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Daniel Hillerström
2021-04-13 23:54:11 +01:00
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@@ -10863,9 +10863,12 @@ the value interpretation function, which turns any source language
value into a corresponding machine value. value into a corresponding machine value.
% %
A continuation $\shk$ is a stack of generalised continuation frames A continuation $\shk$ is a stack of generalised continuation frames
$[\shf_1, \dots, \shf_n]$. Each frame $\shf = (\slk, \chi)$ represents $[\shf_1, \dots, \shf_n]$. As in
pure continuation $\slk$, corresponding to a sequence of let bindings, Section~\ref{sec:generalised-continuations} each continuation frame
inside handler closure $\chi$. $\shf = (\slk, \chi)$ consists of a pure continuation $\slk$,
corresponding to a sequence of let bindings, interpreted under some
handler, which in this context is represented by the handler closure
$\chi$.
% %
A pure continuation is a stack of pure frames. A pure frame A pure continuation is a stack of pure frames. A pure frame
$(\env, x, N)$ closes a let-binding $\Let \;x=[~] \;\In\;N$ over $(\env, x, N)$ closes a let-binding $\Let \;x=[~] \;\In\;N$ over
@@ -10964,6 +10967,11 @@ Section~\ref{subsec:machine-correctness} easier to state.
&\stepsto& \cek{ M \mid \env'[g \mapsto (\env', \Rec\,g^{A \to C}\,x.M), x \mapsto \val{W}{\env}] \mid \shk}, &\stepsto& \cek{ M \mid \env'[g \mapsto (\env', \Rec\,g^{A \to C}\,x.M), x \mapsto \val{W}{\env}] \mid \shk},
&\text{if }\val{V}{\env} = (\env', \Rec\,g^{A \to C}\,x.M) \\ &\text{if }\val{V}{\env} = (\env', \Rec\,g^{A \to C}\,x.M) \\
% TyApp
\mlab{AppType} & \cek{ V\,T \mid \env \mid \shk}
&\stepsto& \cek{ M[T/\alpha] \mid \env' \mid \shk},
&\text{if }\val{V}{\env} = (\env', \Lambda \alpha^K . \, M) \\
% Deep resumption application % Deep resumption application
\mlab{Resume} & \cek{ V\;W \mid \env \mid \shk} \mlab{Resume} & \cek{ V\;W \mid \env \mid \shk}
&\stepsto& \cek{ \Return \; W \mid \env \mid \shk' \concat \shk}, &\stepsto& \cek{ \Return \; W \mid \env \mid \shk' \concat \shk},
@@ -10979,11 +10987,6 @@ Section~\ref{subsec:machine-correctness} easier to state.
\mlab{Resume^\param} & \cek{ V\,\Record{W;W'} \mid \env \mid \shk} \mlab{Resume^\param} & \cek{ V\,\Record{W;W'} \mid \env \mid \shk}
&\stepsto& \cek{ \Return \; W \mid \env \mid (\sigma,(\env'[q \mapsto \val{W'}\env],q.\,H)) \cons \shk' \concat \shk}, &\stepsto& \cek{ \Return \; W \mid \env \mid (\sigma,(\env'[q \mapsto \val{W'}\env],q.\,H)) \cons \shk' \concat \shk},
&\text{if }\val{V}{\env} = (\sigma,(\env',q.\,H)) \cons \shk')^A \\ &\text{if }\val{V}{\env} = (\sigma,(\env',q.\,H)) \cons \shk')^A \\
% TyApp
\mlab{AppType} & \cek{ V\,T \mid \env \mid \shk}
&\stepsto& \cek{ M[T/\alpha] \mid \env' \mid \shk},
&\text{if }\val{V}{\env} = (\env', \Lambda \alpha^K . \, M) \\
% %
\mlab{Split} & \cek{ \Let \; \Record{\ell = x;y} = V \; \In \; N \mid \env \mid \shk} \mlab{Split} & \cek{ \Let \; \Record{\ell = x;y} = V \; \In \; N \mid \env \mid \shk}
&\stepsto& \cek{ N \mid \env[x \mapsto v, y \mapsto w] \mid \shk}, &\stepsto& \cek{ N \mid \env[x \mapsto v, y \mapsto w] \mid \shk},
@@ -10991,12 +10994,15 @@ Section~\ref{subsec:machine-correctness} easier to state.
