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thesis.bib
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@@ -2765,3 +2765,12 @@
note = {Lectures delivered at University of California, Berkeley, California, USA, 1962/63} note = {Lectures delivered at University of California, Berkeley, California, USA, 1962/63}
} }
# Computer architecture
@book{BryantO03,
author = {Randal E. Bryant and
David R. O'Hallaron},
title = {Computer systems - a programmers perspective},
publisher = {Pearson Education},
year = {2003},
edition = {2nd}
}

204
thesis.tex
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@@ -10757,7 +10757,9 @@ continuations (Section~\ref{sec:generalised-continuations}) to
computation that makes program control more apparent than standard computation that makes program control more apparent than standard
reduction semantics. Abstract machines come in different styles and reduction semantics. Abstract machines come in different styles and
flavours, though, a common trait is that they closely model how an flavours, though, a common trait is that they closely model how an
actual computer might go about executing a program. actual computer might go about executing a program, meaning they
embody some high-level abstract models of main memory and the
instruction fetch-execute cycle of processors~\cite{BryantO03}.
\paragraph{Relation to prior work} The work in this chapter is based \paragraph{Relation to prior work} The work in this chapter is based
on work in the following previously published papers. on work in the following previously published papers.
@@ -10768,7 +10770,7 @@ on work in the following previously published papers.
\item \bibentry{HillerstromLA20} \label{en:ch-am-HLA20} \item \bibentry{HillerstromLA20} \label{en:ch-am-HLA20}
\end{enumerate} \end{enumerate}
% %
The particular presentation in this chapter closely follows that of The particular presentation in this chapter follows closely that of
item~\ref{en:ch-am-HLA20}. item~\ref{en:ch-am-HLA20}.
% In this chapter we develop an abstract machine that supports deep and % In this chapter we develop an abstract machine that supports deep and
@@ -10864,52 +10866,52 @@ Figure~\ref{fig:abstract-machine-syntax}.
%% \[ %% \[
%% \shk_0 = [(\nil, (\emptyset, \{\Return\;x \mapsto x\}))] %% \shk_0 = [(\nil, (\emptyset, \{\Return\;x \mapsto x\}))]
%% \] %% \]
%\textbf{Initialisation} $~~\stepsto \subseteq \CompCat \times \MConfCat$
\textbf{Transition function} $~~\stepsto~ \subseteq \MConfCat \times \MConfCat$ %
\begin{displaymath} \[
\bl
\ba{@{}l@{~}r@{~}c@{~}l@{\quad}l@{}} \ba{@{}l@{~}r@{~}c@{~}l@{\quad}l@{}}
% \mlab{Init} & M &\stepsto& \cek{M \mid \emptyset \mid [(\nil, (\emptyset, \{\Return\;x \mapsto \Return\;x\}))]} \\[1ex] \mlab{Init} & M &\stepsto& \cek{M \mid \emptyset \mid [(\nil, (\emptyset, \{\Return\;x \mapsto \Return\;x\}))]} \\
%
%\textbf{Finalisation} $~~\stepsto \subseteq \MConfCat \times \CompCat$
%
\mlab{Halt} & \cek{\Return\;V \mid \env \mid \nil} &\stepsto& \val{V}{\env}\\
%
%\textbf{Transition function} $~~\stepsto~ \subseteq \MConfCat \times \MConfCat$
% App % App
\mlab{AppClosure} & \cek{ V\;W \mid \env \mid \shk} \mlab{AppClosure} & \cek{ V\;W \mid \env \mid \shk}
&\stepsto& \cek{ M \mid \env'[x \mapsto \val{W}{\env}] \mid \shk}, &\stepsto& \cek{ M \mid \env'[x \mapsto \val{W}{\env}] \mid \shk},
&\text{if }\val{V}{\env} = (\env', \lambda x^A.M) \\ \text{if }\val{V}{\env} = (\env', \lambda x^A.M) \\
\mlab{AppRec} & \cek{ V\;W \mid \env \mid \shk} \mlab{AppRec} & \cek{ V\;W \mid \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}, &\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) \\
% App - continuation % App - continuation
\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},
&\text{if }\val{V}{\env} = (\shk')^A \\ \text{if }\val{V}{\env} = (\shk')^A \\
\mlab{Resume^\dagger} & \cek{ V\,W \mid \env \mid (\slk, \chi) \cons \shk} \mlab{Resume^\dagger} & \cek{ V\,W \mid \env \mid (\slk, \chi) \cons \shk}
&\stepsto& &\stepsto&
\cek{\Return\; W \mid \env \mid \shk' \concat ((\slk' \concat \slk, \chi) \cons \shk)}, \cek{\Return\; W \mid \env \mid \shk' \concat ((\slk' \concat \slk, \chi) \cons \shk)},
&\text{if } \val{V}{\env} = (\shk', \slk')^A \\ \text{if } \val{V}{\env} = (\shk', \slk')^A \\
% TyApp % TyApp
\mlab{AppType} & \cek{ V\,T \mid \env \mid \shk} \mlab{AppType} & \cek{ V\,T \mid \env \mid \shk}
&\stepsto& \cek{ M[T/\alpha] \mid \env' \mid \shk}, &\stepsto& \cek{ M[T/\alpha] \mid \env' \mid \shk},
&\text{if }\val{V}{\env} = (\env', \Lambda \alpha^K . \, M) \\[1ex] \text{if }\val{V}{\env} = (\env', \Lambda \alpha^K . \, M) \\[1ex]
\ea \\
\ba{@{}l@{}r@{~}c@{~}l@{\quad}l@{}}
\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},
& \text {if }\val{V}{\env} = \Record{\ell=v; w} \\ \text {if }\val{V}{\env} = \Record{\ell=v; w} \\
% 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& \left\{\ba{@{}l@{}} &\stepsto& \begin{cases}
\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.
