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  1. 12
      thesis.bib
  2. 118
      thesis.tex

12
thesis.bib
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@ -2434,6 +2434,18 @@
year = {2015}
}
# Abstract machines
@article{BiernackaBD03,
author = {Malgorzata Biernacka and Dariusz Biernacki and Olivier Danvy},
title = {An Operational Foundation for Delimited Continuations},
volume = {10},
OPTdoi = {10.7146/brics.v10i41.21809},
number = {41},
journal = {BRICS Report Series},
year = {2003},
OPTmonth = dec
}
# Partial evaluation with control
@inproceedings{LawallD94,
author = {Julia L. Lawall and

118
thesis.tex
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@ -10820,13 +10820,13 @@ on work in the following previously published papers.
The particular presentation in this chapter is adapted from
item~\ref{en:ch-am-HLA20}.
\section{Machine configurations}
\section{Configurations with generalised continuations}
\label{sec:machine-configurations}
The abstract machine is formally defined in terms of configurations. A
configuration $\cek{M \mid \env \mid \shk \circ \shk'}$ is a triple
consisting of a computation term $M \in \CompCat$, an environment
$\env \in \MEnvCat$, and a two generalised continuations
$\kappa,\kappa' \in \MGContCat$.
configuration $\cek{M \mid \env \mid \shk \circ \shk'} \in \MConfCat$
is a triple consisting of a computation term $M \in \CompCat$, an
environment $\env \in \MEnvCat$, and a pair of generalised
continuations $\kappa,\kappa' \in \MGContCat$.
%
The complete abstract machine syntax is given in
Figure~\ref{fig:abstract-machine-syntax}.
@ -10835,38 +10835,69 @@ The control and environment components are completely standard as they
are similar to the components in \citeauthor{FelleisenF86}'s original
CEK machine modulo the syntax of the source language.
%
The structure of the continuation component is new. This component
comprises two generalised continuations, where the latter continuation
is an entirely administrative object that only manifests during effect
forwarding. The continuation component The latter continuation is
always the identity, except when forwarding an operation, in which
case it is used to keep track of the extent to which the operation has
been forwarded.
However, the structure of the continuation component is new. This
component comprises two generalised continuations, where the latter
continuation $\kappa'$ is an entirely administrative object that
materialises only during operation invocations as it is used to
construct the reified segment of the continuation up to an appropriate
enclosing handler. For the most part $\kappa'$ is empty, therefore we
will write $\cek{M \mid \env \mid \shk}$ as syntactic sugar for
$\cek{M \mid \env \mid \shk \circ []}$ where $[]$ is the empty
continuation (an alternative is to syntactically differentiate between
regular and administrative configurations by having both three-place
and four-place configurations as for example as \citet{BiernackaBD03}
do).
%
An environment is either empty, written $\emptyset$, or an extension
of some other environment $\env$, written $\env[x \mapsto v]$, where
$x$ is the name of a variable and $v$ is a machine value.
%
The machine values consist of function closures, recursive function
closures, type function closures, records, variants, and reified
continuations. The three abstraction forms are paired with an
environment that binds the free variables in the their bodies. The
records and variants are transliterated from the value forms of the
source calculi. Figure~\ref{fig:abstract-machine-val-interp} defines
the value interpretation function, which turns any source language
value into a corresponding machine value.
%
A continuation $\shk$ is a stack of generalised continuation frames
$[\shf_1, \dots, \shf_n]$. Each frame $\shf = (\slk, \chi)$ represents
pure continuation $\slk$, corresponding to a sequence of let bindings,
inside handler closure $\chi$.
%
We write $\cek{M \mid \env \mid \shk}$ as syntactic sugar for $\cek{M
\mid \env \mid \shk \circ []}$ where $[]$ is the identity
continuation.
A pure continuation is a stack of pure frames. A pure frame
$(\env, x, N)$ closes a let-binding $\Let \;x=[~] \;\In\;N$ over
environment $\env$. The pure continuation structure is similar to the
continuation structure of \citeauthor{FelleisenF86}'s original CEK
machine.
%
Values consist of function closures, type function closures, records,
variants, and captured continuations.
There are three kinds of handler closures, one for each kind of
handler. A deep handler closure is a pair $(\env, H)$ which closes a
deep handler definition $H$ over environment $\env$. Similarly, a
