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  1. 9
      thesis.bib
  2. 204
      thesis.tex

9
thesis.bib
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@ -2765,3 +2765,12 @@
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
reduction semantics. Abstract machines come in different styles and
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
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}
\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}.
% In this chapter we develop an abstract machine that supports deep and
@ -10864,52 +10866,52 @@ Figure~\ref{fig:abstract-machine-syntax}.
%% \[
%% \shk_0 = [(\nil, (\emptyset, \{\Return\;x \mapsto x\}))]
%% \]
\textbf{Transition function} $~~\stepsto~ \subseteq \MConfCat \times \MConfCat$
\begin{displaymath}
\bl
%\textbf{Initialisation} $~~\stepsto \subseteq \CompCat \times \MConfCat$
%
\[
\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
\mlab{AppClosure} & \cek{ V\;W \mid \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}
&\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
\mlab{Resume} & \cek{ V\;W \mid \env \mid \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}
&\stepsto&
\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
\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) \\[1ex]
\ea \\
\ba{@{}l@{}r@{~}c@{~}l@{\quad}l@{}}
\text{if }\val{V}{\env} = (\env', \Lambda \alpha^K . \, M) \\[1ex]
\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},
& \text {if }\val{V}{\env} = \Record{\ell=v; w} \\
\text {if }\val{V}{\env} = \Record{\ell=v; w} \\
% Case
\mlab{Case} & \cek{ \Case\; V\, \{ \ell~x \mapsto M; y \mapsto N\} \mid \env \mid \shk}
&\stepsto& \left\{\ba{@{}l@{}}
\cek{ M \mid \env[x \mapsto v] \mid \shk}, \\
\cek{ N \mid \env[y \mapsto \ell'\, v] \mid \shk}, \\
\ea \right.
&
\ba{@{}l@{}}
\text{if }\val{V}{\env} = \ell\, v \\
&\stepsto& \begin{cases}
\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},
\text{if }\val{V}{\env} = \ell'\, v \text{ and } \ell \neq \ell' \\
\ea \\[2ex]
\end{cases}\\
% Let - eval M
\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} \\
% 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},
& \text{if } \hret = \{\Return\; x \mapsto M\} \\
% \mlab{RetTop} & \cek{\Return\;V \mid \env \mid \nil} &\stepsto& \val{V}{\env} \\[1ex]
\text{if } \hret = \{\Return\; x \mapsto M\} \\
% Deep
\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},
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\}} \\
&\stepsto& \cek{M \mid \env'[x \mapsto \val{V}{\env},
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\}} \\
% Shallow
\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},}\\
&&&\multicolumn{2}{@{}r@{}}{\text{if } \ell : A \to B \in E \text{ and } \hell = \{\ell\; x \; r \mapsto M\}} \\
\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\}} \\
% Forward
\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)])},
& \text{if } \hell = \emptyset \\
\ea \\
\el
\end{displaymath}
&\stepsto& \cek{ (\Do \; \ell \; V)^E \mid \env \mid \shk \circ (\shk' \concat [(\slk, (\env', H)^\depth)])}, \text{if } \hell = \emptyset \\
\ea
\]
\textbf{Value interpretation} $~\val{-} : \ValCat \times \MEnvCat \to \MValCat$
\begin{displaymath}
@ -11573,7 +11572,7 @@ If $M \reducesto N$ then $\dstrans{M} \reducesto_{\Cong}^+
\end{theorem}
\begin{proof}
By induction on $\reducesto$ using
By case analysis 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
@ -11903,24 +11902,105 @@ $N'$ such that $N' \approxa \sdtrans{N}$ and $M' \reducesto^+ N'$.
\end{theorem}
%
\begin{proof}
By induction on $\reducesto$ using Lemma~\ref{lem:sdtrans-subst} and
Lemma~\ref{lem:sdtrans-admin}. We show only the interesting case
By case analysis 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.
\begin{description}
\item[Case]
$\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H \reducesto
reduction.\smallskip
\noindent\textbf{Case} $\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H \reducesto
N_\ell[V/p,\lambda y.\EC[\Return\;y]/r]$ where
$\ell \notin \BL(\EC)$ and
$H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$.
$H^\ell = \{\OpCase{\ell}{p}{r} \mapsto N_\ell\}$. \smallskip\\
%
% \begin{derivation}
% \end{derivation}
\dhil{Write the proof sketch}
\end{description}
Pick $M' =
\sdtrans{\ShallowHandle\;\EC[\Do\;\ell~V]\;\With\;H}$. Clearly
$M' \approxa \sdtrans{M}$. We compute $N'$ via reduction as follows.
\begin{derivation}
& \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}
\section{Parameterised handlers as ordinary deep handlers}
@ -11934,10 +12014,10 @@ 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. As with the two
previous translations it is formally defined on terms, types, type
environments, and substitutions. We omit the homomorphic cases and
show only the interesting cases.
parameterised handlers into ordinary deep handlers. Formally, the
translation is defined on terms, types, environments, and
substitutions. We omit the homomorphic cases and show only the
interesting cases.
%
\[
\bl
@ -12029,9 +12109,10 @@ If $\Delta; \Gamma \vdash M : C$ then $\PD{\Delta};
%
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.
handlers in Section~\ref{sec:deep-as-shallow}, this simulation is not
quite on the nose as the image simulates the source only up to
congruence due to the need for an application of a pure function to a
variable to be reduced.
%
\begin{theorem}[Simulation up to congruence]
\label{thm:param-simulation}
@ -12039,9 +12120,10 @@ to a variable to be reduced.
\end{theorem}
%
\begin{proof}
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.
By case analysis on the relation $\reducesto$ using
Lemma~\ref{lem:pd-subst}. The interesting case is
\semlab{Op^\param}, which is where we need to reduce under the
$\lambda$-abstraction representing the parameterised resumption.
% \begin{description}
% \item[Case]
% $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
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
control in the form of shift/reset, and monadic reflection using the
notions of typability-preserving macro-expressiveness and untyped

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