doc/guide/localhost/StaticSum

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  26 <h1>StaticSum</h1>
  27 </div>
  28 <div id="content">
  29 <div id="preamble">
  30 <div class="sectionbody">
  31 <div class="paragraph"><p>While SML makes it impossible to write functions whose types would
  32 depend on the values of their arguments, or so called dependently
  33 typed functions, it is possible, and arguably commonplace, to write
  34 functions whose types depend on the types of their arguments.  Indeed,
  35 the types of parametrically polymorphic functions like <span class="monospaced">map</span> and
  36 <span class="monospaced">foldl</span> can be said to depend on the types of their arguments.  What
  37 is less commonplace, however, is to write functions whose behavior
  38 would depend on the types of their arguments.  Nevertheless, there are
  39 several techniques for writing such functions.
  40 <a href="TypeIndexedValues">Type-indexed values</a> and <a href="Fold">fold</a> are two such
  41 techniques.  This page presents another such technique dubbed static
  42 sums.</p></div>
  43 </div>
  44 </div>
  45 <div class="sect1">
  46 <h2 id="_ordinary_sums">Ordinary Sums</h2>
  47 <div class="sectionbody">
  48 <div class="paragraph"><p>Consider the sum type as defined below:</p></div>
  49 <div class="listingblock">
  50 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Sum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
  51 <span class="w">   </span><span class="k">datatype</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">)</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">INL</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="p">|</span><span class="w"> </span><span class="n">INR</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"></span>
  52 <span class="k">end</span><span class="w"></span>
  53 </pre></div></div></div>
  54 <div class="paragraph"><p>While a generic sum type such as defined above is very useful, it has
  55 a number of limitations.  As an example, we could write the function
  56 <span class="monospaced">out</span> to extract the value from a sum as follows:</p></div>
  57 <div class="listingblock">
  58 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;a</span><span class="p">)</span><span class="w"> </span><span class="n">Sum</span><span class="p">.</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">&#39;a</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  59 <span class="w">    </span><span class="k">case</span><span class="w"> </span><span class="n">s</span><span class="w"></span>
  60 <span class="w">     </span><span class="k">of</span><span class="w"> </span><span class="n">Sum</span><span class="p">.</span><span class="n">INL</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"></span>
  61 <span class="w">      </span><span class="p">|</span><span class="w"> </span><span class="n">Sum</span><span class="p">.</span><span class="n">INR</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"></span>
  62 </pre></div></div></div>
  63 <div class="paragraph"><p>As can be seen from the type of <span class="monospaced">out</span>, it is limited in the sense that
  64 it requires both variants of the sum to have the same type.  So, <span class="monospaced">out</span>
  65 cannot be used to extract the value of a sum of two different types,
  66 such as the type <span class="monospaced">(int, real) Sum.t</span>.  As another example of a
  67 limitation, consider the following attempt at a <span class="monospaced">succ</span> function:</p></div>
  68 <div class="listingblock">
  69 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">succ</span><span class="w"> </span><span class="p">(</span><span class="n">s</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">int</span><span class="p">,</span><span class="w"> </span><span class="n">real</span><span class="p">)</span><span class="w"> </span><span class="n">Sum</span><span class="p">.</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">???</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  70 <span class="w">    </span><span class="k">case</span><span class="w"> </span><span class="n">s</span><span class="w"></span>
  71 <span class="w">     </span><span class="k">of</span><span class="w"> </span><span class="n">Sum</span><span class="p">.</span><span class="n">INL</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
  72 <span class="w">      </span><span class="p">|</span><span class="w"> </span><span class="n">Sum</span><span class="p">.</span><span class="n">INR</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">Real</span><span class="p">.</span><span class="n">nextAfter</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="n">Real</span><span class="p">.</span><span class="n">posInf</span><span class="p">)</span><span class="w"></span>
  73 </pre></div></div></div>
  74 <div class="paragraph"><p>The above definition of <span class="monospaced">succ</span> cannot be typed, because there is no
  75 type for the codomain within SML.</p></div>
  76 </div>
  77 </div>
  78 <div class="sect1">
  79 <h2 id="_static_sums">Static Sums</h2>
  80 <div class="sectionbody">
  81 <div class="paragraph"><p>Interestingly, it is possible to define values <span class="monospaced">inL</span>, <span class="monospaced">inR</span>, and
  82 <span class="monospaced">match</span> that satisfy the laws</p></div>
  83 <div class="listingblock">
  84 <div class="content monospaced">
  85 <pre>match (inL x) (f, g) = f x
  86 match (inR x) (f, g) = g x</pre>
  87 </div></div>
  88 <div class="paragraph"><p>and do not suffer from the same limitions.  The definitions are
  89 actually quite trivial:</p></div>
  90 <div class="listingblock">
  91 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
  92 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">inL</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="p">,</span><span class="w"> </span><span class="p">_)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">x</span><span class="w"></span>