shallow handler closure $(\env, H^\dagger)$ closes a shallow handler
definition over environment $\env$. Finally, a parameterised handler
closure $(\env, (q.\,H))$ closes a parameterised handler definition
over environment $\env$. As a syntactic shorthand we write $H^\depth$
to range over deep, shallow, and parameterised handler
definitions. Sometimes $H^\depth$ will range over just two kinds of
handler definitions; it will be clear from the context which handler
definition is omitted.
%
A continuation $\shk$ is a stack of frames $[\shf_1, \dots,
\shf_n]$. We annotate captured continuations with input types in order
to make the results of Section~\ref{subsec:machine-correctness} easier
to state. Each frame $\shf = (\slk, \chi)$ represents pure
continuation $\slk$, corresponding to a sequence of let bindings,
inside handler closure $\chi$.
We extend the clause projection notation to handler closures and
generalised continuation frames, i.e.
%
A pure continuation is a stack of pure frames. A pure frame $(\env, x,
N)$ closes a let-binding $\Let \;x=[~] \;\In\;N$ over environment
$\env$. A handler closure $(\env, H)$ closes a handler definition $H$
over environment $\env$.
\[
\ba{@{~}l@{~}c@{~}l@{~}c@{~}l@{~}l}
\theta^{\mathrm{ret}} &\defas& (\sigma, \chi^{\mathrm{ret}}) &\defas& \hret, &\quad \text{where } \chi = (\env, H^\depth)\\
\theta^{\ell} &\defas& (\sigma, \chi^{\ell}) &\defas& \hell, &\quad \text{where } \chi = (\env, H^\depth)
\el
\]
%
We 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.
Values are annotated with types where appropriate to facilitate type
reconstruction in order to make the results of
Section~\ref{subsec:machine-correctness} easier to state.
%
%
\begin{figure}[t]
@ -10874,13 +10905,15 @@ deconstruct stacks.
\begin{syntax}
\slab{Configurations} & \conf \in \MConfCat &::= & \cek{M \mid \env \mid \shk \circ \shk'} \\
\slab{Value\textrm{ }environments} &\env \in \MEnvCat &::= & \emptyset \mid \env[x \mapsto v] \\
\slab{Values} &v, w \in \MValCat &::= & (\env, \lambda x^A . M) \mid (\env, \Lambda \alpha^K . M) \\
& &\mid& \Record{} \mid \Record{\ell = v; w} \mid (\ell\, v)^R \mid \shk^A \mid (\shk, \slk)^A \medskip\\
\slab{Values} &v, w \in \MValCat &::= & (\env, \lambda x^A . M) \mid (\env, \Rec^{A \to C}\,x.M)\\
& &\mid& (\env, \Lambda \alpha^K . M) \\
& &\mid& \Record{} \mid \Record{\ell = v; w} \mid (\ell\, v)^R \\
& &\mid& \shk^A \mid (\shk, \slk)^A \medskip\\
\slab{Continuations} &\shk \in \MGContCat &::= & \nil \mid \shf \cons \shk \\
\slab{Continuation\textrm{ }frames} &\shf \in \MGFrameCat &::= & (\slk, \chi) \\
\slab{Pure\textrm{ }continuations} &\slk \in \MPContCat &::= & \nil \mid \slf \cons \slk \\
\slab{Pure\textrm{ }continuation\textrm{ }frames} &\slf \in \MPFrameCat &::= & (\env, x, N) \\
\slab{Handler\textrm{ }closures} &\chi \in \MHCloCat &::= & (\env, H) \mid (\env, H)^\dagger \mid (\env, (q.\,H)) \medskip \\
\slab{Handler\textrm{ }closures} &\chi \in \MHCloCat &::= & (\env, H) \mid (\env, H^\dagger) \mid (\env, (q.\,H)) \medskip \\
\end{syntax}
\caption{Abstract machine syntax.}
@ -10910,7 +10943,7 @@ deconstruct stacks.
\end{figure}
%
\section{Machine transitions}
\section{Generalised continuation-based machine semantics}
\label{sec:machine-transitions}
%
\begin{figure}[p]
@ -11013,7 +11046,7 @@ deconstruct stacks.
\[
\bl
\ba{@{~}l@{\quad}l@{~}l}
\multicolumn{2}{l}{\textbf{Identity continuation}}\\
\multicolumn{2}{l}{\textbf{Initial continuation}}\\
\multicolumn{3}{l}{\quad\shk_0 \defas [(\nil, (\emptyset, \{\Return\;x \mapsto x\}))]}
\medskip\\
%
@ -11032,7 +11065,12 @@ deconstruct stacks.
The abstract machine semantics defining the transition function $\stepsto$ is given in
Fig.~\ref{fig:abstract-machine-semantics}.
%
It depends on an interpretation function $\val{-}$ for values.
We 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.
%
The machine is initialised (\mlab{Init}) by placing a term in a
configuration alongside the empty environment and identity
@ -11342,7 +11380,7 @@ type $A$, or get stuck failing to handle an operation appearing in
$E$. We now make the correspondence between the operational semantics
and the abstract machine more precise.
\section{Correctness}
\section{Machine correctness}
\label{subsec:machine-correctness}
Fig.~\ref{fig:config-to-term} defines an inverse mapping $\inv{-}$
@ -11400,6 +11438,8 @@ If $\typc{}{M : A}{E}$ and $M \reducesto^+ N \not\reducesto$, then $M
\stepsto^+ \conf$ with $\inv{\conf} = N$.
\end{corollary}
\section{Related work}
\part{Expressiveness}
\chapter{Interdefinability of effect handlers}

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