/usr/share/doc/mlton/guide/Fold01N is in mlton-doc 20100608-5.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 | <!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<meta name="robots" content="index,nofollow">
<title>Fold01N - MLton Standard ML Compiler (SML Compiler)</title>
<link rel="stylesheet" type="text/css" charset="iso-8859-1" media="all" href="common.css">
<link rel="stylesheet" type="text/css" charset="iso-8859-1" media="screen" href="screen.css">
<link rel="stylesheet" type="text/css" charset="iso-8859-1" media="print" href="print.css">
<link rel="Start" href="Home">
</head>
<body lang="en" dir="ltr">
<script src="http://www.google-analytics.com/urchin.js" type="text/javascript">
</script>
<script type="text/javascript">
_uacct = "UA-833377-1";
urchinTracker();
</script>
<table bgcolor = lightblue cellspacing = 0 style = "border: 0px;" width = 100%>
<tr>
<td style = "
border: 0px;
color: darkblue;
font-size: 150%;
text-align: left;">
<a class = mltona href="Home">MLton MLTONWIKIVERSION</a>
<td style = "
border: 0px;
font-size: 150%;
text-align: center;
width: 50%;">
Fold01N
<td style = "
border: 0px;
text-align: right;">
<table cellspacing = 0 style = "border: 0px">
<tr style = "vertical-align: middle;">
</table>
<tr style = "background-color: white;">
<td colspan = 3
style = "
border: 0px;
font-size:70%;
text-align: right;">
<a href = "Home">Home</a>
<a href = "TitleIndex">Index</a>
</table>
<div id="content" lang="en" dir="ltr">
A common use pattern of <a href="Fold">Fold</a> is to define a variable-arity function that combines multiple arguments together using a binary function. It is slightly tricky to do this directly using fold, because of the special treatment required for the case of zero or one argument. Here is a structure, <tt>Fold01N</tt>, that solves the problem once and for all, and eases the definition of such functions.
<pre class=code>
<B><FONT COLOR="#0000FF">structure</FONT></B> Fold01N =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">fun</FONT></B> fold {finish, start, zero} =
Fold.fold ((id, finish, <B><FONT COLOR="#A020F0">fn</FONT></B> () => zero, start),
<B><FONT COLOR="#A020F0">fn</FONT></B> (finish, _, p, _) => finish (p ()))
<B><FONT COLOR="#A020F0">fun</FONT></B> step0 {combine, input} =
Fold.step0 (<B><FONT COLOR="#A020F0">fn</FONT></B> (_, finish, _, f) =>
(finish,
finish,
<B><FONT COLOR="#A020F0">fn</FONT></B> () => f input,
<B><FONT COLOR="#A020F0">fn</FONT></B> x' => combine (f input, x')))
<B><FONT COLOR="#A020F0">fun</FONT></B> step1 {combine} z input =
step0 {combine = combine, input = input} z
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
If one has a value <tt>zero</tt>, and functions <tt>start</tt>, <tt>c</tt>, and <tt>finish</tt>, then one can define a variable-arity function <tt>f</tt> and stepper <tt>`</tt> as follows.
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> f = <B><FONT COLOR="#A020F0">fn</FONT></B> z => Fold01N.fold {finish = finish, start = start, zero = zero} z
<B><FONT COLOR="#A020F0">val</FONT></B> ` = <B><FONT COLOR="#A020F0">fn</FONT></B> z => Fold01N.step1 {combine = c} z
</PRE>
<p>
</p>
<p>
One can then use the fold equation to prove the following equations.
<pre class=code>
f $ = zero
f `a1 $ = finish (start a1)
f `a1 `a2 $ = finish (c (start a1, a2))
f `a1 `a2 `a3 $ = finish (c (c (start a1, a2), a3))
...
</PRE>
</p>
<p>
For an example of <tt>Fold01N</tt>, see <a href="VariableArityPolymorphism">VariableArityPolymorphism</a>.
