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Row polymorphism

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88
thesis.bib
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@@ -592,16 +592,6 @@
year = {2017}
}
@inproceedings{Leijen05,
author = {Daan Leijen},
title = {Extensible records with scoped labels},
booktitle = {Trends in Functional Programming},
series = {Trends in Functional Programming},
volume = {6},
pages = {179--194},
publisher = {Intellect},
year = {2005}
}
# Helium
@article{BiernackiPPS18,
@@ -1205,6 +1195,50 @@
year = {2006}
}
# Row polymorphism
@inproceedings{Wand87,
author = {Mitchell Wand},
title = {Complete Type Inference for Simple Objects},
booktitle = {{LICS}},
pages = {37--44},
publisher = {{IEEE} Computer Society},
year = {1987}
}
@inbook{Remy94,
author = {Didier R\'{e}my},
title = {Type Inference for Records in Natural Extension of ML},
year = 1994,
OPTisbn = {026207155X},
publisher = {MIT Press},
address = {Cambridge, MA, USA},
booktitle = {Theoretical Aspects of Object-Oriented Programming: Types, Semantics, and Language Design},
pages = {67--95},
numpages = {29}
}
@inproceedings{Leijen05,
author = {Daan Leijen},
title = {Extensible records with scoped labels},
booktitle = {Trends in Functional Programming},
series = {Trends in Functional Programming},
volume = {6},
pages = {179--194},
publisher = {Intellect},
year = {2005}
}
@article{MorrisM19,
author = {J. Garrett Morris and
James McKinna},
title = {Abstracting extensible data types: or, rows by any other name},
journal = {Proc. {ACM} Program. Lang.},
volume = {3},
number = {{POPL}},
pages = {12:1--12:28},
year = {2019}
}
# fancy row typing systems that support shapes
@inproceedings{BerthomieuS95,
author = {Bernard Berthomieu and Camille le Moniès de Sagazan},
@@ -1221,7 +1255,6 @@
year = {1993},
}
# Zipper data structure.
@article{Huet97,
author = {G{\'{e}}rard P. Huet},
@@ -3497,6 +3530,26 @@
year = {1999}
}
@inproceedings{BentonB07,
author = {Nick Benton and
Peter Buchlovsky},
title = {Semantics of an effect analysis for exceptions},
booktitle = {{TLDI}},
pages = {15--26},
publisher = {{ACM}},
year = {2007}
}
@article{BentonK99,
author = {Nick Benton and
Andrew Kennedy},
title = {Monads, Effects and Transformations},
journal = {Electron. Notes Theor. Comput. Sci.},
volume = {26},
pages = {3--20},
year = {1999}
}
# Intensional type theory
@book{MartinLof84,
author = {Per Martin{-}L{\"{o}}f},
@@ -3628,3 +3681,16 @@
publisher = {Springer},
year = {2014}
}
# TT lifting
@inproceedings{LindleyS05,
author = {Sam Lindley and
Ian Stark},
title = {Reducibility and TT-Lifting for Computation Types},
booktitle = {{TLCA}},
series = {Lecture Notes in Computer Science},
volume = {3461},
pages = {262--277},
publisher = {Springer},
year = {2005}
}

154
thesis.tex
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@@ -123,6 +123,7 @@
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{claim}[theorem]{Claim}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
% Example environment.
@@ -682,6 +683,7 @@ operations and separate the from their handling.
\dhil{Maybe expand this a bit more to really sell it}
\section{State of effectful programming}
\label{sec:state-of-effprog}
Functional programmers tend to view programs as impenetrable black
boxes, whose outputs are determined entirely by their
@@ -1535,6 +1537,7 @@ The free monad brings us close to the essence of programming with
effect handlers.
\subsection{Back to direct-style}
\label{sec:back-to-directstyle}
Monads do not freely compose, because monads must satisfy a
distributive property in order to combine~\cite{KingW92}. Alas, not
every monad has a distributive property.
