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@@ -743,8 +743,7 @@ control, where programmers can give distinct names to control reifying
operations and separate them from their handling. Throughout this operations and separate them from their handling. Throughout this
dissertation we will see numerous examples of how effect handlers dissertation we will see numerous examples of how effect handlers
makes programming with delimited structured (c.f. the following makes programming with delimited structured (c.f. the following
section, Chapter~\ref{ch:continuations}, and section, Chapter~\ref{ch:ehop}, and Chapter~\ref{ch:continuations}.).
Chapter~\ref{ch:unary-handlers}).
% %
\section{State of effectful programming} \section{State of effectful programming}
@@ -1890,7 +1889,7 @@ handler to run the delimited control variant of $\incrEven$.
% %
Effect handlers come into their own when multiple effects are Effect handlers come into their own when multiple effects are
combined. Throughout the dissertation we will see multiple examples of combined. Throughout the dissertation we will see multiple examples of
handlers in action (e.g. Chapter~\ref{ch:unary-handlers}). handlers in action (e.g. Chapter~\ref{ch:ehop}).
\subsubsection{Effect tracking} \subsubsection{Effect tracking}
% \dhil{Cite \citet{GiffordL86}, \citet{LucassenG88}, \citet{TalpinJ92}, % \dhil{Cite \citet{GiffordL86}, \citet{LucassenG88}, \citet{TalpinJ92},
@@ -1946,7 +1945,7 @@ function under a handler that interprets at least $\Get$ and $\Put$.
Some form of polymorphism is necessary to make an effect system Some form of polymorphism is necessary to make an effect system
extensible and useful in practice. Otherwise effect annotations end up extensible and useful in practice. Otherwise effect annotations end up
pervading the entire program in a similar fashion as monads do. In pervading the entire program in a similar fashion as monads do. In
Chapter~\ref{ch:base-language} we will develop an extensible effect Chapter~\ref{ch:handler-calculi} we will develop an extensible effect
system based on row polymorphism. system based on row polymorphism.
\section{Scope} \section{Scope}
@@ -2078,7 +2077,7 @@ part.
\end{itemize} \end{itemize}
Another contribution worth noting is the continuation literature Another contribution worth noting is the continuation literature
review in Appendix~\ref{ch:continuation}, which provides a review in Appendix~\ref{ch:continuations}, which provides a
comprehensive operational characterisation of various notions of comprehensive operational characterisation of various notions of
continuations and first-class control phenomena. continuations and first-class control phenomena.
@@ -2089,20 +2088,23 @@ this dissertation.
\paragraph{Programming} \paragraph{Programming}
\begin{itemize} \begin{itemize}
\item Chapter~\ref{ch:base-language} introduces a polymorphic fine-grain \item Chapter~\ref{ch:ehop} showcases effect handler oriented
call-by-value core calculus, $\BCalc$, which makes key use of programming in practice by implementing a small operating system
\citeauthor{Remy93}-style row polymorphism to implement polymorphic dubbed \OSname{} based on \citeauthor{RitchieT74}'s original
variants, structural records, and a structural effect system. The \UNIX{}. The implementation starts from a basic notion of file i/o,
calculus distils the essence of the core of the Links programming which evolves into a feature-rich operating system with full file
language. i/o, multiple user environments, multi-tasking, and more, by
composing ever more effect handlers.
\item Chapter~\ref{ch:unary-handlers} presents three extensions of $\BCalc$, \item Chapter~\ref{ch:handler-calculi} introduces a polymorphic
which are $\HCalc$ that adds deep handlers, $\SCalc$ that adds shallow fine-grain call-by-value core calculus, $\BCalc$, which makes key
handlers, and $\HPCalc$ that adds parameterised handlers. The chapter use of \citeauthor{Remy93}-style row polymorphism to implement
also contains a running case study that demonstrates effect handler polymorphic variants, structural records, and a structural effect
oriented programming in practice by implementing a small operating system. The calculus distils the essence of the core of the Links
system dubbed \OSname{} based on \citeauthor{RitchieT74}'s original programming language. The chapter also presents three extensions of
\UNIX{}. $\BCalc$, which are $\HCalc$ that adds deep handlers, $\SCalc$ that
adds shallow handlers, and $\HPCalc$ that adds parameterised
handlers.