% Case % Case
\mlab{Case} & \cek{ \Case\; V\, \{ \ell~x \mapsto M; y \mapsto N\} \mid \env \mid \shk} \mlab{Case} & \cek{ \Case\; V\, \{ \ell~x \mapsto M; y \mapsto N\} \mid \env \mid \shk}
&\stepsto& \begin{cases} &\stepsto& \left\{\ba{@{}l@{}}
\cek{ M \mid \env[x \mapsto v] \mid \shk}, & \cek{ M \mid \env[x \mapsto v] \mid \shk}, \\
\text{ if }\val{V}{\env} = \ell\, v\\ \cek{ N \mid \env[y \mapsto \ell'\, v] \mid \shk}, \\
\cek{ N \mid \env[y \mapsto \ell'\, v] \mid \shk}, & \ea \right.
\text{ if }\val{V}{\env} = \ell'\, v \text{ and } \ell \neq \ell' &
\end{cases}\\ \ba{@{}l@{}}
\text{if }\val{V}{\env} = \ell\, v \\
\text{if }\val{V}{\env} = \ell'\, v \text{ and } \ell \neq \ell' \\
\ea \\
% Let - eval M % Let - eval M
\mlab{Let} & \cek{ \Let \; x \revto M \; \In \; N \mid \env \mid (\slk, \chi) \cons \shk} \mlab{Let} & \cek{ \Let \; x \revto M \; \In \; N \mid \env \mid (\slk, \chi) \cons \shk}
@@ -11010,22 +11016,22 @@ Section~\ref{subsec:machine-correctness} easier to state.
&\stepsto& \cek{ M \mid \env \mid (\nil, (\env[q \mapsto \val{W}\env], H)) \cons \shk} \\ &\stepsto& \cek{ M \mid \env \mid (\nil, (\env[q \mapsto \val{W}\env], H)) \cons \shk} \\
% Return - let binding % Return - let binding
\mlab{PureCont} &\cek{ \Return \; V \mid \env \mid ((\env_H,x,N) \cons \slk, \chi) \cons \shk} \mlab{PureCont} &\cek{ \Return \; V \mid \env \mid ((\env',x,N) \cons \slk, \chi) \cons \shk}
&\stepsto& \cek{ N \mid \env_H[x \mapsto \val{V}{\env}] \mid (\slk, \chi) \cons \shk} \\ &\stepsto& \cek{ N \mid \env'[x \mapsto \val{V}{\env}] \mid (\slk, \chi) \cons \shk} \\
% Return - handler % Return - handler
\mlab{GenCont} & \cek{ \Return \; V \mid \env \mid (\nil, (\env_H,H^\delta)) \cons \shk} \mlab{GenCont} & \cek{ \Return \; V \mid \env \mid (\nil, (\env',H^\delta)) \cons \shk}
&\stepsto& \cek{ M \mid \env_H[x \mapsto \val{V}{\env}] \mid \shk}, &\stepsto& \cek{ M \mid \env'[x \mapsto \val{V}{\env}] \mid \shk},
&\text{if } \hret = \{\Return\; x \mapsto M\} \\ &\text{if } \hret = \{\Return\; x \mapsto M\} \\
% Deep % Deep
\mlab{Do^\depth} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env_H, H^\depth)) \cons \shk) \circ \shk'} \mlab{Do^\depth} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env', H^\depth)) \cons \shk) \circ \shk'}
&\stepsto& \cek{M \mid \env_H[p \mapsto \val{V}{\env}, &\stepsto& \cek{M \mid \env'[p \mapsto \val{V}{\env},
r \mapsto (\shk' \concat [(\slk, (\env_H, H^\depth))])^B] \mid \shk},\\ r \mapsto (\shk' \concat [(\slk, (\env', H^\depth))])^B] \mid \shk},\\
&&&\quad\text{if } \ell : A \to B \in E \text{ and } \hell = \{\OpCase{\ell}{p}{r} \mapsto M\} \\ &&&\quad\text{if } \ell : A \to B \in E \text{ and } \hell = \{\OpCase{\ell}{p}{r} \mapsto M\} \\
% Shallow % Shallow
\mlab{Do^\dagger} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env_H, H)^\dagger) \cons \shk) \circ \shk'} &\stepsto& \cek{M \mid \env_H[p \mapsto \val{V}{\env}, \mlab{Do^\dagger} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env', H)^\dagger) \cons \shk) \circ \shk'} &\stepsto& \cek{M \mid \env'[p \mapsto \val{V}{\env},
r \mapsto (\shk', \slk)^B] \mid \shk},\\ r \mapsto (\shk', \slk)^B] \mid \shk},\\
&&&\quad\text{if } \ell : A \to B \in E \text{ and } \hell = \{\OpCase{\ell}{p}{r} \mapsto M\} \\ &&&\quad\text{if } \ell : A \to B \in E \text{ and } \hell = \{\OpCase{\ell}{p}{r} \mapsto M\} \\
@@ -11062,23 +11068,75 @@ Section~\ref{subsec:machine-correctness} easier to state.