@@ -5813,10 +5816,10 @@ contexts.
\end{proof}
%
The calculus enjoys a rather strong \emph{progress} property, which
states that \emph{every} closed computation term reduces to a trivial
computation term $\Return\;V$ for some value $V$. In other words, any
realisable function in \BCalc{} is effect-free and total.
The calculus satisfies the standard \emph{progress} property, which
states that \emph{every} closed computation term either reduces to a
trivial computation term $\Return\;V$ for some value $V$, or there
exists some $N$ such that $M \reducesto N$.
%
\begin{definition}[Computation normal form]\label{def:base-language-comp-normal}
A computation $M \in \CompCat$ is said to be \emph{normal} if it is
@@ -5841,29 +5844,41 @@ which states that if some computation $M$ is well typed and reduces to
some other computation $M'$, then $M'$ is also well typed.
%
\begin{theorem}[Subject reduction]\label{thm:base-language-preservation}
Suppose $\typ{\Gamma}{M : C}$ and $M \reducesto M'$, then
$\typ{\Gamma}{M' : C}$.
Suppose $\typ{\Delta;\Gamma}{M : C}$ and $M \reducesto N$, then
$\typ{\Delta;\Gamma}{N : C}$.
\end{theorem}
%
\begin{proof}
By induction on the typing derivations.
\end{proof}
Even more the calculus, as specified, is \emph{strongly normalising},
meaning that every closed computation term reduces to a trivial
computation term. In other words, any realisable function in $\BCalc$
is effect-free and total.
%
\begin{claim}[Type soundness]
If $\typ{}{M : C}$, then there exists $\typ{}{N : C}$ such that
$M \reducesto^\ast N \not\reducesto$ and $N$ is normal.
\end{claim}
%
We will not prove this theorem here as the required proof gadgetry is
rather involved~\cite{LindleyS05}, and we will soon dispense of this
property.
\section{A primitive effect: recursion}
\label{sec:base-language-recursion}
%
As evident from Theorem~\ref{thm:base-language-progress} \BCalc{}
admit no computational effects. As a consequence every realisable
program is terminating. Thus the calculus provide a solid and minimal
basis for studying the expressiveness of any extension, and in
particular, which primitive effects any such extension may sneak into
the calculus.
As $\BCalc$ is (claimed to be) strongly normalising it provides a
solid and minimal basis for studying the expressiveness of any
extension, and in particular, which primitive effects any such
extension may sneak into the calculus.
However, we cannot write many (if any) interesting programs in
\BCalc{}. The calculus is simply not expressive enough. In order to
bring it closer to the ML-family of languages we endow the calculus
with a fixpoint operator which introduces recursion as a primitive
effect. We dub the resulting calculus \BCalcRec{}.
bring the calculus closer to the ML-family of languages we endow the
calculus with a fixpoint operator which introduces recursion as a
primitive effect. % We dub the resulting calculus \BCalcRec{}.
%
First we augment the syntactic category of values with a new
@@ -5908,7 +5923,7 @@ progress theorem as some programs may now diverge.
\subsection{Tracking divergence via the effect system}
\label{sec:tracking-div}
%
With the $\Rec$-operator in \BCalcRec{} we can now implement the
With the $\Rec$-operator in place we can now implement the
factorial function.
%
\[
@@ -5953,16 +5968,17 @@ In this typing rule we have chosen to inject an operation named
$\dec{Div}$ into the effect row of the computation type on the
recursive binder $f$. The operation is primitive, because it can never
be directly invoked, rather, it occurs as a side-effect of applying a
$\Rec$-definition.
$\Rec$-definition (this is also why we ascribe it type $\ZeroType$,
though, we may use whatever type we please).
%
Using this typing rule we get that
$\dec{fac} : \Int \to \Int \eff \{\dec{Div}:\ZeroType\}$. Consequently,
every application-site of $\dec{fac}$ must now permit the $\dec{Div}$
operation in order to type check.