&
\ba{@{}l@{}}
\text{if }\val{V}{\env} = \ell\, v \\
\text{if }\val{V}{\env} = \ell'\, v \text{ and } \ell \neq \ell' \\ \text{if }\val{V}{\env} = \ell'\, v \text{ and } \ell \neq \ell' \\
\ea \\[2ex] \end{cases}\\
% 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}
@@ -10924,29 +10926,26 @@ Figure~\ref{fig:abstract-machine-syntax}.
&\stepsto& \cek{ N \mid \env'[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{RetHandler} & \cek{ \Return \; V \mid \env \mid (\nil, (\env',H)) \cons \shk} \mlab{AppGenCont} & \cek{ \Return \; V \mid \env \mid (\nil, (\env',H)) \cons \shk}
&\stepsto& \cek{ M \mid \env'[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\} \\
% \mlab{RetTop} & \cek{\Return\;V \mid \env \mid \nil} &\stepsto& \val{V}{\env} \\[1ex]
% Deep % Deep
\mlab{Do} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env', H)) \cons \shk) \circ \shk'} \mlab{Do} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env', H)) \cons \shk) \circ \shk'}
&\stepsto& \multicolumn{2}{@{}l@{}}{\cek{M \mid \env'[x \mapsto \val{V}{\env}, &\stepsto& \cek{M \mid \env'[x \mapsto \val{V}{\env},
r \mapsto (\shk' \concat [(\slk, (\env', H))])^B] \mid \shk},} \\ r \mapsto (\shk' \concat [(\slk, (\env', H))])^B] \mid \shk}, \\
&&& \multicolumn{2}{@{}r@{}}{\text{if } \ell : A \to B \in E \text{ and } \hell = \{\ell\; x \; r \mapsto M\}} \\ && \multicolumn{2}{@{}r@{}}{\text{if } \ell : A \to B \in E \text{ and } \hell = \{\ell\; x \; r \mapsto M\}} \\
% Shallow % Shallow
\mlab{Do^\dagger} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\gamma', H)^\dagger) \cons \shk) \circ \shk'} &\stepsto& \mlab{Do^\dagger} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\gamma', H)^\dagger) \cons \shk) \circ \shk'} &\stepsto&
\multicolumn{2}{@{}l@{}}{\cek{M \mid \env'[x \mapsto \val{V}{\env}, r \mapsto (\shk', \slk)^B] \mid \shk},}\\ \cek{M \mid \env'[x \mapsto \val{V}{\env}, r \mapsto (\shk', \slk)^B] \mid \shk},\\
&&&\multicolumn{2}{@{}r@{}}{\text{if } \ell : A \to B \in E \text{ and } \hell = \{\ell\; x \; r \mapsto M\}} \\ &&\multicolumn{2}{@{}r@{}}{\text{if } \ell : A \to B \in E \text{ and } \hell = \{\ell\; x \; r \mapsto M\}} \\
% Forward % Forward
\mlab{Forward} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env', H)^\depth) \cons \shk) \circ \shk'} \mlab{Forward} & \cek{ (\Do \; \ell \; V)^E \mid \env \mid ((\slk, (\env', H)^\depth) \cons \shk) \circ \shk'}
&\stepsto& \cek{ (\Do \; \ell \; V)^E \mid \env \mid \shk \circ (\shk' \concat [(\slk, (\env', H)^\depth)])}, &\stepsto& \cek{ (\Do \; \ell \; V)^E \mid \env \mid \shk \circ (\shk' \concat [(\slk, (\env', H)^\depth)])}, \text{if } \hell = \emptyset \\
& \text{if } \hell = \emptyset \\ \ea
\ea \\ \]
\el
\end{displaymath}
\textbf{Value interpretation} $~\val{-} : \ValCat \times \MEnvCat \to \MValCat$ \textbf{Value interpretation} $~\val{-} : \ValCat \times \MEnvCat \to \MValCat$
\begin{displaymath} \begin{displaymath}
@@ -11573,7 +11572,7 @@ If $M \reducesto N$ then $\dstrans{M} \reducesto_{\Cong}^+
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
By induction on $\reducesto$ using By case analysis on $\reducesto$ using
