Mercurial > hg > Papers > 2015 > atton-icfp

annotate delta.tex @ 5:87bb6169c80d default tip

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Writing ...
author Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
date Fri, 27 Feb 2015 17:07:58 +0900
parents ecebdd7bfa48
children
Ignore whitespace changes - Everywhere: Within whitespace: At end of lines:
rev   line source
3
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
1 \documentclass[preprint]{sigplanconf}
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
2
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
3 % The following \documentclass options may be useful:
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
4
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
5 % preprint Remove this option only once the paper is in final form.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
6 % 10pt To set in 10-point type instead of 9-point.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
7 % 11pt To set in 11-point type instead of 9-point.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
8 % authoryear To obtain author/year citation style instead of numeric.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
9
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
10 \usepackage{amsmath}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
11
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
12
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
13 \begin{document}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
14
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
15 \special{papersize=8.5in,11in}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
16 \setlength{\pdfpageheight}{\paperheight}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
17 \setlength{\pdfpagewidth}{\paperwidth}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
18
1
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
19 \conferenceinfo{ICFP '15}{August 31 - September 2 , 2015, Vancouver, British Columbia, Canada}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
20 \copyrightyear{2015}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
21 \copyrightdata{978-1-nnnn-nnnn-n/yy/mm}
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
22 \doi{nnnnnnn.nnnnnnn}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
23
1
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
24 % Uncomment one of the following two, if you are not going for the
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
25 % traditional copyright transfer agreement.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
26
1
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
27 %\exclusivelicense % ACM gets exclusive license to publish,
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
28 % you retain copyright
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
29
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
30 %\permissiontopublish % ACM gets nonexclusive license to publish
1
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
31 % (paid open-access papers,
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
32 % short abstracts)
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
33
3
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
34 %\titlebanner{banner above paper title} % These are ignored unless
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
35 \preprintfooter{Formalization of Program Modifications using Monad} % 'preprint' option specified.
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
36
1
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
37 \title{Formalization of Program Modifications using Monad}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
38
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
39 \subtitle{}
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
40
1
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
41 \authorinfo{Yasutaka HIGA}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
42 {Department of Information Engineering \\ University of the Ryukyus}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
43 {atton@cr.ie.u-ryukyu.ac.jp}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
44 \authorinfo{Shiji KONO}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
45 {University of the Ryukyus}
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
46 {kono@ie.u-ryukyu.ac.jp}
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
47
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
48 \maketitle
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
49
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
50 \begin{abstract}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
51 This is the text of the abstract.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
52 \end{abstract}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
53
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
54 \category{CR-number}{subcategory}{third-level}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