%
\begin{example}
We will use the following suspended computation to demonstrate
effect tracking in action.
% \begin{example}
To illustrate effect tracking in action consider the following
suspended computation.
%
\[
\bl
@@ -5972,10 +5988,11 @@ operation in order to type check.
%
The computation calculates $3!$ when forced.
%
We will now give a typing derivation for this computation to
illustrate how the application of $\dec{fac}$ causes its effect row
to be propagated outwards. Let
$\Gamma = \{\dec{fac} : \Int \to \Int \eff \{\dec{Div}:\ZeroType\}\}$.
The typing derivation for this computation illustrates how the
application of $\dec{fac}$ causes its effect row to be propagated
outwards. Let
$\Gamma = \{\dec{fac} : \Int \to \Int \eff
\{\dec{Div}:\ZeroType\}\}$.
%
\begin{mathpar}
\inferrule*[Right={\tylab{Lam}}]
@@ -5990,8 +6007,8 @@ operation in order to type check.
%
The information that the computation applies a possibly divergent
function internally gets reflected externally in its effect
signature.
\end{example}
signature.\medskip
%
A possible inconvenience of the current formulation of
$\tylab{Rec}^\ast$ is that it recursion cannot be mixed with other
@@ -6027,15 +6044,24 @@ The term `pure' is heavily overloaded in the programming literature.
%
\dhil{In this thesis we use the Haskell notion of purity.}
\section{Row polymorphism}
\label{sec:row-polymorphism}
\dhil{A discussion of alternative row systems}
\section{Related work}
\paragraph{Row polymorphism} Row polymorphism was originally
introduced by \citet{Wand87} as a typing discipline for extensible
records. The style of row polymorphism used in this chapter was
originally developed by \citet{Remy94}.
\citet{MorrisM19} have developed a unifying theory of rows, which
collects the aforementioned row systems under one umbrella. Their
system provides a general account of record extension and projection,
and dually, variant injection and branching.
\paragraph{Effect tracking} As mentioned in
Section~\ref{sec:back-to-directstyle} the original effect system
was developed by \citet{LucassenG88} to provide a lightweight facility
for static concurrency analysis. \citet{NielsonN99} \citet{TofteT97}
\citet{BentonK99} \citet{BentonB07}
% \section{Type and effect inference}
% \dhil{While I would like to detail the type and effect inference, it
% may not be worth the effort. The reason I would like to do this goes
% back to 2016 when Richard Eisenberg asked me about how we do effect
% inference in Links.}
\chapter{Effect handler oriented programming}
\label{ch:unary-handlers}
@@ -6073,17 +6099,17 @@ interpret the effect signature of the whole program.
% trees. Dually, \emph{shallow} handlers are defined as case-splits over
% computation trees.
%
The purpose of the chapter is to augment the base calculi \BCalc{} and
\BCalcRec{} with effect handlers, and demonstrate their practical
versatility by way of a programming case study. The primary focus is
on so-called \emph{deep} and \emph{shallow} variants of handlers. In
The purpose of the chapter is to augment the base calculi \BCalc{}
with effect handlers, and demonstrate their practical versatility by
way of a programming case study. The primary focus is on so-called
\emph{deep} and \emph{shallow} variants of handlers. In
Section~\ref{sec:unary-deep-handlers} we endow \BCalc{} with deep
handlers, which we put to use in
Section~\ref{sec:deep-handlers-in-action} where we implement a
\UNIX{}-style operating system. In
Section~\ref{sec:unary-shallow-handlers} we extend \BCalcRec{} with
shallow handlers, and subsequently we use them to extend the
functionality of the operating system example. Finally, in
Section~\ref{sec:unary-shallow-handlers} we extend \BCalc{} with shallow
handlers, and subsequently we use them to extend the functionality of
the operating system example. Finally, in
Section~\ref{sec:unary-parameterised-handlers} we will look at
\emph{parameterised} handlers, which are a refinement of ordinary deep
handlers.
