[hcoop/debian/mlton.git] / doc / guide / localhost / PolymorphicEquality

<!DOCTYPE html>\r
<html lang="en">\r
<head>\r
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">\r
<meta name="generator" content="AsciiDoc 8.6.9">\r
<title>PolymorphicEquality</title>\r
<link rel="stylesheet" href="./asciidoc.css" type="text/css">\r
<link rel="stylesheet" href="./pygments.css" type="text/css">\r
\r
\r
<script type="text/javascript" src="./asciidoc.js"></script>\r
<script type="text/javascript">\r
/*<![CDATA[*/\r
asciidoc.install(2);\r
/*]]>*/\r
</script>\r
<link rel="stylesheet" href="./mlton.css" type="text/css">\r
</head>\r
<body class="article">\r
<div id="banner">\r
<div id="banner-home">\r
<a href="./Home">MLton 20180207</a>\r
</div>\r
</div>\r
<div id="header">\r
<h1>PolymorphicEquality</h1>\r
<div id="toc">
  <div id="toctitle">Table of Contents</div>
  <noscript><p><b>JavaScript must be enabled in your browser to display the table of contents.</b></p></noscript>
</div>\r
</div>\r
<div id="content">\r
<div id="preamble">\r
<div class="sectionbody">\r
<div class="paragraph"><p>Polymorphic equality is a built-in function in\r
<a href="StandardML">Standard ML</a> that compares two values of the same type\r
for equality.  It is specified as</p></div>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;&#39;a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">bool</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>The <span class="monospaced">''a</span> in the specification are\r
<a href="EqualityTypeVariable">equality type variables</a>, and indicate that\r
polymorphic equality can only be applied to values of an\r
<a href="EqualityType">equality type</a>.  It is not allowed in SML to rebind\r
<span class="monospaced">=</span>, so a programmer is guaranteed that <span class="monospaced">=</span> always denotes polymorphic\r
equality.</p></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_equality_of_ground_types">Equality of ground types</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>Ground types like <span class="monospaced">char</span>, <span class="monospaced">int</span>, and <span class="monospaced">word</span> may be compared (to values\r
of the same type).  For example, <span class="monospaced">13 = 14</span> is type correct and yields\r
<span class="monospaced">false</span>.</p></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_equality_of_reals">Equality of reals</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>The one ground type that can not be compared is <span class="monospaced">real</span>.  So,\r
<span class="monospaced">13.0 = 14.0</span> is not type correct.  One can use <span class="monospaced">Real.==</span> to compare\r
reals for equality, but beware that this has different algebraic\r
properties than polymorphic equality.</p></div>\r
<div class="paragraph"><p>See <a href="http://standardml.org/Basis/real.html">http://standardml.org/Basis/real.html</a> for a discussion of why\r
<span class="monospaced">real</span> is not an equality type.</p></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_equality_of_functions">Equality of functions</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>Comparison of functions is not allowed.</p></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_equality_of_immutable_types">Equality of immutable types</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>Polymorphic equality can be used on <a href="Immutable">immutable</a> values like\r
tuples, records, lists, and vectors.  For example,</p></div>\r
<div class="listingblock">\r
<div class="content monospaced">\r
<pre>(1, 2, 3) = (4, 5, 6)</pre>\r
</div></div>\r
<div class="paragraph"><p>is a type-correct expression yielding <span class="monospaced">false</span>, while</p></div>\r
<div class="listingblock">\r
<div class="content monospaced">\r
<pre>[1, 2, 3] = [1, 2, 3]</pre>\r
</div></div>\r
<div class="paragraph"><p>is type correct and yields <span class="monospaced">true</span>.</p></div>\r
<div class="paragraph"><p>Equality on immutable values is computed by structure, which means\r
that values are compared by recursively descending the data structure\r
until ground types are reached, at which point the ground types are\r
compared with primitive equality tests (like comparison of\r