  93 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">inR</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">g</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">g</span><span class="w"> </span><span class="n">x</span><span class="w"></span>
  94 <span class="w">   </span><span class="k">fun</span><span class="w"> </span><span class="n">match</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">x</span><span class="w"></span>
  95 <span class="k">end</span><span class="w"></span>
  96 </pre></div></div></div>
  97 <div class="paragraph"><p>Now, given the <span class="monospaced">succ</span> function defined as</p></div>
  98 <div class="listingblock">
  99 <div class="content"><div class="highlight"><pre><span class="k">fun</span><span class="w"> </span><span class="n">succ</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 100 <span class="w">    </span><span class="n">StaticSum</span><span class="p">.</span><span class="n">match</span><span class="w"> </span><span class="n">s</span><span class="w"></span>
 101 <span class="w">       </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">i</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="mi">1</span><span class="p">,</span><span class="w"></span>
 102 <span class="w">        </span><span class="k">fn</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">Real</span><span class="p">.</span><span class="n">nextAfter</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="n">Real</span><span class="p">.</span><span class="n">posInf</span><span class="p">))</span><span class="w"></span>
 103 </pre></div></div></div>
 104 <div class="paragraph"><p>we get</p></div>
 105 <div class="listingblock">
 106 <div class="content"><div class="highlight"><pre><span class="n">succ</span><span class="w"> </span><span class="p">(</span><span class="n">StaticSum</span><span class="p">.</span><span class="n">inL</span><span class="w"> </span><span class="mi">1</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="mi">2</span><span class="w"></span>
 107 <span class="n">succ</span><span class="w"> </span><span class="p">(</span><span class="n">StaticSum</span><span class="p">.</span><span class="n">inR</span><span class="w"> </span><span class="n">Real</span><span class="p">.</span><span class="n">maxFinite</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Real</span><span class="p">.</span><span class="n">posInf</span><span class="w"></span>
 108 </pre></div></div></div>
 109 <div class="paragraph"><p>To better understand how this works, consider the following signature
 110 for static sums:</p></div>
 111 <div class="listingblock">
 112 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"> </span><span class="p">:&gt;</span><span class="w"> </span><span class="k">sig</span><span class="w"></span>
 113 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;dR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 114 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">inL</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;dL</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;dR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 115 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">inR</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;dR</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;dR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 116 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">match</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;dR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dL</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dR</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;c</span><span class="w"></span>
 117 <span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
 118 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;dR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dL</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;cL</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;dR</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;cR</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;c</span><span class="w"></span>
 119 <span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>
 120 <span class="k">end</span><span class="w"></span>
 121 </pre></div></div></div>
 122 <div class="paragraph"><p>Above, <span class="monospaced">'d</span> stands for domain and <span class="monospaced">'c</span> for codomain.  The key
 123 difference between an ordinary sum type, like <span class="monospaced">(int, real) Sum.t</span>, and
 124 a static sum type, like <span class="monospaced">(int, real, real, int, real) StaticSum.t</span>, is
 125 that the ordinary sum type says nothing about the type of the result
 126 of deconstructing a sum while the static sum type specifies the type.</p></div>
 127 <div class="paragraph"><p>With the sealed static sum module, we get the type</p></div>
 128 <div class="listingblock">
 129 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">succ</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">int</span><span class="p">,</span><span class="w"> </span><span class="n">int</span><span class="p">,</span><span class="w"> </span><span class="n">real</span><span class="p">,</span><span class="w"> </span><span class="n">real</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;a</span><span class="p">)</span><span class="w"> </span><span class="n">StaticSum</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"></span>
 130 </pre></div></div></div>
 131 <div class="paragraph"><p>for the previously defined <span class="monospaced">succ</span> function.  The type specifies that
 132 <span class="monospaced">succ</span> maps a left <span class="monospaced">int</span> to an <span class="monospaced">int</span> and a right <span class="monospaced">real</span> to a <span class="monospaced">real</span>.