Lemma~\ref{lem:dstrans-subst}. The interesting case is Lemma~\ref{lem:dstrans-subst}. The interesting case is
$\semlab{Op}$, which is where we apply a single $\beta$-reduction, $\semlab{Op}$, which is where we apply a single $\beta$-reduction,
renaming a variable, under the $\lambda$-abstraction representing renaming a variable, under the $\lambda$-abstraction representing
@@ -11903,24 +11902,105 @@ $N'$ 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 By case analysis on $\reducesto$ using Lemma~\ref{lem:sdtrans-subst}
Lemma~\ref{lem:sdtrans-admin}. We show only the interesting case and Lemma~\ref{lem:sdtrans-admin}. We show only the interesting case
$\semlab{Op^\dagger}$, which uses Lemma~\ref{lem:sdtrans-admin} to $\semlab{Op^\dagger}$, which uses Lemma~\ref{lem:sdtrans-admin} to
approximate the body of the resumption up to administrative approximate the body of the resumption up to administrative
reduction. reduction.\smallskip
\begin{description}
\item[Case] \noindent\textbf{Case} $\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H \reducesto
$\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H \reducesto
N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ where N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ where
$\ell \notin \BL(\EC)$ and $\ell \notin \BL(\EC)$ and
$H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$. $H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$. \smallskip\\
% %
% \begin{derivation} Pick $M' =
\sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. Clearly
% \end{derivation} $M' \approxa \sdtrans{M}$. We compute $N'$ via reduction as follows.
\dhil{Write the proof sketch} \begin{derivation}
\end{description} & \sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}\\
=& \reason{definition of $\sdtrans{-}$}\\
&\bl
\Let\;z \revto \Handle\;\sdtrans{\EC}[\Do\;\ell~\sdtrans{V}]\;\With\;\sdtrans{H}\;\In\\
\Let\;\Record{f;g} = z\;\In\;g\,\Unit
\el\\
\reducesto^+& \reason{\semlab{Op} using assumption $\ell \notin \BL(\sdtrans{\EC})$, \semlab{Let}, \semlab{Let}}\\
% &\bl
% \Let\;z \revto
% (\bl
% \Let\;r \revto \lambda x.\Let\;z \revto r~x\;\In\;\Let\;\Record{f;g} = z\;\In\;f\,\Unit\;\In\\
% \Return\;\Record{
% \bl
% \lambda\Unit.\Let\;x \revto \Do\;\ell~p\;\In\;r~x;\\
% \lambda\Unit.\sdtrans{N_\ell}})[
% \bl
% \sdtrans{V}/p,\\
% \lambda y.\Handle\;\sdtrans{\EC}[\Return\;y]\;\With\;\sdtrans{H}/r]
% \el
% \el
% \el\\
% \In\;\Let\;\Record{f;g} = z\;\In\;g\,\Unit
% \el\\
% \reducesto^+& \reason{\semlab{Let}, \semlab{Let}}\\
&\bl
\Let\;\Record{f;g} = \Record{
\bl
\lambda\Unit.\Let\;x \revto \Do\;\ell~\sdtrans{V}\;\In\;r~x;\\
\lambda\Unit.\sdtrans{N_\ell}}[\lambda x.
\bl
\Let\;z \revto (\lambda y.\Handle\;\sdtrans{E}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\
\Let\;\Record{f;g} = z\;\In\;f\,\Unit/r,\sdtrans{V}/p]\;\In\; g\,\Unit
\el
\el\\
\el\\
\reducesto^+ &\reason{\semlab{Split}, \semlab{App}}\\
&\sdtrans{N_\ell}[\lambda x.
\bl
\Let\;z \revto (\lambda y.\Handle\;\sdtrans{E}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\
\Let\;\Record{f;g} = z\;\In\;f\,\Unit/r,\sdtrans{V}/p]
\el\\
=& \reason{by Lemma~\ref{lem:sdtrans-subst}}\\
&\sdtrans{N_\ell[\lambda x.