characters).  So, the expression</p></div>\r
<div class="listingblock">\r
<div class="content monospaced">\r
<pre>[1, 2, 3] = [1, 1 + 1, 1 + 1 + 1]</pre>\r
</div></div>\r
<div class="paragraph"><p>is guaranteed to yield <span class="monospaced">true</span>, even though the lists may occupy\r
different locations in memory.</p></div>\r
<div class="paragraph"><p>Because of structural equality, immutable values can only be compared\r
if their components can be compared.  For example, <span class="monospaced">[1, 2, 3]</span> can be\r
compared, but <span class="monospaced">[1.0, 2.0, 3.0]</span> can not.  The SML type system uses\r
<a href="EqualityType">equality types</a> to ensure that structural equality is\r
only applied to valid values.</p></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_equality_of_mutable_values">Equality of mutable values</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>In contrast to immutable values, polymorphic equality of\r
<a href="Mutable">mutable</a> values (like ref cells and arrays) is performed by\r
pointer comparison, not by structure.  So, the expression</p></div>\r
<div class="listingblock">\r
<div class="content monospaced">\r
<pre>ref 13 = ref 13</pre>\r
</div></div>\r
<div class="paragraph"><p>is guaranteed to yield <span class="monospaced">false</span>, even though the ref cells hold the\r
same contents.</p></div>\r
<div class="paragraph"><p>Because equality of mutable values is not structural, arrays and refs\r
can be compared <em>even if their components are not equality types</em>.\r
Hence, the following expression is type correct (and yields true).</p></div>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="k">let</span><span class="w"></span>\r
<span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="mf">13.0</span><span class="w"></span>\r
<span class="k">in</span><span class="w"></span>\r
<span class="w">   </span><span class="n">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span><span class="w"></span>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_equality_of_datatypes">Equality of datatypes</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>Polymorphic equality of datatypes is structural.  Two values of the\r
same datatype are equal if they are of the same <a href="Variant">variant</a> and\r
if the <a href="Variant">variant</a>'s arguments are equal (recursively).  So,\r
with the datatype</p></div>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">|</span><span class="w"> </span><span class="n">B</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>then <span class="monospaced">B (B A) = B A</span> is type correct and yields <span class="monospaced">false</span>, while <span class="monospaced">A = A</span>\r
and <span class="monospaced">B A = B A</span> yield <span class="monospaced">true</span>.</p></div>\r
<div class="paragraph"><p>As polymorphic equality descends two values to compare them, it uses\r
pointer equality whenever it reaches a mutable value.  So, with the\r
datatype</p></div>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">|</span><span class="w"> </span><span class="p">...</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>then <span class="monospaced">A (ref 13) = A (ref 13)</span> is type correct and yields <span class="monospaced">false</span>,\r
because the pointer equality on the two ref cells yields <span class="monospaced">false</span>.</p></div>\r
<div class="paragraph"><p>One weakness of the SML type system is that datatypes do not inherit\r
the special property of the <span class="monospaced">ref</span> and <span class="monospaced">array</span> type constructors that\r
allows them to be compared regardless of their component type.  For\r
example, after declaring</p></div>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">ref</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>one might expect to be able to compare two values of type <span class="monospaced">real t</span>,\r
because pointer comparison on a ref cell would suffice.\r
Unfortunately, the type system can only express that a user-defined\r
datatype <a href="AdmitsEquality">admits equality</a> or not.  In this case, <span class="monospaced">t</span>\r