 133 For example, the type of <span class="monospaced">StaticSum.inL 1</span> is
 134 <span class="monospaced">(int, 'cL, 'dR, 'cR, 'cL) StaticSum.t</span>.  Unifying this with the
 135 argument type of <span class="monospaced">succ</span> gives the type <span class="monospaced">(int, int, real, real, int)
 136 StaticSum.t -&gt; int</span>.</p></div>
 137 <div class="paragraph"><p>The <span class="monospaced">out</span> function is quite useful on its own.  Here is how it can be
 138 defined:</p></div>
 139 <div class="listingblock">
 140 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
 141 <span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>
 142 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;a</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;c</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;c</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 143 <span class="w">    </span><span class="k">fn</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">match</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">x</span><span class="p">)</span><span class="w"></span>
 144 <span class="k">end</span><span class="w"></span>
 145 </pre></div></div></div>
 146 <div class="paragraph"><p>Due to the value restriction, lack of first class polymorphism and
 147 polymorphic recursion, the usefulness and convenience of static sums
 148 is somewhat limited in SML.  So, don&#8217;t throw away the ordinary sum
 149 type just yet.  Static sums can nevertheless be quite useful.</p></div>
 150 <div class="sect2">
 151 <h3 id="_example_send_and_receive_with_argument_type_dependent_result_types">Example: Send and Receive with Argument Type Dependent Result Types</h3>
 152 <div class="paragraph"><p>In some situations it would seem useful to define functions whose
 153 result type would depend on some of the arguments.  Traditionally such
 154 functions have been thought to be impossible in SML and the solution
 155 has been to define multiple functions.  For example, the
 156 <a href="http://www.standardml.org/Basis/socket.html"><span class="monospaced">Socket</span> structure</a> of the
 157 Basis library defines 16 <span class="monospaced">send</span> and 16 <span class="monospaced">recv</span> functions.  In contrast,
 158 the Net structure
 159 (<a href="https://github.com/MLton/mltonlib/blob/master/com/sweeks/basic/unstable/net.sig"><span class="monospaced">net.sig</span></a>) of the
 160 Basic library designed by Stephen Weeks defines only a single <span class="monospaced">send</span>
 161 and a single <span class="monospaced">receive</span> and the result types of the functions depend on
 162 their arguments.  The implementation
 163 (<a href="https://github.com/MLton/mltonlib/blob/master/com/sweeks/basic/unstable/net.sml"><span class="monospaced">net.sml</span></a>) uses
 164 static sums (with a slighly different signature:
 165 <a href="https://github.com/MLton/mltonlib/blob/master/com/sweeks/basic/unstable/static-sum.sig"><span class="monospaced">static-sum.sig</span></a>).</p></div>
 166 </div>
 167 <div class="sect2">
 168 <h3 id="_example_picking_monad_results">Example: Picking Monad Results</h3>
 169 <div class="paragraph"><p>Suppose that we need to write a parser that accepts a pair of integers
 170 and returns their sum given a monadic parsing combinator library.  A
 171 part of the signature of such library could look like this</p></div>
 172 <div class="listingblock">
 173 <div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">PARSING</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">sig</span><span class="w"></span>
 174 <span class="w">   </span><span class="k">include</span><span class="w"> </span><span class="n">MONAD</span><span class="w"></span>
 175 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 176 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">lparen</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 177 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">rparen</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 178 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">comma</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 179 <span class="w">   </span><span class="cm">(* ... *)</span><span class="w"></span>
 180 <span class="k">end</span><span class="w"></span>
 181 </pre></div></div></div>
 182 <div class="paragraph"><p>where the <span class="monospaced">MONAD</span> signature could be defined as</p></div>
 183 <div class="listingblock">
 184 <div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">MONAD</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">sig</span><span class="w"></span>
 185 <span class="w">   </span><span class="k">type</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 186 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">return</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 187 <span class="w">   </span><span class="k">val</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">:</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="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">t</span><span class="p">)</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 188 <span class="k">end</span><span class="w"></span>
 189 <span class="k">infix</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"></span>
 190 </pre></div></div></div>
 191 <div class="paragraph"><p>The straightforward, but tedious, way to write the desired parser is:</p></div>
 192 <div class="listingblock">
 193 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">lparen</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 194 <span class="w">        </span><span class="n">int</span><span class="w">    </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 195 <span class="w">        </span><span class="n">comma</span><span class="w">  </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 196 <span class="w">        </span><span class="n">int</span><span class="w">    </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 197 <span class="w">        </span><span class="n">rparen</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 198 <span class="w">        </span><span class="n">return</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">y</span><span class="p">))))))</span><span class="w"></span>