@@ -6122,7 +6148,7 @@ without the recursion operator and extend it with deep handlers to
yield the calculus \HCalc{}. We elect to do so because deep handlers
do not require the power of an explicit fixpoint operator to be a
practical programming abstraction. Building \HCalc{} on top of
\BCalcRec{} require no change in semantics.
\BCalc{} with the recursion operator requires no change in semantics.
%
Deep handlers~\cite{PlotkinP09,Pretnar10} are defined by folds
(specifically \emph{catamorphisms}~\cite{MeijerFP91}) over computation
@@ -9067,9 +9093,9 @@ in terms of two mutually recursive functions (specifically
implement with deep
handlers~\cite{KammarLO13,HillerstromL18,HillerstromLA20}.
In this section we take $\BCalcRec$ as our starting point and extend
it with shallow handlers, resulting in the calculus $\SCalc$. The
calculus borrows some syntax and semantics from \HCalc{}, whose
In this section we take the full $\BCalc$ as our starting point and
extend it with shallow handlers, resulting in the calculus $\SCalc$.
The calculus borrows some syntax and semantics from \HCalc{}, whose
presentation will not be duplicated in this section.
% Often deep handlers are attractive because they are semantically
% well-behaved and provide appropriate structure for efficient
@@ -10500,7 +10526,7 @@ labels ($\ell$).
Computations ($M$) comprise values ($V$), applications ($M~V$), pair
elimination ($\Let\; \Record{x, y} = V \;\In\; N$), label elimination
($\Case\; V \;\{\ell \mapsto M; x \mapsto N\}$), and explicit marking
of unreachable code ($\Absurd$). A key difference from $\BCalcRec$ is
of unreachable code ($\Absurd$). A key difference from $\BCalc$ is
that the function position of an application is allowed to be a
computation (i.e., the application form is $M~W$ rather than
$V~W$). Later, when we refine the initial CPS translation we will be
@@ -16844,25 +16870,25 @@ predicates. We now prove that no implementation in $\BPCF$ can match
this: in fact, we establish a lower bound of $\Omega(n2^n)$ for the
runtime of any counting program on \emph{any} $n$-standard predicate.
This mathematically rigorous characterisation of the efficiency gap
between languages with and without first-class control constructs is
the central contribution of the paper.
between languages with and without effect handlers is the objective of
this chapter.
One might ask at this point whether the claimed lower bound could not
be obviated by means of some known continuation passing style (CPS) or
monadic transform of effect handlers
\cite{HillerstromLAS17,Leijen17}. This can indeed be done, but only by
dint of changing the type of our predicates $P$ which would defeat the
purpose of our enquiry. We want to investigate the relative power of
various languages for manipulating predicates that are given to us in
a certain way which we do not have the luxury of choosing.
% One might ask at this point whether the claimed lower bound could not
% be obviated by means of some known continuation passing style (CPS) or
% monadic transform of effect handlers
% \cite{HillerstromLAS17,Leijen17}. This can indeed be done, but only by
% dint of changing the type of our predicates $P$ which would defeat the
% purpose of our enquiry. We want to investigate the relative power of
% various languages for manipulating predicates that are given to us in
% a certain way which we do not have the luxury of choosing.
To get a feel for the issues that our proof must address, let us
consider how one might construct a counting program in
$\BPCF$. The \naive approach, of course, would be simply to apply the
given predicate $P$ to all $2^n$ possible $n$-points in turn, keeping
a count of those on which $P$ yields true. It is a routine exercise to
implement this approach in $\BPCF$, yielding (parametrically in $n$)
a program
To get a feel for the issues that the proof must address, let us
consider how one might construct a counting program in $\BPCF$. The
\naive approach, of course, would be simply to apply the given
predicate $P$ to all $2^n$ possible $n$-points in turn, keeping a
count of those on which $P$ yields true. It is a routine exercise to
implement this approach in $\BPCF$, yielding (parametrically in $n$) a
program
%
{
\[