\end{itemize} \end{itemize}
\paragraph{Implementation} \paragraph{Implementation}
@@ -6411,11 +6413,11 @@ Therefore let us formally characterise tail calls.
For our purposes, the most robust characterisation is a syntactic For our purposes, the most robust characterisation is a syntactic
characterisation, as opposed to a semantic characterisation, because characterisation, as opposed to a semantic characterisation, because
in the presence of control effects (which we will add in in the presence of control effects (which we will add in
Chapter~\ref{ch:unary-handlers}) it surprisingly tricky to describe Section~\ref{sec:unary-deep-handlers}) it surprisingly tricky to
tail calls in terms of control flow such as ``the last thing to occur describe tail calls in terms of control flow such as ``the last thing
before returning from the enclosing function'' as a function may to occur before returning from the enclosing function'' as a function
return multiple times. In particular, the effects of a function may be may return multiple times. In particular, the effects of a function
replayed several times. may be replayed several times.
% %
For this reason we will adapt a syntactic characterisation of tail For this reason we will adapt a syntactic characterisation of tail
@@ -6842,11 +6844,12 @@ context ($[~]$) and let expressions ($\Let\;x \revto \EC \;\In\;N$).
The choices of using fine-grain call-by-value and evaluation contexts The choices of using fine-grain call-by-value and evaluation contexts
may seem odd, if not arbitrary at this point; the reader may wonder may seem odd, if not arbitrary at this point; the reader may wonder
with good reason why we elect to use fine-grain call-by-value over with good reason why we elect to use fine-grain call-by-value over
ordinary call-by-value. In Chapter~\ref{ch:unary-handlers} we will ordinary call-by-value. In
reap the benefits from our design choices, as we shall see that the Sections~\ref{sec:unary-deep-handlers}--\ref{sec:unary-parameterised-handlers}
combination of fine-grain call-by-value and evaluation contexts we will reap the benefits from our design choices, as we shall see
provide the basis for a convenient, simple semantic framework for that the combination of fine-grain call-by-value and evaluation
working with continuations. contexts provide the basis for a convenient, simple semantic framework
for working with continuations.
\paragraph{Syntactic sugar} \paragraph{Syntactic sugar}
We will adopt a few conventions to make the notation more convenient We will adopt a few conventions to make the notation more convenient
@@ -8334,8 +8337,9 @@ explicit. In CPS every function takes an additional function-argument
called the \emph{continuation}, which represents the next computation called the \emph{continuation}, which represents the next computation
in evaluation position. CPS is canonical in the sense that it is in evaluation position. CPS is canonical in the sense that it is
definable in pure $\lambda$-calculus without any further definable in pure $\lambda$-calculus without any further
primitives. As an informal illustration of CPS consider again the primitives. As an informal illustration of CPS let us consider the
ever-green factorial function from Section~\ref{sec:tracking-div}. ever-green factorial function, which may be implemented in $\BCalc$ as
follows.
% %
\[ \[
\bl \bl
@@ -10841,6 +10845,21 @@ machine.
% embody some high-level abstract models of main memory and the % embody some high-level abstract models of main memory and the
% instruction fetch-execute cycle of processors~\cite{BryantO03}. % instruction fetch-execute cycle of processors~\cite{BryantO03}.
\paragraph{Chapter outline}
\begin{description}
\item[Section~\ref{sec:machine-configurations}] augments the standard
CEK notion of configurations to accommodate generalised
continuations.
\item[Section~\ref{sec:machine-transitions}] gives the reduction rules for the CEK
machine with generalised continuations.
\item[Section~\ref{subsec:machine-realisability}] discusses ways to
realise the machine and potential efficiency pitfalls.
\item[Section~\ref{subsec:machine-correctness}] shows that the machine with generalised
continuations simulates the substitution-based reduction semantics
of $\HCalc$.
\item[Section~\ref{sec:related-work-abstract-machine}] discusses related work.
\end{description}
\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.
% %
@@ -11939,6 +11958,7 @@ By repeated application of Lemma~\ref{lem:machine-simulation}.
