doc/guide/localhost/PolymorphicEquality

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