 199 </pre></div></div></div>
 200 <div class="paragraph"><p>In Haskell, the parser could be written using the <span class="monospaced">do</span> notation
 201 considerably less verbosely as:</p></div>
 202 <div class="listingblock">
 203 <div class="content"><div class="highlight"><pre><span class="nf">p</span> <span class="ow">=</span> <span class="kr">do</span> <span class="p">{</span> <span class="n">lparen</span> <span class="p">;</span> <span class="n">x</span> <span class="ow">&lt;-</span> <span class="n">int</span> <span class="p">;</span> <span class="n">comma</span> <span class="p">;</span> <span class="n">y</span> <span class="ow">&lt;-</span> <span class="n">int</span> <span class="p">;</span> <span class="n">rparen</span> <span class="p">;</span> <span class="n">return</span> <span class="o">$</span> <span class="n">x</span> <span class="o">+</span> <span class="n">y</span> <span class="p">}</span>
 204 </pre></div></div></div>
 205 <div class="paragraph"><p>SML doesn&#8217;t provide a <span class="monospaced">do</span> notation, so we need another solution.</p></div>
 206 <div class="paragraph"><p>Suppose we would have a "pick" notation for monads that would allows
 207 us to write the parser as</p></div>
 208 <div class="listingblock">
 209 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">`lparen</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">\int</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">`comma</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">\int</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">`rparen</span><span class="w"> </span><span class="n">@</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w"></span>
 210 </pre></div></div></div>
 211 <div class="paragraph"><p>using four auxiliary combinators: <span class="monospaced">&grave;</span>, <span class="monospaced">\</span>, <span class="monospaced">^</span>, and <span class="monospaced">@</span>.</p></div>
 212 <div class="paragraph"><p>Roughly speaking</p></div>
 213 <div class="ulist"><ul>
 214 <li>
 215 <p>
 216 <span class="monospaced">&grave;p</span> means that the result of <span class="monospaced">p</span> is dropped,
 217 </p>
 218 </li>
 219 <li>
 220 <p>
 221 <span class="monospaced">\p</span> means that the result of <span class="monospaced">p</span> is taken,
 222 </p>
 223 </li>
 224 <li>
 225 <p>
 226 <span class="monospaced">p ^ q</span> means that results of <span class="monospaced">p</span> and <span class="monospaced">q</span> are taken as a product, and
 227 </p>
 228 </li>
 229 <li>
 230 <p>
 231 <span class="monospaced">p @ a</span> means that the results of <span class="monospaced">p</span> are passed to the function <span class="monospaced">a</span> and that result is returned.
 232 </p>
 233 </li>
 234 </ul></div>
 235 <div class="paragraph"><p>The difficulty is in implementing the concatenation combinator <span class="monospaced">^</span>.
 236 The type of the result of the concatenation depends on the types of
 237 the arguments.</p></div>
 238 <div class="paragraph"><p>Using static sums and the <a href="ProductType">product type</a>, the pick
 239 notation for monads can be implemented as follows:</p></div>
 240 <div class="listingblock">
 241 <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">MkMonadPick</span><span class="w"> </span><span class="p">(</span><span class="k">include</span><span class="w"> </span><span class="n">MONAD</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">let</span><span class="w"></span>
 242 <span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>
 243 <span class="k">in</span><span class="w"></span>
 244 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
 245 <span class="w">      </span><span class="k">fun</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">inL</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">return</span><span class="w"> </span><span class="p">()))</span><span class="w"></span>
 246 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">\</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">inR</span><span class="w"></span>
 247 <span class="w">      </span><span class="k">fun</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">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="n">return</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
 248 <span class="w">      </span><span class="k">fun</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">b</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 249 <span class="w">          </span><span class="p">(</span><span class="n">match</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">match</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"></span>
 250 <span class="w">             </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 251 <span class="w">                 </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inL</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="p">)),</span><span class="w"></span>
 252 <span class="w">                  </span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inR</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="p">))),</span><span class="w"></span>
 253 <span class="w">              </span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
 254 <span class="w">                 </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inR</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">return</span><span class="w"> </span><span class="n">a</span><span class="p">))),</span><span class="w"></span>
 255 <span class="w">                  </span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inR</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">return</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">b</span><span class="p">))))))</span><span class="w"></span>
 256 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 257 <span class="k">end</span><span class="w"></span>