\end{proof} \end{proof}
\section{Related work} \section{Related work}
\label{sec:related-work-abstract-machine}
The literature on abstract machines is vast and rich. I describe here The literature on abstract machines is vast and rich. I describe here
the basic structure of some selected abstract machines from the the basic structure of some selected abstract machines from the
literature. literature.
@@ -12132,6 +12152,20 @@ reductions. % The
% latter construction generalises the example of pipes implemented % latter construction generalises the example of pipes implemented
% using deep handlers that we gave in Section~\ref{sec:pipes}. % using deep handlers that we gave in Section~\ref{sec:pipes}.
% %
\paragraph{Chapter outline}
\begin{description}
\item[Section~\ref{sec:deep-as-shallow}] develops an encoding of deep
handlers in terms of shallow handlers.
\item[Section~\ref{sec:shallow-as-deep}] shows an encoding going the
other way, i.e. shallow handlers encoded using deep handlers.
\item[Section~\ref{sec:param-desugaring}] demonstrates that
parameterised handlers are encodable using ordinary deep handlers
with the state-passing technique.
\item[Section~\ref{sec:related-work-interdefinability}] discusses
related work.
\end{description}
\paragraph{Relation to prior work} The results in this chapter has \paragraph{Relation to prior work} The results in this chapter has
been published previously in the following papers. been published previously in the following papers.
% %
@@ -13090,6 +13124,7 @@ variable to be reduced.
% \end{description} % \end{description}
\end{proof} \end{proof}
\section{Related work} \section{Related work}
\label{sec:related-work-interdefinability}
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
@@ -13387,6 +13422,26 @@ languages without such control features.
% these results can be extended to base languages with other features % these results can be extended to base languages with other features
% such as mutable state. % such as mutable state.
\paragraph{Chapter outline}
\begin{description}
\item[Section~\ref{sec:calculi}] introduces the core calculi
$\BPCF \subset \HPCF$, which are essentially simply-typed variations
of $\BCalc$ and $\HCalc$, respectively.
\item[Section~\ref{sec:abstract-machine-semantics}] develops an
abstract machine for each core calculus. Both machines are simpler
variations of the machine developed in
Chapter~\ref{ch:abstract-machine}.
\item[Section~\ref{sec:generic-search}] presents the gadgetry required
to set up and proof the results for the generic count problem. It
also shows that there exists an efficient implementation of generic
count in $\HPCF$.
\item[Section~\ref{sec:pure-counting}] establishes the lower bound
runtime result for implementations of generic count in $\BPCF$.
\item[Section~\ref{sec:robustness}] discusses extensions and variations of the phenomenon.
\item[Section~\ref{sec:experiments}] investigates an empirical evaluation of the phenomenon.
\item[Section~\ref{sec:related-work-efficiency}] discusses related work.
\end{description}
\paragraph{Relation to prior work} This chapter is based entirely on \paragraph{Relation to prior work} This chapter is based entirely on
the following previously published paper. the following previously published paper.
\begin{enumerate} \begin{enumerate}
@@ -13454,13 +13509,11 @@ with effect handlers $\HPCF$, both of which amounts to simply-typed
variations of $\BCalc$ and $\HCalc$, variations of $\BCalc$ and $\HCalc$,
respectively. Sections~\ref{sec:base-calculus}--\ref{sec:handler-machine} respectively. Sections~\ref{sec:base-calculus}--\ref{sec:handler-machine}
essentially recast the developments of essentially recast the developments of
Chapters~\ref{ch:base-language}, \ref{ch:unary-handlers}, and Chapters~\ref{ch:handler-calculi} and \ref{ch:abstract-machine} to fit
\ref{ch:abstract-machine} to fit the calculi $\BPCF$ and $\HPCF$. I the calculi $\BPCF$ and $\HPCF$. I will quickly glance over the
reproduce the details here, even though, they are mostly the same as details here, only highlighting the key differences as I am making use
in the previous chapters save for a few tricks such as a crucial of a crucial design decision in Section~\ref{sec:handlers-calculus} to
design decision in Section~\ref{sec:handlers-calculus} which makes it make continuation reification a constant time operation.
possible to implement continuation reification as a constant time
operation.