admits equality, which means that <span class="monospaced">int t</span> can be compared but that\r
<span class="monospaced">real t</span> can not.  We can confirm this with the program</p></div>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">ref</span><span class="w"></span>\r
<span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">y</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>on which MLton reports the following error.</p></div>\r
<div class="listingblock">\r
<div class="content monospaced">\r
<pre>Error: z.sml 2.32-2.36.\r
  Function applied to incorrect argument.\r
    expects: [&lt;equality&gt;] t * [&lt;equality&gt;] t\r
    but got: [real] t * [real] t\r
    in: = (x, y)</pre>\r
</div></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_implementation">Implementation</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>Polymorphic equality is implemented by recursively descending the two\r
values being compared, stopping as soon as they are determined to be\r
unequal, or exploring the entire values to determine that they are\r
equal.  Hence, polymorphic equality can take time proportional to the\r
size of the smaller value.</p></div>\r
<div class="paragraph"><p>MLton uses some optimizations to improve performance.</p></div>\r
<div class="ulist"><ul>\r
<li>\r
<p>\r
When computing structural equality, first do a pointer comparison.\r
If the comparison yields <span class="monospaced">true</span>, then stop and return <span class="monospaced">true</span>, since\r
the structural comparison is guaranteed to do so.  If the pointer\r
comparison fails, then recursively descend the values.\r
</p>\r
</li>\r
<li>\r
<p>\r
If a datatype is an enum (e.g. <span class="monospaced">datatype t = A | B | C</span>), then a\r
single comparison suffices to compare values of the datatype.  No case\r
dispatch is required to determine whether the two values are of the\r
same <a href="Variant">variant</a>.\r
</p>\r
</li>\r
<li>\r
<p>\r
When comparing a known constant non-value-carrying\r
<a href="Variant">variant</a>, use a single comparison.  For example, the\r
following code will compile into a single comparison for <span class="monospaced">A = x</span>.\r
</p>\r
<div class="listingblock">\r
<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">|</span><span class="w"> </span><span class="n">B</span><span class="w"> </span><span class="p">|</span><span class="w"> </span><span class="n">C</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="p">...</span><span class="w"></span>\r
<span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="k">then</span><span class="w"> </span><span class="p">...</span><span class="w"></span>\r
</pre></div></div></div>\r
</li>\r
<li>\r
<p>\r
When comparing a small constant <span class="monospaced">IntInf.int</span> to another\r
<span class="monospaced">IntInf.int</span>, use a single comparison against the constant.  No case\r
dispatch is required.\r
</p>\r
</li>\r
</ul></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_also_see">Also see</h2>\r
<div class="sectionbody">\r
<div class="ulist"><ul>\r
<li>\r
<p>\r
<a href="AdmitsEquality">AdmitsEquality</a>\r
</p>\r
</li>\r
<li>\r
<p>\r
<a href="EqualityType">EqualityType</a>\r
</p>\r
</li>\r
<li>\r
<p>\r
<a href="EqualityTypeVariable">EqualityTypeVariable</a>\r
</p>\r
</li>\r
</ul></div>\r
</div>\r
</div>\r
</div>\r
<div id="footnotes"><hr></div>\r
<div id="footer">\r
<div id="footer-text">\r
</div>\r
<div id="footer-badges">\r
</div>\r
</div>\r
</body>\r
</html>\r
Commit	Line	Data
7f918cf1 CE	1	<!DOCTYPE html>\r
	2	<html lang="en">\r
	3	<head>\r
	4	<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">\r
	5	<meta name="generator" content="AsciiDoc 8.6.9">\r
	6	<title>PolymorphicEquality</title>\r
	7	<link rel="stylesheet" href="./asciidoc.css" type="text/css">\r
	8	<link rel="stylesheet" href="./pygments.css" type="text/css">\r
	9	\r
	10	\r
	11	<script type="text/javascript" src="./asciidoc.js"></script>\r
	12	<script type="text/javascript">\r
	13	/<![CDATA[/\r
	14	asciidoc.install(2);\r
	15	/]]>/\r
	16	</script>\r
	17	<link rel="stylesheet" href="./mlton.css" type="text/css">\r
	18	</head>\r
	19	<body class="article">\r
	20	<div id="banner">\r
	21	<div id="banner-home">\r
	22	<a href="./Home">MLton 20180207</a>\r
	23	</div>\r