 258 </pre></div></div></div>
 259 <div class="paragraph"><p>The above implementation is inefficient, however.  It uses many more
 260 bind operations, <span class="monospaced">&gt;&gt;=</span>, than necessary.  That can be solved with an
 261 additional level of abstraction:</p></div>
 262 <div class="listingblock">
 263 <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">MkMonadPick</span><span class="w"> </span><span class="p">(</span><span class="k">include</span><span class="w"> </span><span class="n">MONAD</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">let</span><span class="w"></span>
 264 <span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>
 265 <span class="k">in</span><span class="w"></span>
 266 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
 267 <span class="w">      </span><span class="k">fun</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">inL</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">()))</span><span class="w"></span>
 268 <span class="w">      </span><span class="k">fun</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">inR</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"> </span><span class="n">b</span><span class="p">)</span><span class="w"></span>
 269 <span class="w">      </span><span class="k">fun</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">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">out</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">(</span><span class="n">return</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">f</span><span class="p">)</span><span class="w"></span>
 270 <span class="w">      </span><span class="k">fun</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">b</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 271 <span class="w">          </span><span class="p">(</span><span class="n">match</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="n">o</span><span class="w"> </span><span class="n">match</span><span class="w"> </span><span class="n">a</span><span class="p">)</span><span class="w"></span>
 272 <span class="w">             </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inL</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="n">c</span><span class="p">)),</span><span class="w"></span>
 273 <span class="w">                       </span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inR</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="n">c</span><span class="p">))),</span><span class="w"></span>
 274 <span class="w">              </span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inR</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="n">a</span><span class="p">))),</span><span class="w"></span>
 275 <span class="w">                       </span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">inR</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">a</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">b</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">c</span><span class="w"> </span><span class="p">(</span><span class="n">a</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">b</span><span class="p">))))))</span><span class="w"></span>
 276 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 277 <span class="k">end</span><span class="w"></span>
 278 </pre></div></div></div>
 279 <div class="paragraph"><p>After instantiating and opening either of the above monad pick
 280 implementations, the previously given definition of <span class="monospaced">p</span> can be
 281 compiled and results in a parser whose result is of type <span class="monospaced">int</span>.  Here
 282 is a functor to test the theory:</p></div>
 283 <div class="listingblock">
 284 <div class="content"><div class="highlight"><pre><span class="k">functor</span><span class="w"> </span><span class="n">Test</span><span class="w"> </span><span class="p">(</span><span class="n">Arg</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">PARSING</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">struct</span><span class="w"></span>
 285 <span class="w">   </span><span class="k">local</span><span class="w"></span>
 286 <span class="w">      </span><span class="k">structure</span><span class="w"> </span><span class="n">Pick</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">MkMonadPick</span><span class="w"> </span><span class="p">(</span><span class="n">Arg</span><span class="p">)</span><span class="w"></span>
 287 <span class="w">      </span><span class="k">open</span><span class="w"> </span><span class="n">Pick</span><span class="w"> </span><span class="n">Arg</span><span class="w"></span>
 288 <span class="w">   </span><span class="k">in</span><span class="w"></span>
 289 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 290 <span class="w">          </span><span class="n">`lparen</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">\int</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">`comma</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">\int</span><span class="w"> </span><span class="n">^</span><span class="w"> </span><span class="n">`rparen</span><span class="w"> </span><span class="n">@</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">y</span><span class="p">)</span><span class="w"></span>
 291 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 292 <span class="k">end</span><span class="w"></span>
 293 </pre></div></div></div>
 294 </div>
 295 </div>
 296 </div>
 297 <div class="sect1">
 298 <h2 id="_also_see">Also see</h2>
 299 <div class="sectionbody">
 300 <div class="paragraph"><p>There are a number of related techniques.  Here are some of them.</p></div>
 301 <div class="ulist"><ul>
 302 <li>
 303 <p>
 304 <a href="Fold">Fold</a>
 305 </p>
 306 </li>
 307 <li>
 308 <p>
 309 <a href="TypeIndexedValues">TypeIndexedValues</a>
 310 </p>
 311 </li>
 312 </ul></div>
 313 </div>
 314 </div>
 315 </div>
 316 <div id="footnotes"><hr></div>
 317 <div id="footer">
 318 <div id="footer-text">
 319 </div>
 320 <div id="footer-badges">
 321 </div>
 322 </div>
 323 </body>
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