\bl
\Let\;z \revto (\lambda y.\Handle\;E[\Return\;y]\;\With\;H)~x\;\In\\
\Let\;\Record{f;g} = z\;\In\;f\,\Unit/r,V/p]}
\el
\end{derivation}
% %
We take the above computation term to be our $N'$. If
$r \notin \FV(N_\ell)$ then the two terms $N'$ and
$\sdtrans{N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]}$ are the
identical, and thus by reflexivity of the $\approxa$-relation it
follows that
$N' \approxa \sdtrans{N_\ell[V/p,\lambda
y.\EC[\Return\;y]/r]}$. Otherwise $N'$ approximates
$N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ at least up to a use of
$r$. We need to show that the approximation remains faithful during
any application of $r$. Specifically, we proceed to show that for
any value $W \in \ValCat$
%
\[
(\bl
\lambda x.\Let\;z \revto (\lambda y.\Handle\;\sdtrans{E}[\Return\;y]\;\With\;\sdtrans{H})~x\;\In\\
\Let\;\Record{f;g} = z \;\In\;f\,\Unit)~W \approxa (\lambda y.\sdtrans{\EC}[\Return\;y])~W.
\el
\]
%
The right hand side reduces to $\sdtrans{\EC}[\Return\;W]$. Two
applications of \semlab{App} on the left hand side yield the term
%
\[
\Let\;z \revto \Handle\;\sdtrans{E}[\Return\;W]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;g} = z \;\In\;f\,\Unit.
\]
%
Define
%
$\EC' \defas \Let\;z \revto \Handle\; [\,]\;\With\;\sdtrans{H}\;\In\;\Let\;\Record{f;g} = z \;\In\;f\,\Unit$
%
such that $\EC'$ is an administrative evaluation context by
Lemma~\ref{lem:sdtrans-admin}. Then it follows by
Defintion~\ref{def:approx-admin} that
$\EC'[\sdtrans{\EC}[\Return\;W]] \approxa
\sdtrans{\EC}[\Return\;W]$.
\end{proof} \end{proof}
\section{Parameterised handlers as ordinary deep handlers} \section{Parameterised handlers as ordinary deep handlers}
@@ -11934,10 +12014,10 @@ show formally that parameterised handlers are special instances of
ordinary deep handlers. ordinary deep handlers.
% %
We define a local transformation $\PD{-}$ which translates We define a local transformation $\PD{-}$ which translates
parameterised handlers into ordinary deep handlers. As with the two parameterised handlers into ordinary deep handlers. Formally, the
previous translations it is formally defined on terms, types, type translation is defined on terms, types, environments, and
environments, and substitutions. We omit the homomorphic cases and substitutions. We omit the homomorphic cases and show only the
show only the interesting cases. interesting cases.
% %
\[ \[
\bl \bl
@@ -12029,9 +12109,10 @@ If $\Delta; \Gamma \vdash M : C$ then $\PD{\Delta};
% %
This translation of parameterised handlers simulates the native This translation of parameterised handlers simulates the native
semantics. As with the simulation of deep handlers via shallow semantics. As with the simulation of deep handlers via shallow
handlers in Section~\ref{sec:deep-as-shallow}, this simulation is only handlers in Section~\ref{sec:deep-as-shallow}, this simulation is not
up to congruence due to the need for an application of a pure function quite on the nose as the image simulates the source only up to
to a variable to be reduced. congruence due to the need for an application of a pure function to a
variable to be reduced.
% %
\begin{theorem}[Simulation up to congruence] \begin{theorem}[Simulation up to congruence]
\label{thm:param-simulation} \label{thm:param-simulation}
@@ -12039,9 +12120,10 @@ to a variable to be reduced.
\end{theorem} \end{theorem}
% %
\begin{proof} \begin{proof}
By induction on $M$. The interesting case is \semlab{Op^\param}, By case analysis on the relation $\reducesto$ using
which is where we need to reduce under the $\lambda$-abstraction Lemma~\ref{lem:pd-subst}. The interesting case is
representing the resumption. \semlab{Op^\param}, which is where we need to reduce under the
$\lambda$-abstraction representing the parameterised resumption.
% \begin{description} % \begin{description}
% \item[Case] % \item[Case]
% $M = \ParamHandle\; \Return\;V\;\With\;(q.~H)(W) \reducesto % $M = \ParamHandle\; \Return\;V\;\With\;(q.~H)(W) \reducesto
@@ -12099,7 +12181,9 @@ to a variable to be reduced.
Precisely how effect handlers fit into the landscape of programming Precisely how effect handlers fit into the landscape of programming
language features is largely unexplored in the literature. The most language features is largely unexplored in the literature. The most
relevant work in this area is due to \citet{ForsterKLP17}, who relevant related work in this area is due to my collaborators and
myself on the inherited efficiency of effect handlers
(c.f. Chapter~\ref{ch:handlers-efficiency}) and \citet{ForsterKLP17}, who
investigate various relationships between effect handlers, delimited investigate various relationships between effect handlers, delimited
control in the form of shift/reset, and monadic reflection using the control in the form of shift/reset, and monadic reflection using the
notions of typability-preserving macro-expressiveness and untyped notions of typability-preserving macro-expressiveness and untyped