\subsection{Base calculus} \subsection{Base calculus}
\label{sec:base-calculus} \label{sec:base-calculus}
@@ -13965,17 +14018,15 @@ expressiveness.
\section{A practical model of computation} \section{A practical model of computation}
\label{sec:abstract-machine-semantics} \label{sec:abstract-machine-semantics}
Thus far we have introduced the base calculus $\BPCF$ and its
extension with effect handlers $\HPCF$. The calculi $\BPCF$ and $\HPCF$ use a substitution model for
% evaluation. Whilst this model is semantically pleasing, it falls short
For each calculus we have given a \emph{small-step operational of providing a realistic account of practical computation as
semantics} which uses a substitution model for evaluation. Whilst substitution is an expensive operation. Instead we shall use a
this model is semantically pleasing, it falls short of providing a slightly simpler variation of the abstract machine from
realistic account of practical computation as substitution is an Chapter~\ref{ch:abstract-machine} as it provides a more practical
expensive operation. Instead we shall use a slightly simpler variation model of computation (it is simpler, because the source language is
of the abstract machine from Chapter~\ref{ch:abstract-machine} as it simpler).
provides a more practical model of computation (it is simpler, because
the source language is simpler).
\subsection{Base machine} \subsection{Base machine}
\label{sec:base-abstract-machine} \label{sec:base-abstract-machine}
@@ -13984,7 +14035,13 @@ the source language is simpler).
\newcommand{\EConf}{\dec{EConf}} \newcommand{\EConf}{\dec{EConf}}
\newcommand{\MVal}{\dec{MVal}} \newcommand{\MVal}{\dec{MVal}}
The base machine operates on configurations of the form The base machine is similar to the abstract machine from
Chapter~\ref{ch:abstract-machine}. The main differences are that the
base machine does not feature support for effect handlers, and
therefore it has a considerably simpler continuation structure, and it
interprets the base language $\BPCF$ rather than $\BCalc$.
Formally, the base machine operates on configurations of the form
$\cek{M \mid \gamma \mid \sigma}$. The first component contains the $\cek{M \mid \gamma \mid \sigma}$. The first component contains the
computation currently being evaluated. The second component contains computation currently being evaluated. The second component contains
the environment $\gamma$ which binds free variables. The third the environment $\gamma$ which binds free variables. The third
@@ -14157,10 +14214,9 @@ is the same. The full details are published in Appendix A of
\label{sec:handler-machine} \label{sec:handler-machine}
\newcommand{\HClosure}{\dec{HClo}} \newcommand{\HClosure}{\dec{HClo}}
% %
We now extend the $\BPCF$ machine with functionality to evaluate The abstract machine for $\HPCF$ closely follows the machine
$\HPCF$ terms. The resulting machine is almost the same as the machine definition in Chapter~\ref{ch:abstract-machine}, though, this machine
in Chapter~\ref{ch:abstract-machine}, though, this machine supports supports only deep handlers.
only deep handlers.
% %
The syntax is extended as follows. The syntax is extended as follows.
% %
@@ -14173,14 +14229,14 @@ The syntax is extended as follows.
\slab{Machine\textrm{ }values} &v, w \in \MValCat &::=& \cdots \mid \rho \\ \slab{Machine\textrm{ }values} &v, w \in \MValCat &::=& \cdots \mid \rho \\
\end{syntax}}% \end{syntax}}%
% %
The notion of configurations changes slightly in that the continuation The notion of configurations changes slightly as the continuation
component is replaced by a generalised continuation component is replaced by a generalised continuation
$\kappa \in \MGContCat$; a continuation is now a list of $\kappa \in \MGContCat$, just as described in Chapter~\ref{ch:abstract-machine}% ; a continuation is now a list of
resumptions. A resumption is a pair of a pure continuation (as in the % resumptions. A resumption is a pair of a pure continuation (as in the
base machine) and a handler closure ($\chi$). % base machine) and a handler closure ($\chi$).
% %
A handler closure consists of an environment and a handler definition, % A handler closure consists of an environment and a handler definition,
where the former binds the free variables that occur in the latter. % where the former binds the free variables that occur in the latter.
