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Note on higher order functions

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Daniel Hillerström
2021-05-24 18:51:40 +01:00
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6
macros.tex
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@@ -382,6 +382,12 @@
% configurations
\newcommand{\conf}{\mathcal{C}}
% effect sugar
\newcommand{\trcomp}[1]{\sembr{#1}}
\newcommand{\trval}[1]{\sembr{#1}}
\newcommand{\treff}[1]{\sembr{#1}}
\newcommand{\pp}{\Theta}
% UNIX example
\newcommand{\UNIX}{UNIX}
\newcommand{\OSname}[0]{Tiny UNIX}

14
thesis.bib
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@@ -3574,4 +3574,16 @@
publisher = {Springer Netherlands},
pages = {25--63},
OPTisbn = {978-94-017-1291-0}
}
}
# Sixth order function example
@article{Okasaki98,
author = {Chris Okasaki},
title = {Functional Pearl: Even Higher-Order Functions for Parsing},
journal = {J. Funct. Program.},
volume = {8},
number = {2},
pages = {195--199},
year = {1998}
}

66
thesis.tex
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@@ -610,17 +610,18 @@ efficiency.
\section{Why first-class control matters}
From the perspective of programmers first-class control is a valuable
programming feature because it enables them to implement their own
control idioms as if they were native to the programming
language. More important, with first-class control programmer-defined
control idioms are local phenomena which can be encapsulated in a
library such that the rest of the program does not need to be made
aware of their existence. Conversely, without first-class control some
control idioms can only be implemented using global program
restructuring techniques such as continuation passing style.
control idioms as if they were native to the programming language (in
fact this is the very definition of first-class control). More
important, with first-class control programmer-defined control idioms
are local phenomena which can be encapsulated in a library such that
the rest of the program does not need to be made aware of their
existence. Conversely, without first-class control some control idioms
can only be implemented using global program restructuring techniques
such as continuation passing style.
From the perspective of compiler engineers first-class control is
valuable because it unifies several control-related constructs under
one single construct. First-class control can be beneficial for
one single construct. First-class control can even be beneficial for
implementing programming languages which have no notion of first-class
control in source language. A runtime with support for first-class
control can considerably simplify and ease maintainability of an
@@ -839,7 +840,7 @@ return type $R$ with state of type $S$, we transform the function
signature as follows.
%
\[
\val{A_1 \to \cdots \to A_n \to R}_S
\sembr{A_1 \to \cdots \to A_n \to R}_S
\defas A_1 \to \cdots \to A_n \to S \to R \times S
\]
%
@@ -6496,6 +6497,53 @@ getting stuck on an unhandled operation.
By induction on the typing derivations.
\end{proof}
\subsection{Effect sugar}
\label{sec:effect-sugar}
The class of second-order functions capture many familiar and useful
functions (although, it should be noted that there exist useful
functions at higher order, e.g. in
Chapter~\ref{ch:handlers-efficiency} we shall use third-order
functions; for an example of a sixth order function see
\citet{Okasaki98}).
%
\[
\bl
\pp : \EffectCat \to \EffectCat\\
\ba{@{}l@{~}c@{~}l}
\pp(\{\cdot\}) &\defas& \{\cdot\}\\
\pp(\{\varepsilon\}) &\defas& \{\varepsilon\}\\
\pp(\{\ell : A \opto B\} \cup E) &\defas& \{\ell : \theta \} \cup \pp(E),\quad \text{fresh}~\theta
\ea\\
\el
\]
%
\[
\bl
\trcomp{-} : \ValTypeCat \times \EffectCat \to \ValTypeCat\\
\trcomp{A \to B}_E \defas \trval{A}_{E} \to \trval{B}_{\{\varepsilon\}} \eff E,\quad \text{fresh}~\varepsilon\\
\trcomp{A \to B \eff E}_{E'} \defas \trval{A}_{E \cup E'} \to \trval{B}_{\{\varepsilon\}} \eff E \cup E',\quad \text{fresh}~\varepsilon \smallskip\\
\trcomp{-} : \HandlerTypeCat \to \HandlerTypeCat\\
\trcomp{A \Harrow B} \defas \trcomp{B}_{\{\varepsilon\}} \eff \{\varepsilon\} \Harrow \trcomp{B}_{\{\varepsilon\}} \eff \{\varepsilon\},\quad \text{fresh}~\varepsilon\\
\trcomp{A \eff E_A \Harrow B} \defas \trcomp{A}_{E_A} \Harrow \trcomp{B}_{\{\varepsilon\}} \eff \pp(E),\quad \text{fresh}~\varepsilon\\
\trcomp{A \Harrow B \eff E_B} \defas \trcomp{A}_{E_B} \Harrow \trcomp{B}_{\{\varepsilon\}} \eff E_B\\
\trcomp{A \eff E_A \Harrow B \eff E_B} \defas \trcomp{A}_{E_A \cup E_B} \eff E_A \cup E_B \Harrow \trcomp{B}_{\{\varepsilon\}} \eff \pp(E_A) \cup E_B,\quad \text{fresh}~\varepsilon\\
% \trcomp{(A_1 \to B_1 \eff E_1) \to B_2 \eff E_2} \defas (\trval{A_1} \to \trval{B_1} \eff E_1 \cup E_2) \to \trval{B_2} \eff E_1 \cup E_2
% \trcomp{A \to B \eff \varepsilon} \defas A \to B \eff \{\varepsilon\}\\
% \trcomp{A \to B \eff E} \defas A' \to \trval{B}_{\{\varepsilon\}} \eff E''\\
% \quad\where~\bl
% (E', A') = \trval{A}_{\treff{E}}\;\keyw{and}\; E'' = \treff{E \cup E'}
% \el\\
% \val{A \to B \eff E} \defas A' \to B' \eff E'\\
% ~\where~\bl
% (E_A, A') = \val{A}\\
% (E_B,
% \ea
\map : \Record{\alpha \to \beta;\List~\alpha} \to \List~\beta\\
% \map : \Record{\alpha \to \beta \eff \{\varepsilon\};\List~\alpha} \to \List~\beta \eff \{\varepsilon\}
\el
\]
\section{Composing \UNIX{} with effect handlers}
\label{sec:deep-handlers-in-action}