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
|
<!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>FirstClassPolymorphism - 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%;">
FirstClassPolymorphism
<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">
First-class polymorphism is the ability to treat polymorphic functions just like other values: pass them as arguments, store them in data structures, etc. Although <a href="StandardML">Standard ML</a> does have polymorphic functions, it does not support first-class polymorphism. <p>
For example, the following declares and uses the polymorphic function <tt>id</tt>.
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> id = <B><FONT COLOR="#A020F0">fn</FONT></B> x => x
<B><FONT COLOR="#A020F0">val</FONT></B> _ = id <B><FONT COLOR="#5F9EA0">13</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> _ = id <B><FONT COLOR="#BC8F8F">"foo"</FONT></B>
</PRE>
</p>
<p>
If SML supported first-class polymorphism, we could write the following.
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">fun</FONT></B> useId id = (id <B><FONT COLOR="#5F9EA0">13</FONT></B>; id <B><FONT COLOR="#BC8F8F">"foo"</FONT></B>)
</PRE>
<p>
</p>
<p>
However, this does not type check. MLton reports the following error.
</p>
<pre>Error: z.sml 1.24.
Function applied to incorrect argument.
expects: [int]
but got: [string]
in: id "foo"
</pre><p>
The error message arises because MLton infers from <tt>id 13</tt> that <tt>id</tt> accepts an integer argument, but that <tt>id "foo"</tt> is passing a string. Using explicit types sheds some light on the problem.
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">fun</FONT></B> useId (id: 'a -> 'a) = (id <B><FONT COLOR="#5F9EA0">13</FONT></B>; id <B><FONT COLOR="#BC8F8F">"foo"</FONT></B>)
</PRE>
<p>
</p>
<p>
On this, MLton reports the following errors.
</p>
<pre>Error: z.sml 1.29.
Function applied to incorrect argument.
expects: ['a]
but got: [int]
in: id 13
Error: z.sml 1.36.
Function applied to incorrect argument.
expects: ['a]
but got: [string]
in: id "foo"
</pre><p>
The errors arise because the argument <tt>id</tt> is <em>not</em> polymorphic; rather, it is monomorphic, with type <tt>'a -> 'a</tt>. It is perfectly valid to apply <tt>id</tt> to a value of type <tt>'a</tt>, as in the following
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">fun</FONT></B> useId (id: 'a -> 'a, x: 'a) = id x <I><FONT COLOR="#B22222">(* type correct *)</FONT></I>
</PRE>
<p>
</p>
<p>
So, what is the difference between the type specification on <tt>id</tt> in the following two declarations?
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> id: 'a -> 'a = <B><FONT COLOR="#A020F0">fn</FONT></B> x => x
<B><FONT COLOR="#A020F0">fun</FONT></B> useId (id: 'a -> 'a) = (id <B><FONT COLOR="#5F9EA0">13</FONT></B>; id <B><FONT COLOR="#BC8F8F">"foo"</FONT></B>)
</PRE>
<p>
</p>
<p>
While the type specifications on <tt>id</tt> look identical, they mean different things. The difference can be made clearer by explicitly <a href="TypeVariableScope">scoping the type variables</a>.
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> <B><FONT COLOR="#228B22">'a</FONT></B> id: 'a -> 'a = <B><FONT COLOR="#A020F0">fn</FONT></B> x => x
<B><FONT COLOR="#A020F0">fun</FONT></B> <B><FONT COLOR="#228B22">'a</FONT></B> useId (id: 'a -> 'a) = (id <B><FONT COLOR="#5F9EA0">13</FONT></B>; id <B><FONT COLOR="#BC8F8F">"foo"</FONT></B>) <I><FONT COLOR="#B22222">(* type error *)</FONT></I>
</PRE>
<p>
</p>
<p>
In <tt>val 'a id</tt>, the type variable scoping means that for any <tt>'a</tt>, <tt>id</tt> has type <tt>'a -> 'a</tt>. Hence, <tt>id</tt> can be applied to arguments of type <tt>int</tt>, <tt>real</tt>, etc. Similarly, in <tt>fun 'a useId</tt>, the scoping means that <tt>useId</tt> is a polymorphic function that for any <tt>'a</tt> takes a function of type <tt>'a -> 'a</tt> and does something. Thus, <tt>useId</tt> could be applied to a function of type <tt>int -> int</tt>, <tt>real -> real</tt>, etc.
</p>
<p>
One could imagine an extension of SML that allowed scoping of type variables at places other than <tt>fun</tt> or <tt>val</tt> declarations, as in the following.
</p>
<pre>fun useId (id: ('a).'a -> 'a) = (id 13; id "foo") (* not SML *)
</pre><p>
Such an extension would need to be thought through very carefully, as it could cause significant complications with <a class="nonexistent" href="TypeInference">TypeInference</a>, possible even undecidability.
</p>
</div>
<p>
<hr>
Last edited on 2005-12-01 04:14:09 by <span title="ppp-71-139-183-221.dsl.snfc21.pacbell.net"><a href="StephenWeeks">StephenWeeks</a></span>.
</body></html>
|
