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Daniel Hillerström
2021-03-15 23:40:25 +00:00
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thesis.tex
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@@ -3022,22 +3022,22 @@ Section~\ref{sec:base-language-type-rules}.
% %\slab{Type variables} &\alpha, \rho, \theta& \\
% \slab{Type environments} &\Gamma &::=& \cdot \mid \Gamma, x:A \\
% \slab{Kind environments} &\Delta &::=& \cdot \mid \Delta, \alpha:K
\slab{Value types} &A,B \in \ValTypeCat &::= & A \to C
\slab{Value\mathrm{~}types} &A,B \in \ValTypeCat &::= & A \to C
\mid \forall \alpha^K.C
\mid \Record{R} \mid [R]
\mid \alpha \\
\slab{Computation types\!\!}
\slab{Computation\mathrm{~}types\!\!}
&C,D \in \CompTypeCat &::= & A \eff E \\
\slab{Effect types} &E \in \EffectCat &::= & \{R\}\\
\slab{Row types} &R \in \RowCat &::= & \ell : P;R \mid \rho \mid \cdot \\
\slab{Presence types\!\!\!\!\!} &P \in \PresenceCat &::= & \Pre{A} \mid \Abs \mid \theta\\
\slab{Effect\mathrm{~}types} &E \in \EffectCat &::= & \{R\}\\
\slab{Row\mathrm{~}types} &R \in \RowCat &::= & \ell : P;R \mid \rho \mid \cdot \\
\slab{Presence\mathrm{~}types\!\!\!\!\!} &P \in \PresenceCat &::= & \Pre{A} \mid \Abs \mid \theta\\
\\
\slab{Types} &T \in \TypeCat &::= & A \mid C \mid E \mid R \mid P \\
\slab{Kinds} &K \in \KindCat &::= & \Type \mid \Comp \mid \Effect \mid \Row_\mathcal{L} \mid \Presence \\
\slab{Label sets} &\mathcal{L} \in \LabelCat &::=& \emptyset \mid \{\ell\} \uplus \mathcal{L}\\\\
\slab{Label\mathrm{~}sets} &\mathcal{L} \in \LabelCat &::=& \emptyset \mid \{\ell\} \uplus \mathcal{L}\\\\
\slab{Type environments} &\Gamma \in \TyEnvCat &::=& \cdot \mid \Gamma, x:A \\
\slab{Kind environments} &\Delta \in \KindEnvCat &::=& \cdot \mid \Delta, \alpha:K \\
\slab{Type\mathrm{~}environments} &\Gamma \in \TyEnvCat &::=& \cdot \mid \Gamma, x:A \\
\slab{Kind\mathrm{~}environments} &\Delta \in \KindEnvCat &::=& \cdot \mid \Delta, \alpha:K \\
\end{syntax}
\caption{Syntax of types, kinds, and their environments.}
\label{fig:base-language-types}
@@ -10649,11 +10649,12 @@ continuation argument per handler in scope~\cite{SchusterBO20}. The
idea of iterated CPS is due to \citet{DanvyF90}, who used it to give
develop a CPS transform for shift and reset.
%
\citet{XieBHSL20} has devised an \emph{evidence-passing translation}
\citet{XieBHSL20} have devised an \emph{evidence-passing translation}
for deep effect handlers. The basic idea is similar to
capability-passing style as evidence for handlers are passed downwards
to their operations in shape of a vector containing the handlers in
scope through computations.
scope through computations. \citet{XieL20} have realised handlers by
evidence-passing style as a Haskell library.
There are clear connections between the CPS translations presented in
this chapter and the continuation monad implementation of
@@ -10730,7 +10731,10 @@ then in the image $\cps{M}\,k \reducesto^\ast k\,\cps{V}$.
% \paragraph{Partial evaluation}
\paragraph{ANF vs CPS}
\paragraph{Selective CPS transforms}
\citet{Nielsen01} \citet{DanvyH92} \citet{DanvyH93} \citet{Leijen17}
\chapter{Abstract machine semantics}
\label{ch:abstract-machine}
@@ -11349,7 +11353,7 @@ If $\typc{}{M : A}{E}$ and $M \reducesto^+ N \not\reducesto$, then $M
\part{Expressiveness}
\chapter{Interdefinability of deep and shallow handlers}
\chapter{Interdefinability of effect handlers}
\label{ch:deep-vs-shallow}
In this section we show that shallow handlers and general recursion
@@ -11373,18 +11377,34 @@ resumption has its body wrapped in a call to this recursive function.