% %
The identity continuation is a singleton list containing the identity The identity continuation is a singleton list containing the identity
resumption, which is an empty pure continuation paired with the resumption, which is an empty pure continuation paired with the
@@ -14191,15 +14247,16 @@ identity handler closure:
\kappa_0 \defas [(\nil, (\emptyset, \{\Return\;x \mapsto x\}))] \kappa_0 \defas [(\nil, (\emptyset, \{\Return\;x \mapsto x\}))]
\]}% \]}%
% %
Machine values are augmented to include resumptions as an operation Besides the structure of the continuation component, the primary
difference between the base machine and this machine is that the
machine values are augmented to include resumptions as an operation
invocation causes the topmost frame of the machine continuation to be invocation causes the topmost frame of the machine continuation to be
reified (and bound to the resumption parameter in the operation reified (and bound to the resumption parameter in the operation
clause). clause).
% %
In addition, the handler machine adds transition rules for handlers,
The handler machine adds transition rules for handlers, and modifies and modifies $(\mlab{Let})$ and $(\mlab{PureCont})$ from the base
$(\mlab{Let})$ and $(\mlab{PureCont})$ from the base machine to account machine to account for the richer continuation
for the richer continuation
structure. Figure~\ref{fig:abstract-machine-semantics-handlers} structure. Figure~\ref{fig:abstract-machine-semantics-handlers}
depicts the new and modified rules. depicts the new and modified rules.
% %
@@ -16345,6 +16402,7 @@ However, the effectful implementation runs between 2 and 14 times as
fast with SML/NJ compared with MLton. fast with SML/NJ compared with MLton.
\section{Related work} \section{Related work}
\label{sec:related-work-efficiency}
There are relatively little work in the present literature on There are relatively little work in the present literature on
expressivity that has focused on complexity difference. expressivity that has focused on complexity difference.
% %
@@ -16400,19 +16458,19 @@ I will begin this chapter with a brief summary of this
dissertation. The following sections each elaborates and spells out dissertation. The following sections each elaborates and spells out
directions for future work. directions for future work.
In Part~\ref{p:background} of this dissertation I have compiled an % In Part~\ref{p:background} of this dissertation I have compiled an
extensive survey of first-class control. In this survey I characterise % extensive survey of first-class control. In this survey I characterise
the various kinds of control phenomena that appear in the literature % the various kinds of control phenomena that appear in the literature
as well as providing an overview of the operational characteristics of % as well as providing an overview of the operational characteristics of
control operators appearing in the literature. To the best of my % control operators appearing in the literature. To the best of my
knowledge this survey is the only of its kind in the present % knowledge this survey is the only of its kind in the present
literature (other studies survey a particular control phenomenon, % literature (other studies survey a particular control phenomenon,
e.g. \citet{FelleisenS99} survey undelimited continuations as provided % e.g. \citet{FelleisenS99} survey undelimited continuations as provided
by call/cc in programming practice, \citet{DybvigJS07} classify % by call/cc in programming practice, \citet{DybvigJS07} classify
delimited control operators according to their delimiter placement, % delimited control operators according to their delimiter placement,
\citet{Brachthauser20} provides a delimited control perspective on % \citet{Brachthauser20} provides a delimited control perspective on
effect handlers, and \citet{MouraI09} survey the use of first-class % effect handlers, and \citet{MouraI09} survey the use of first-class
control operators to implement coroutines). % control operators to implement coroutines).
In Part~\ref{p:design} I have presented the design of a ML-like In Part~\ref{p:design} I have presented the design of a ML-like
programming language equipped an effect-and-type system and a programming language equipped an effect-and-type system and a
@@ -16437,11 +16495,11 @@ asymptotic improvement in runtime performance for some class of
programs. programs.
