'/r]\\
\lambda \Record{x,q'}. (\Handle\;\PD{\EC}[\Return\;x]\;\With\; \PD{H}_q)~q'/r]\\
\el \\
\reducesto_{\Cong}& \reason{$\semlab{App}$}\\
&\PD{N}[\PD{V}/p,\PD{W}/q,\lambda \Record{x,q'}. (\Handle\;\EC[\Return\;x]\;\With\; \PD{H}_q)~q'/r]\\
=& \reason{definition of $\PD{-}$}\\
&\PD{N}[\PD{V}/p,\PD{W}/q,\lambda \Record{x,q'}. \PD{\ParamHandle\;\EC[\Return\;x]\;\With\;(q.~H)(q')}/r]\\
=& \reason{lemma~\ref{lem:pd-subst}}\\
=& \reason{definition of $\PD{-}$ and Lemma~\ref{lem:pd-subst}}\\
&\PD{N[V/p,W/q,\lambda \Record{x,q'}. \ParamHandle\;\EC[\Return\;x]\;\With\; (q.~H)(q')/r]}
\end{derivation}
\end{description}
% \end{description}
\end{proof}
\section{Related work}
\citet{ForsterKLP17,ForsterKLP19} \citet{PirogPS19}, \citet{Shan04}
Precisely how effect handlers fit into the landscape of programming
language features is largely unexplored in the literature. The most
notable work in this area is due to \citet{ForsterKLP17}, who
investigate various relationships between effect handlers, delimited
control in the form of shift/reset, and monadic reflection using the
notions of typability-preserving macro-expressiveness and untyped
macro-expressiveness~\cite{ForsterKLP17,ForsterKLP19}. \citet{PirogPS19}
build upon the work of \citeauthor{ForsterKLP17} as they show that
with sufficient polymorphism effect handlers and delimited control á
la shift/reset can simulate one another. \citet{Shan04}
% \chapter{Computability, complexity, and expressivness}
% \label{ch:expressiveness}

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