	24	</div>\r
	25	<div id="header">\r
	26	<h1>PolymorphicEquality</h1>\r
	27	<div id="toc">
	28	<div id="toctitle">Table of Contents</div>
	29	<noscript><p><b>JavaScript must be enabled in your browser to display the table of contents.</b></p></noscript>
	30	</div>\r
	31	</div>\r
	32	<div id="content">\r
	33	<div id="preamble">\r
	34	<div class="sectionbody">\r
	35	<div class="paragraph"><p>Polymorphic equality is a built-in function in\r
	36	<a href="StandardML">Standard ML</a> that compares two values of the same type\r
	37	for equality. It is specified as</p></div>\r
	38	<div class="listingblock">\r
	39	<div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">''a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">bool</span><span class="w"></span>\r
	40	</pre></div></div></div>\r
	41	<div class="paragraph"><p>The <span class="monospaced">''a</span> in the specification are\r
	42	<a href="EqualityTypeVariable">equality type variables</a>, and indicate that\r
	43	polymorphic equality can only be applied to values of an\r
	44	<a href="EqualityType">equality type</a>. It is not allowed in SML to rebind\r
	45	<span class="monospaced">=</span>, so a programmer is guaranteed that <span class="monospaced">=</span> always denotes polymorphic\r
	46	equality.</p></div>\r
	47	</div>\r
	48	</div>\r
	49	<div class="sect1">\r
	50	<h2 id="_equality_of_ground_types">Equality of ground types</h2>\r
	51	<div class="sectionbody">\r
	52	<div class="paragraph"><p>Ground types like <span class="monospaced">char</span>, <span class="monospaced">int</span>, and <span class="monospaced">word</span> may be compared (to values\r
	53	of the same type). For example, <span class="monospaced">13 = 14</span> is type correct and yields\r
	54	<span class="monospaced">false</span>.</p></div>\r
	55	</div>\r
	56	</div>\r
	57	<div class="sect1">\r
	58	<h2 id="_equality_of_reals">Equality of reals</h2>\r
	59	<div class="sectionbody">\r
	60	<div class="paragraph"><p>The one ground type that can not be compared is <span class="monospaced">real</span>. So,\r
	61	<span class="monospaced">13.0 = 14.0</span> is not type correct. One can use <span class="monospaced">Real.==</span> to compare\r
	62	reals for equality, but beware that this has different algebraic\r
	63	properties than polymorphic equality.</p></div>\r
	64	<div class="paragraph"><p>See <a href="http://standardml.org/Basis/real.html">http://standardml.org/Basis/real.html</a> for a discussion of why\r
65	<span class="monospaced">real</span> is not an equality type.</p></div>\r
66	</div>\r
67	</div>\r
68	<div class="sect1">\r
69	<h2 id="_equality_of_functions">Equality of functions</h2>\r
70	<div class="sectionbody">\r
71	<div class="paragraph"><p>Comparison of functions is not allowed.</p></div>\r
72	</div>\r
73	</div>\r
74	<div class="sect1">\r
75	<h2 id="_equality_of_immutable_types">Equality of immutable types</h2>\r
76	<div class="sectionbody">\r
77	<div class="paragraph"><p>Polymorphic equality can be used on <a href="Immutable">immutable</a> values like\r
78	tuples, records, lists, and vectors. For example,</p></div>\r
79	<div class="listingblock">\r
80	<div class="content monospaced">\r
81	<pre>(1, 2, 3) = (4, 5, 6)</pre>\r
82	</div></div>\r
83	<div class="paragraph"><p>is a type-correct expression yielding <span class="monospaced">false</span>, while</p></div>\r
84	<div class="listingblock">\r
85	<div class="content monospaced">\r
86	<pre>[1, 2, 3] = [1, 2, 3]</pre>\r
87	</div></div>\r
88	<div class="paragraph"><p>is type correct and yields <span class="monospaced">true</span>.</p></div>\r
89	<div class="paragraph"><p>Equality on immutable values is computed by structure, which means\r
90	that values are compared by recursively descending the data structure\r
91	until ground types are reached, at which point the ground types are\r
92	compared with primitive equality tests (like comparison of\r
93	characters). So, the expression</p></div>\r
94	<div class="listingblock">\r
95	<div class="content monospaced">\r
96	<pre>[1, 2, 3] = [1, 1 + 1, 1 + 1 + 1]</pre>\r
97	</div></div>\r
98	<div class="paragraph"><p>is guaranteed to yield <span class="monospaced">true</span>, even though the lists may occupy\r
99	different locations in memory.</p></div>\r