%
Formally, the translation $\dstrans{-}$ is defined as the homomorphic
extension of the following equations to all terms.
\begin{equations}
\dstrans{\Handle \; M \; \With \; H} &=&
(\Rec~h~f.\ShallowHandle\; f\,\Unit \; \With \; \dstrans{H} h)\,(\lambda \Unit{}.\dstrans{M}) \\
\dstrans{H}h &=& \dstrans{\hret}h \uplus \dstrans{\hops}h \\
\dstrans{\{ \Return \; x \mapsto N\}} h &=&
\{ \Return \; x \mapsto \dstrans{N} \}\\
\dstrans{\{ \ell \; p \; r \mapsto N_\ell \}_{\ell \in \mathcal{L}}} h &=&
\{ \ell \; p \; r \mapsto
\Let \; r \revto \Return \; \lambda x.h\,(\lambda \Unit{}.r\,x) \; \In \;
\dstrans{N_\ell} \}_{\ell \in \mathcal{L}}
\end{equations}
%
\[
\bl
\dstrans{-} : \CompCat \to \CompCat\\
\dstrans{\Handle \; M \; \With \; H} \defas
(\Rec~h~f.\ShallowHandle\; f\,\Unit \; \With \; \dstrans{H}_h)\,(\lambda \Unit{}.\dstrans{M}) \medskip\\
% \dstrans{H}h &=& \dstrans{\hret}h \uplus \dstrans{\hops}h \\
\dstrans{-} : \HandlerCat \times \ValCat \to \HandlerCat\\
\ba{@{}l@{~}c@{~}l}
\dstrans{\{ \Return \; x \mapsto N\}}_h &\defas&
\{ \Return \; x \mapsto \dstrans{N} \}\\
\dstrans{\{ \OpCase{\ell}{p}{r} \mapsto N_\ell \}_{\ell \in \mathcal{L}}}_h &\defas&
\{ \OpCase{\ell}{p}{r} \mapsto
\bl
\Let \; r \revto \Return \; \lambda x.h\,(\lambda \Unit{}.r\,x)\\
\In\;\dstrans{N_\ell} \}_{\ell \in \mathcal{L}}
\el
\ea
\el
\]
%
The translation of $\Handle$ uses a $\Rec$-abstraction to introduce a
fresh name $h$ for the handler $H$. This name is used by the
translation of the handler definitions. The translation of
$\Return$-clauses is the identity, and thus ignores the handler
name. However, the translation of operation clauses uses the name to
simulate a deep resumption by guarding invocations of the shallow
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
@@ -11467,28 +11487,36 @@ translation on handler types as well as handler terms.
Formally, the translation $\sdtrans{-}$ is defined as the homomorphic
extension of the following equations to all types, terms, and type
environments.
\begin{equations}
\sdtrans{C \Rightarrow D} &=&
\sdtrans{C} \Rightarrow \Record{\UnitType \to \sdtrans{C}; \UnitType \to \sdtrans{D}} \medskip \\
\sdtrans{\ShallowHandle \; M \; \With \; H} &=&
%
\[
\bl
\sdtrans{-} : \TypeCat \to \TypeCat\\
\sdtrans{C \Rightarrow D} \defas
\sdtrans{C} \Rightarrow \Record{\UnitType \to \sdtrans{C}; \UnitType \to \sdtrans{D}} \medskip \medskip\\
\sdtrans{-} : \CompCat \to \CompCat\\
\sdtrans{\ShallowHandle \; M \; \With \; H} \defas
\ba[t]{@{}l}
\Let\;z \revto \Handle \; \sdtrans{M} \; \With \; \sdtrans{H} \; \In\\
\Let\;\Record{f; g} = z \;\In\;
g\,\Unit
\ea \\
\sdtrans{H} &=& \sdtrans{\hret} \uplus \sdtrans{\hops} \\
\sdtrans{\{\Return \; x \mapsto N\}} &=&
\{\Return \; x \mapsto \Return\; \Record{\lambda \Unit.\Return\; x; \lambda \Unit.\sdtrans{N}}\} \\
\sdtrans{\{\ell\; p\; r \mapsto N\}_{\ell \in \mathcal{L}}} &=& \{
\ea \medskip\\
\sdtrans{-} : \HandlerCat \to \HandlerCat\\
% \sdtrans{H} &=& \sdtrans{\hret} \uplus \sdtrans{\hops} \\
\ba{@{}l@{~}c@{~}l}
\sdtrans{\{\Return \; x \mapsto N\}} &\defas&
\{\Return \; x \mapsto \Return\; \Record{\lambda \Unit.\Return\; x; \lambda \Unit.\sdtrans{N}}\} \\
\sdtrans{\{\OpCase{\ell}{p}{r} \mapsto N\}_{\ell \in \mathcal{L}}} &\defas& \{
\bl
\ell\; p\; r \mapsto \\