</p>
<h2 id="head-c8f728eef389fe7d14bbb9f38f947c3019d75348">Typing Fold01N</h2>
<p>
Here is the signature for <tt>Fold01N</tt>. We use a trick to avoid having to duplicate the definition of some rather complex types in both the signature and the structure. We first define the types in a structure. Then, we define them via type re-definitions in the signature, and via <tt>open</tt> in the full structure.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">structure</FONT></B> Fold01N =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> ('input, 'accum1, 'accum2, 'answer, 'zero,
'a, 'b, 'c, 'd, 'e) t </FONT></B>=<B><FONT COLOR="#228B22">
(('zero -> 'zero)
* ('accum2 -> 'answer)
* (unit -> 'zero)
* ('input -> 'accum1),
('a -> 'b) * 'c * (unit -> 'a) * 'd,
'b,
'e) Fold.t
</FONT></B><B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> ('input1, 'accum1, 'input2, 'accum2,
'a, 'b, 'c, 'd, 'e, 'f) step0 </FONT></B>=<B><FONT COLOR="#228B22">
('a * 'b * 'c * ('input1 -> 'accum1),
'b * 'b * (unit -> 'accum1) * ('input2 -> 'accum2),
'd, 'e, 'f) Fold.step0
</FONT></B><B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> ('accum1, 'input, 'accum2,
'a, 'b, 'c, 'd, 'e, 'f, 'g) step1 </FONT></B>=<B><FONT COLOR="#228B22">
('a,
'b * 'c * 'd * ('a -> 'accum1),
'c * 'c * (unit -> 'accum1) * ('input -> 'accum2),
'e, 'f, 'g) Fold.step1
</FONT></B><B><FONT COLOR="#0000FF">end</FONT></B>
<B><FONT COLOR="#0000FF">signature</FONT></B> FOLD_01N =
<B><FONT COLOR="#0000FF">sig</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> ('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) t </FONT></B>=<B><FONT COLOR="#228B22">
('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) Fold01N.t
</FONT></B><B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> ('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) step0 </FONT></B>=<B><FONT COLOR="#228B22">
('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) Fold01N.step0
</FONT></B><B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> ('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) step1 </FONT></B>=<B><FONT COLOR="#228B22">
('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i, 'j) Fold01N.step1
</FONT></B><B><FONT COLOR="#A020F0">val</FONT></B> fold:
{finish: 'accum2 -> 'answer,
start: 'input -> 'accum1,
zero: 'zero}
-> ('input, 'accum1, 'accum2, 'answer, 'zero,
'a, 'b, 'c, 'd, 'e) t
<B><FONT COLOR="#A020F0">val</FONT></B> step0:
{combine: 'accum1 * 'input2 -> 'accum2,
input: 'input1}
-> ('input1, 'accum1, 'input2, 'accum2,
'a, 'b, 'c, 'd, 'e, 'f) step0
<B><FONT COLOR="#A020F0">val</FONT></B> step1:
{combine: 'accum1 * 'input -> 'accum2}
-> ('accum1, 'input, 'accum2,
'a, 'b, 'c, 'd, 'e, 'f, 'g) step1
<B><FONT COLOR="#0000FF">end</FONT></B>
<B><FONT COLOR="#0000FF">structure</FONT></B> Fold01N: FOLD_01N =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#0000FF">open</FONT></B> Fold01N
<B><FONT COLOR="#A020F0">fun</FONT></B> fold {finish, start, zero} =
Fold.fold ((id, finish, <B><FONT COLOR="#A020F0">fn</FONT></B> () => zero, start),
<B><FONT COLOR="#A020F0">fn</FONT></B> (finish, _, p, _) => finish (p ()))
<B><FONT COLOR="#A020F0">fun</FONT></B> step0 {combine, input} =
Fold.step0 (<B><FONT COLOR="#A020F0">fn</FONT></B> (_, finish, _, f) =>
(finish,
finish,
<B><FONT COLOR="#A020F0">fn</FONT></B> () => f input,
<B><FONT COLOR="#A020F0">fn</FONT></B> x' => combine (f input, x')))
<B><FONT COLOR="#A020F0">fun</FONT></B> step1 {combine} z input =
step0 {combine = combine, input = input} z
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
</div>
<p>
<hr>
Last edited on 2006-03-21 23:10:39 by <span title="cs180212.pp.htv.fi"><a href="VesaKarvonen">VesaKarvonen</a></span>.
</body></html>
|