55
1
7ae8b1797711 Add title and names
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 0
diff changeset
56 % general terms are not compulsory anymore,
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
57 % you may leave them out
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
58 \terms
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
59 term1, term2
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
60
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
61 \keywords
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
62 keyword1, keyword2
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
63
3
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
64 \section{Formalization of Modifications}
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
65 In software development, programs was modified for implementing features, refactoring, and more.
4
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
66 However, defective modifies decreases reliability of program.
3
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
67 Formalization of program modifications proposed to improves reliability.
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
68 Especially, We formalized modifications using Monad.
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
69 Monad provides notions of meta computations (partiality, nondeterminism, side-effects, ...) in functional program.
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
70 We define meta computation notated modifications list like structure named Delta Monad.
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
71
3
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
72 Delta Monad represents modifications by accumulates all versions of a program.
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
73 Accumulated modifications can computes by meta computation.
11a86cf869e2 Wrote introduction
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 1
diff changeset
74 In this paper, We propose meta computation execute a program includes modifications simultaneously on Delta.
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
75
4
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
76 \section{Programs and Monads}
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
77 Programs notated typed lambda calculus constructed values and abstractions.
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
78 Abstractions maps value to value.
5
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
79 Abstraction f applies to x notated $ f x $.
4
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
80 Every lambda term has a type.
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
81 Value x has type A notated $ x : A $.
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
82 Abstraction f has a argument of type A and return value of type B notated $ f : A \rightarrow B $.
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
83
5
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
84 \begin{eqnarray*}
4
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
85 x : A \\
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
86 f : A \rightarrow B \\
5
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
87 f x : B \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
88 \end{eqnarray*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
89
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
90 Type matched abstractions can be composed by operator ';'.
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
91 Order of composition are commutative.
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
92
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
93 \begin{eqnarray*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
94 f : A \rightarrow B \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
95 g : B \rightarrow C \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
96 f;g : A \rightarrow C \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
97 \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
98 h : C \rightarrow D \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
99 (f;g);h = f;(g;h)
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
100 \end{eqnarray*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
101
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
102 Abstractions can be extended using Monad.
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
103
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
104 Monad is $ triple (T, \eta, \mu) $ satisfies laws(Figure) % TODO :commutative diagram
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
105 Various meta computations represents by definition of triple.
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
106 Monad has another description Kleisli Triple $ (T, \eta, \_^{*}) $.
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
107 Kleisli triple are following equations hold:
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
108
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
109 \begin{itemize}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
110 \item $ \eta^{*}_A = id_{T A} $
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
111 \item $ \eta;f^{*} = f \mbox{ for } f : A \rightarrow T B $
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
112 \item $ f^{*} ; g^{*} = (f; g^{*})^{*} \mbox{ for } f : A \rightarrow T B \mbox{ and } g : B \rightarrow T C $
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
113 \end{itemize}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
114
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
115 Kleisli Triple can build from Monad ($ (T, \eta, \mu) $).
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
116
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
117 For Example, definition of diverging computation using Monad are shown.
4
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
118
5
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
119 \begin{itemize}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
120 \item $ T A = A_{\bot} (= A + \{\bot\}) $
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
121 \item $ \eta_A $ is the inclusion of A into $ A_{\bot} $
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
122 \item if $ f : A \rightarrow T B $, then $ f^{*} (\bot) = \bot $ and $ f^{*}(a) = f (a) $ when $ a $ has type A.
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
123 \end{itemize}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
124
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
125 \section{Modification using Monad}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
126
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
127 \begin{equation*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
128 T A = A_0 \times A_1 \times \dots \times A_n
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
129 \end{equation*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
130
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
131 \begin{eqnarray*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
132 x : A \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
133 x^{*} : T A \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
134 x^{*} : A_0 \times A_1 \times \dots \times A_n \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
135 \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
136 f : A \rightarrow B \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
137 f^{*} : A \rightarrow T B \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
138 f^{*} : (A_0 \rightarrow B_0) \times (A_1 \rightarrow B_1) \times \dots \times (A_n \rightarrow B_n)
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
139 \end{eqnarray*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
140
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
141 \begin{eqnarray*}
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
142 f^{*} : A \rightarrow T B \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
143 f^{*} = f_0 \times f_1 \times \dots f_n \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
144 \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
145 g^{*} : B \rightarrow T C \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
146 g^{*} = g_0 \times g_1 \times \dots g_n \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
147 \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
148 g^{*} \circ f^{*} : A \rightarrow T C \\
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
149 g^{*} \circ f^{*} = (g_0 \circ f_0) \times (g_1 \circ f_1) \times \dots \times (g_n \circ f_n)
87bb6169c80d Writing ...
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 4
diff changeset
150 \end{eqnarray*}
4
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
151
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
152
ecebdd7bfa48 Temporary save
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents: 3
diff changeset
153
0
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
154 \appendix
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
155 \section{Appendix Title}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
156
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
157 This is the text of the appendix, if you need one.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
158
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
159 \acks
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
160
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
161 Acknowledgments, if needed.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
162
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
163 % We recommend abbrvnat bibliography style.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
164
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
165 \bibliographystyle{abbrvnat}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
166
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
167 % The bibliography should be embedded for final submission.
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
168
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
169 \begin{thebibliography}{}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
170 \softraggedright
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
171
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
172 \bibitem[Smith et~al.(2009)Smith, Jones]{smith02}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
173 P. Q. Smith, and X. Y. Jones. ...reference text...
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
174
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
175 \end{thebibliography}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
176
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
177
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
178 \end{document}
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
179
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
180 % Revision History
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
181 % -------- -------
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
182 % Date Person Ver. Change
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
183 % ---- ------ ---- ------
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
184
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
185 % 2013.06.29 TU 0.1--4 comments on permission/copyright notices
32308ce01324 Add Template : http://www.sigplan.org/Resources/Author/
Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp>
parents:
diff changeset
186