100	<div class="paragraph"><p>Because of structural equality, immutable values can only be compared\r
101	if their components can be compared. For example, <span class="monospaced">[1, 2, 3]</span> can be\r
102	compared, but <span class="monospaced">[1.0, 2.0, 3.0]</span> can not. The SML type system uses\r
103	<a href="EqualityType">equality types</a> to ensure that structural equality is\r
104	only applied to valid values.</p></div>\r
105	</div>\r
106	</div>\r
107	<div class="sect1">\r
108	<h2 id="_equality_of_mutable_values">Equality of mutable values</h2>\r
109	<div class="sectionbody">\r
110	<div class="paragraph"><p>In contrast to immutable values, polymorphic equality of\r
111	<a href="Mutable">mutable</a> values (like ref cells and arrays) is performed by\r
112	pointer comparison, not by structure. So, the expression</p></div>\r
113	<div class="listingblock">\r
114	<div class="content monospaced">\r
115	<pre>ref 13 = ref 13</pre>\r
116	</div></div>\r
117	<div class="paragraph"><p>is guaranteed to yield <span class="monospaced">false</span>, even though the ref cells hold the\r
118	same contents.</p></div>\r
119	<div class="paragraph"><p>Because equality of mutable values is not structural, arrays and refs\r
120	can be compared <em>even if their components are not equality types</em>.\r
121	Hence, the following expression is type correct (and yields true).</p></div>\r
122	<div class="listingblock">\r
123	<div class="content"><div class="highlight"><pre><span class="k">let</span><span class="w"></span>\r
124	<span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="mf">13.0</span><span class="w"></span>\r
125	<span class="k">in</span><span class="w"></span>\r
126	<span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span><span class="w"></span>\r
127	<span class="k">end</span><span class="w"></span>\r
128	</pre></div></div></div>\r
129	</div>\r
130	</div>\r
131	<div class="sect1">\r
132	<h2 id="_equality_of_datatypes">Equality of datatypes</h2>\r
133	<div class="sectionbody">\r
134	<div class="paragraph"><p>Polymorphic equality of datatypes is structural. Two values of the\r
135	same datatype are equal if they are of the same <a href="Variant">variant</a> and\r
136	if the <a href="Variant">variant</a>'s arguments are equal (recursively). So,\r
137	with the datatype</p></div>\r
138	<div class="listingblock">\r
139	<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">\|</span><span class="w"> </span><span class="n">B</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r
140	</pre></div></div></div>\r
141	<div class="paragraph"><p>then <span class="monospaced">B (B A) = B A</span> is type correct and yields <span class="monospaced">false</span>, while <span class="monospaced">A = A</span>\r
142	and <span class="monospaced">B A = B A</span> yield <span class="monospaced">true</span>.</p></div>\r
143	<div class="paragraph"><p>As polymorphic equality descends two values to compare them, it uses\r
144	pointer equality whenever it reaches a mutable value. So, with the\r
145	datatype</p></div>\r
146	<div class="listingblock">\r
147	<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">\|</span><span class="w"> </span><span class="p">...</span><span class="w"></span>\r
148	</pre></div></div></div>\r
149	<div class="paragraph"><p>then <span class="monospaced">A (ref 13) = A (ref 13)</span> is type correct and yields <span class="monospaced">false</span>,\r
150	because the pointer equality on the two ref cells yields <span class="monospaced">false</span>.</p></div>\r
151	<div class="paragraph"><p>One weakness of the SML type system is that datatypes do not inherit\r
152	the special property of the <span class="monospaced">ref</span> and <span class="monospaced">array</span> type constructors that\r
153	allows them to be compared regardless of their component type. For\r
154	example, after declaring</p></div>\r
155	<div class="listingblock">\r
156	<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">ref</span><span class="w"></span>\r
157	</pre></div></div></div>\r
158	<div class="paragraph"><p>one might expect to be able to compare two values of type <span class="monospaced">real t</span>,\r
159	because pointer comparison on a ref cell would suffice.\r
160	Unfortunately, the type system can only express that a user-defined\r