\caption{Machine initialisation and finalisation.} \caption{Machine initialisation and finalisation.}
\end{figure} \end{figure}
% %
The abstract machine semantics defining the transition function $\stepsto$ is given in The semantics of the abstract machine is defined in terms of a
Fig.~\ref{fig:abstract-machine-semantics}. transition relation $\stepsto \subseteq \MConfCat \times \MConfCat$ on
machine configurations. The definition of the transition relation is
given in Figure~\ref{fig:abstract-machine-semantics}.
% %
We write $\nil$ for an empty stack, $x \cons s$ for the result of A fair amount of the transition rules involve manipulating the
pushing $x$ on top of stack $s$, and $s \concat s'$ for the continuation. We adopt the same stack notation conventions used in the
concatenation of stack $s$ on top of $s'$. We use pattern matching to CPS translation with generalised continuations
deconstruct stacks. (Section~\ref{sec:cps-gen-conts}) and write $\nil$ for an empty stack,
$x \cons s$ for the result of pushing $x$ on top of stack $s$, and
$s \concat s'$ for the concatenation of stack $s$ on top of $s'$. We
use pattern matching to deconstruct stacks.
% %
% It depends on an interpretation function $\val{-}$ for values. % It depends on an interpretation function $\val{-}$ for values.
% %
The machine is initialised (\mlab{Init}) by placing a term in a % The machine is initialised (\mlab{Init}) by placing a term in a
configuration alongside the empty environment and identity % configuration alongside the empty environment and identity
continuation. % continuation.
% %
The rules (\mlab{AppClosure}), (\mlab{AppRec}), (\mlab{AppCont}),
(\mlab{AppCont^\dagger}), (\mlab{AppType}), (\mlab{Split}), and The first eight rules \mlab{App}, \mlab{AppRec}, \mlab{AppType},
(\mlab{Case}) enact the elimination of values. \mlab{Resume}, \mlab{Resume}, \mlab{Resume^\param}, \mlab{Split}, and
\mlab{Case} enact the elimination of values.
%
The closure rules (\mlab{App}, \mlab{AppRec}, \mlab{AppType}) all
essentially work the same by interpreting the abstractor $V$ in
environment $\env$ using the value interpretation function $\val{-}$
to obtain the closure. The body $M$ of closure gets put into the
control component. Before the closure environment $\env'$ gets
installed as the new machine environment, it gets extended with a
binding of the formal parameter of the abstraction to the
interpretation of argument $W$ in the previous environment $\env$
(except in the case of $\mlab{AppType}$ where the argument is a type
$T$ which gets substituted directly into the body). Note that in
either rule continuation component remains untouched.
The resumption rules (\mlab{Resume}, \mlab{Resume^\dagger},
\mlab{Resume^\param}), however, manipulate the continuation component
as they implement the context restorative behaviour of deep, shallow,
and parameterised resumption application respectively. The shape of a
deep resumption is the same as the machine continuation as it is a
reified segment of some machine continuation, hence deep resumption
invocation is realised by concatenating the reified continuation
$\kappa'$ with the current machine continuation $\kappa$. A shallow
resumption is a pair consisting of some dangling pure continuation
$\sigma'$ and reified continuation $\kappa'$. During shallow
resumption invocation the dangling pure continuation gets appended
onto the pure continuation of the current machine continuation. A
parameterised resumption is similar to an ordinary deep resumption,
except that the handler closure contains a parameterised handler
definition. During invocation the handler parameter $q$ gets rebound
to the interpretation of the argument $W'$ in the handler closure
environment $\env'$ to make the updated parameter value available
during the next activation of the handler. Following the environment
update the reified continuation gets reconstructed and appended onto
the current machine continuation.
The rules $\mlab{Split}$ and $\mlab{Case}$ concern record destructing
and variant scrutinising, respectively. Record destructing binds both
the variable $x$ to the value $v$ at label $\ell$ in the record $V$
and the variable $y$ to the tail of the record in current environment
$\env$.
%
Case splitting dispatches to the first branch with the variable $x$
bound to the variant payload in the environment if the label of the
variant $V$ matches $\ell$, otherwise it dispatches to the second
branch with the variable $y$ bound to the interpretation of $V$ in the
environment.
% %
The rules (\mlab{Let}) and (\mlab{Handle}) extend the current The rules (\mlab{Let}) and (\mlab{Handle}) extend the current
continuation with let bindings and handlers respectively. continuation with let bindings and handlers respectively.