\quad \Let \;r \revto
\lambda x. \Let\; z \revto r\,x\; \In\;
\Let\; \Record{f; g} = z\; \In\; f\,\Unit
\; \In\\
\quad \Return\;\Record{\lambda \Unit.\Let\; x \revto \Do\;\ell\,p\; \In\; r\,x; \lambda \Unit.\sdtrans{N}}\}_{\ell \in \mathcal{L}}
\OpCase{\ell}{p}{r} \mapsto\\
\qquad\bl\Let \;r \revto
\lambda x. \Let\; z \revto r~x\;\In\;
\Let\; \Record{f; g} = z\; \In\; f\,\Unit\;\In\\
\Return\;\Record{\lambda \Unit.\Let\; x \revto \Do\;\ell~p\; \In\; r~x; \lambda \Unit.\sdtrans{N}}\}_{\ell \in \mathcal{L}}
\el
\end{equations}
\el
\ea
\el
\]
%
Each shallow handler is encoded as a deep handler that returns a pair
of thunks. The first forwards all operations, acting as the identity
@@ -11626,6 +11654,58 @@ $\semlab{Op^\dagger}$, which uses Lemma~\ref{lem:sdtrans-admin} to
approximate the body of the resumption up to administrative reduction.
\end{proof}
\section{Parameterised handlers as ordinary deep handlers}
\label{sec:param-desugaring}
\newcommand{\PD}[1]{\mathcal{P}\cps{#1}}
%
As mentioned in Section~\ref{sec:unary-parameterised-handlers},
parameterised handlers codify the parameter-passing idiom. They may be
seen as an optimised form of parameter-passing deep handlers. We now
show formally that parameterised handlers are special instances of
ordinary deep handlers.
%
We define a local transformation $\PD{-}$ which translates
parameterised handlers into ordinary deep handlers. We omit the
homomorphic cases and show only the interesting cases.
%
\[
\bl
\PD{-} : \TypeCat \to \TypeCat\\
\PD{\Record{C, A} \Rightarrow^\param B \eff E} \defas \PD{C} \Rightarrow (\PD{A} \to \PD{B \eff E})\eff \PD{E} \medskip\\
\PD{-} : \CompCat \to \CompCat\\
\PD{\ParamHandle\;M\;\With\;(q.\,H)(W)} \defas \left(\Handle\; \PD{M}\; \With\; \PD{H}_q\right)~\PD{W} \medskip\\
\PD{-} : \HandlerCat \times \ValCat \to \HandlerCat\\
\ba{@{}l@{~}c@{~}l}
\PD{\{\Return\;x \mapsto M\}}_q &\defas& \{\Return\;x \mapsto \lambda q. \PD{M}\}\\
\PD{\{\OpCase{\ell}{p}{r} \mapsto M\}}_q &\defas&
\{\OpCase{\ell}{p}{r} \mapsto \lambda q.
\Let\; r \revto \Return\;\lambda \Record{x,q'}. r~x~q'\;
\In\; \PD{M}\}
\ea
\el
\]
%
The parameterised $\ParamHandle$ construct becomes an application of a
$\Handle$ construct to the translation of the parameter. The bodies of
return and operation clauses are each enclosed in a lambda abstraction
whose formal parameter is the handler parameter $q$. As a result the
ordinary deep resumption $r$ is a curried function. However, the uses
of $r$ in $M$ expects a binary function. To repair this discrepancy,
we construct an uncurried interface of $r$ via the function $r'$.
This translation of parameterised handlers simulates the native
semantics. As with the simulation of deep handlers via shallow
handlers in Section~\ref{sec:deep-as-shallow}, this simulation is only up
to congruence due to the need for an application of a pure function to
a variable to be reduced. The interesting cases of the proof appear in
Appendix~\ref{sec:param-proof}.
\begin{theorem}[Simulation of parameterised handlers by deep
handlers]
\label{thm:param-simulation}
If $M \reducesto N$ then $\PD{M} \reducesto^+_{\mathrm{cong}} \PD{N}$.
\end{theorem}
% \chapter{Computability, complexity, and expressivness}
% \label{ch:expressiveness}
% \section{Notions of expressiveness}