161	datatype <a href="AdmitsEquality">admits equality</a> or not. In this case, <span class="monospaced">t</span>\r
162	admits equality, which means that <span class="monospaced">int t</span> can be compared but that\r
163	<span class="monospaced">real t</span> can not. We can confirm this with the program</p></div>\r
164	<div class="listingblock">\r
165	<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">ref</span><span class="w"></span>\r
166	<span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">y</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">y</span><span class="w"></span>\r
167	</pre></div></div></div>\r
168	<div class="paragraph"><p>on which MLton reports the following error.</p></div>\r
169	<div class="listingblock">\r
170	<div class="content monospaced">\r
171	<pre>Error: z.sml 2.32-2.36.\r
172	Function applied to incorrect argument.\r
173	expects: [<equality>] t * [<equality>] t\r
174	but got: [real] t * [real] t\r
175	in: = (x, y)</pre>\r
176	</div></div>\r
177	</div>\r
178	</div>\r
179	<div class="sect1">\r
180	<h2 id="_implementation">Implementation</h2>\r
181	<div class="sectionbody">\r
182	<div class="paragraph"><p>Polymorphic equality is implemented by recursively descending the two\r
183	values being compared, stopping as soon as they are determined to be\r
184	unequal, or exploring the entire values to determine that they are\r
185	equal. Hence, polymorphic equality can take time proportional to the\r
186	size of the smaller value.</p></div>\r
187	<div class="paragraph"><p>MLton uses some optimizations to improve performance.</p></div>\r
188	<div class="ulist"><ul>\r
189	<li>\r
190	<p>\r
191	When computing structural equality, first do a pointer comparison.\r
192	If the comparison yields <span class="monospaced">true</span>, then stop and return <span class="monospaced">true</span>, since\r
193	the structural comparison is guaranteed to do so. If the pointer\r
194	comparison fails, then recursively descend the values.\r
195	</p>\r
196	</li>\r
197	<li>\r
198	<p>\r
199	If a datatype is an enum (e.g. <span class="monospaced">datatype t = A \| B \| C</span>), then a\r
200	single comparison suffices to compare values of the datatype. No case\r
201	dispatch is required to determine whether the two values are of the\r
202	same <a href="Variant">variant</a>.\r
203	</p>\r
204	</li>\r
205	<li>\r
206	<p>\r
207	When comparing a known constant non-value-carrying\r
208	<a href="Variant">variant</a>, use a single comparison. For example, the\r
209	following code will compile into a single comparison for <span class="monospaced">A = x</span>.\r
210	</p>\r
211	<div class="listingblock">\r
212	<div class="content"><div class="highlight"><pre><span class="k">datatype</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">\|</span><span class="w"> </span><span class="n">B</span><span class="w"> </span><span class="p">\|</span><span class="w"> </span><span class="n">C</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="p">...</span><span class="w"></span>\r
213	<span class="k">fun</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">A</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="k">then</span><span class="w"> </span><span class="p">...</span><span class="w"></span>\r
214	</pre></div></div></div>\r
215	</li>\r
216	<li>\r
217	<p>\r
218	When comparing a small constant <span class="monospaced">IntInf.int</span> to another\r
219	<span class="monospaced">IntInf.int</span>, use a single comparison against the constant. No case\r
220	dispatch is required.\r
221	</p>\r
222	</li>\r
223	</ul></div>\r
224	</div>\r
225	</div>\r
226	<div class="sect1">\r
227	<h2 id="_also_see">Also see</h2>\r
228	<div class="sectionbody">\r
229	<div class="ulist"><ul>\r
230	<li>\r
231	<p>\r
232	<a href="AdmitsEquality">AdmitsEquality</a>\r
233	</p>\r
234	</li>\r
235	<li>\r
236	<p>\r
237	<a href="EqualityType">EqualityType</a>\r
238	</p>\r
239	</li>\r
240	<li>\r
241	<p>\r
242	<a href="EqualityTypeVariable">EqualityTypeVariable</a>\r
243	</p>\r
244	</li>\r
245	</ul></div>\r
246	</div>\r
247	</div>\r
248	</div>\r
249	<div id="footnotes"><hr></div>\r
250	<div id="footer">\r
251	<div id="footer-text">\r
252	</div>\r
253	<div id="footer-badges">\r
254	</div>\r
255	</div>\r
256	</body>\r
257	</html>\r