source: Papers/polymorphic-variants-2012/polymorphic-variants.tex @ 2409

Last change on this file since 2409 was 2409, checked in by mulligan, 7 years ago

Some text about algebraic data types and their limitations. Needs to be finished.

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1\documentclass[smallextended]{svjour3}
2
3\smartqed
4
5\usepackage[english]{babel}
6\usepackage[colorlinks]{hyperref}
7\usepackage{microtype}
8
9
10\author{Dominic P. Mulligan \and Claudio Sacerdoti Coen}
11\title{Polymorphic variants in dependent type theory\thanks{The project CerCo acknowledges the financial support of the Future and Emerging Technologies (FET) programme within the Seventh Framework Programme for Research of the European Commission, under FET-Open grant number: 243881.}}
12
13\institute{
14  Dominic P. Mulligan \at
15  Computer Laboratory,\\
16  University of Cambridge.
17  \email{dominic.p.mulligan@gmail.com} \and
18  Claudio Sacerdoti Coen \at
19  Dipartimento di Scienze dell'Informazione,\\
20  Universit\`a di Bologna.
21  \email{sacerdot@cs.unibo.it}
22}
23
24\begin{document}
25
26\maketitle
27
28\begin{abstract}
29
30Big long abstract introducing the work
31
32\keywords{Polymorphic variants \and dependent type theory \and Matita theorem prover}
33
34\end{abstract}
35
36%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
37% Section
38%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
39\section{Introduction}
40\label{sect.introduction}
41
42%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
43% Section
44%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
45\section{Polymorphic variants}
46\label{sect.polymorphic.variants}
47
48In this section we provide a self-contained \emph{pr\'ecis} of polymorphic variants.
49For a more complete summary, we refer the reader to Garrigue's publications on the subject~\cite{dpm: todo}.
50
51Most mainstream functional programming languages, such as OCaml and Haskell, have mechanisms for inductively defining types through the use of \emph{algebraic data types}.
52Each algebraic data type can be described as a sum-of-products, wherein we associate a fixed number of distinct \emph{constructors} to the type being introduced, all of whom expect a product of arguments.
53Inductive data, modelled as an algebraic data type, is built incrementally from the ground-up using the constructors of that type.
54Quotidian data structures---such as lists, trees, heaps, zippers, and so forth---can all be introduced using this now familiar mechanism.
55
56Having built data using constructors, to complete the picture we now need some facility for picking said data apart.
57Functional languages employ \emph{pattern matching} for this task.
58Given any inhabitant of an inductive type, by the aforementioned sum-of-products property, we know that it must consist of some constructor of that type applied to various arguments.
59Using pattern matching we can therefore deconstruct algebraic data by performing a case analysis on the constructors of a given type.
60
61The combination of algebraic data types and pattern matching is powerful, and is arguably the main branching mechanism for most functional programming languages.
62Further, using pattern matching it is easy to define new functions that consume algebraic data---the set of operations that can be defined for any given algebraic type is unbounded.
63Unfortunately, when it comes to extending algebraic data types with new constructors these types are essentially `closed'.
64We cannot simply extend an algebraic type with a new constructor.
65We must introduce a new algebraic type with the additional constructor, lifting the old type---and any functions defined over it---into this type.
66
67Moving sideways, we can compare and contrast functional programming languages' use of algebraic data paired with pattern matching with the approach taken by object-oriented languages.
68In mainstream object-oriented languages such as Java algebraic data types correspond to class hierarchies, each constructor corresponding to a subclass in this hierarchy.
69Pattern matching is emulated using a dynamic dispatch mechanism.
70Using the object-oriented approach it is easy to add a new `case' to a given hierarchy: just introduce a new subclass.
71
72[dpm: reword the above --- hard to phrase precisely ]
73
74
75\begin{itemize}
76 \item General introduction, motivations
77 \item Bounded vs not-bounded.
78\end{itemize}
79
80\subsection{Subtyping as instantiation vs subtyping as safe static cast}
81
82\subsection{Syntax \& type checking rules}
83The ones of Guarrigue + casts, but also for the bounded case?
84Casts support both styles of subtyping.
85
86\subsection{Examples}
87The weird function types that only work in subtyping as instantiation
88
89\subsection{Solution to the expression problem}
90Our running example in pseudo-OCaml syntax
91
92\section{Bounded polymorphic variants via dependent types}
93Requirements (i.e. O(1) pattern-matching, natural extracted code, etc.)
94\subsection{Simulation (reduction + type checking)}
95\subsection{Examples}
96The weird function types redone
97\subsection{Subtyping as instantiation vs subtyping as safe static cast}
98Here we show/discuss how our approach supports both styles at once.
99\subsection{Solution to the expression problem, I}
100Using subtyping as cast, the file I have produced
101\subsection{Solution to the expression problem, II}
102Using subtyping as instantiation, comparisons, pros vs cons
103\subsection{Negative encoding (??)}
104The negative encoding and application to the expression problem
105\subsection{Other encodings (??)}
106Hints to other possible encodings
107
108\section{Extensible records (??)}
109
110\section{Comparison to related work and alternatives}
111\begin{itemize}
112 \item Disjoint unions: drawbacks
113 \item Encoding the unbounded case: drawbacks
114\end{itemize}
115
116\section{Appendix: interface of library functions used to implement everything}
117
118\end{document}
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