\section{Programming with effect handlers} \section{Programming with effect handlers}
Chapters~\ref{ch:base-language} and \ref{ch:unary-handlers} present Chapter~\ref{ch:handler-calculi} presents the design of a core
the design of a core calculus that forms the basis for Links, which is calculus that forms the basis for Links, which is a practical
a practical programming language with deep, shallow, and parameterised programming language with deep, shallow, and parameterised effect
effect handlers. A distinguishing feature of the core calculus is that handlers. A distinguishing feature of the core calculus is that it is
it is based on a structural notion of data and effects, whereas other based on a structural notion of data and effects, whereas other
literature predominantly consider nominal data and effects. In the literature predominantly consider nominal data and effects. In the
setting of structural effects the effect system play a pivotal role in setting of structural effects the effect system play a pivotal role in
ensuring that the standard safety and soundness properties of ensuring that the standard safety and soundness properties of
@@ -16461,7 +16519,7 @@ two functions may introduce arbitrary effects that need to be handled
accordingly. Alternatively, a composition of any two functions may accordingly. Alternatively, a composition of any two functions may
inadvertently eliminate arbitrary effects, and as such, programming inadvertently eliminate arbitrary effects, and as such, programming
with effect handlers without an effect system is prone to error. The with effect handlers without an effect system is prone to error. The
\UNIX{} case study in Chapter~\ref{ch:unary-handlers} demonstrates how \UNIX{} case study in Chapter~\ref{ch:ehop} demonstrates how
the effect system assists to ensure that effectful function the effect system assists to ensure that effectful function
compositions are meaningful. compositions are meaningful.
@@ -16578,7 +16636,7 @@ Other work has attempted to exploit the algebraic structure of (deep)
effect handlers to fuse nested handlers~\cite{WuS15}. effect handlers to fuse nested handlers~\cite{WuS15}.
% %
An obvious idea is to apply these lines of work to the handler calculi An obvious idea is to apply these lines of work to the handler calculi
of Chapter~\ref{ch:unary-handlers}. of Chapter~\ref{ch:handler-calculi}.
% %
Moreover, I hypothesise there is untapped potential in the combination Moreover, I hypothesise there is untapped potential in the combination
of effect-dependent analysis with respect to equational theories to of effect-dependent analysis with respect to equational theories to
@@ -16623,7 +16681,7 @@ resumptions, e.g. suppose some channel first expects an integer
followed by a boolean, then the running the program followed by a boolean, then the running the program
$\Do\;\Fork\,\Unit;\keyw{send}~42;\Absurd\;\Do\;\Fail\,\Unit$ under $\Do\;\Fork\,\Unit;\keyw{send}~42;\Absurd\;\Do\;\Fail\,\Unit$ under
the composition of the nondeterminism handler and default failure the composition of the nondeterminism handler and default failure
handler from Chapter~\ref{ch:unary-handlers} will cause the primitive handler from Chapter~\ref{ch:ehop} will cause the primitive
$\keyw{send}$ operation to supply two integers in succession, thus $\keyw{send}$ operation to supply two integers in succession, thus
breaking the session protocol. Figuring out how to safely combine breaking the session protocol. Figuring out how to safely combine
linear resources, such as channels, and handlers with multi-shot linear resources, such as channels, and handlers with multi-shot
@@ -18964,9 +19022,8 @@ that the type of the payload is compatible with the declared type.
This particular presentation is nominal, because operations are This particular presentation is nominal, because operations are
declared up front. Nominal typing is the only sound option in the declared up front. Nominal typing is the only sound option in the
absence of an effect system (unless we restrict operations to work absence of an effect system (unless we restrict operations to work
over a fixed type, say, an integer). In over a fixed type, say, an integer). Chapter~\ref{ch:handler-calculi}
Chapter~\ref{ch:unary-handlers} we see a different presentation based provides a different presentation based on structural typing.
on structural typing.
The dynamic semantics of effect handlers are similar to that of The dynamic semantics of effect handlers are similar to that of
$\fcontrol$, though, the $\slab{Value}$ rule is more interesting. $\fcontrol$, though, the $\slab{Value}$ rule is more interesting.
@@ -19149,8 +19206,8 @@ continuation without the handler.
% \end{reductions} % \end{reductions}
% %
Chapter~\ref{ch:unary-handlers} contains further examples of deep and Chapter~\ref{ch:ehop} contains further examples of deep and shallow
shallow handlers in action. handlers in action.
% %
% \dhil{Consider whether to present the below encodings\dots} % \dhil{Consider whether to present the below encodings\dots}
% % % %