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<a href="./Home">MLton 20180207</a>\r
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<div id="header">\r
<h1>StaticSum</h1>\r
</div>\r
<div id="content">\r
<div id="preamble">\r
<div class="sectionbody">\r
<div class="paragraph"><p>While SML makes it impossible to write functions whose types would\r
depend on the values of their arguments, or so called dependently\r
typed functions, it is possible, and arguably commonplace, to write\r
functions whose types depend on the types of their arguments.  Indeed,\r
the types of parametrically polymorphic functions like <span class="monospaced">map</span> and\r
<span class="monospaced">foldl</span> can be said to depend on the types of their arguments.  What\r
is less commonplace, however, is to write functions whose behavior\r
would depend on the types of their arguments.  Nevertheless, there are\r
several techniques for writing such functions.\r
<a href="TypeIndexedValues">Type-indexed values</a> and <a href="Fold">fold</a> are two such\r
techniques.  This page presents another such technique dubbed static\r
sums.</p></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_ordinary_sums">Ordinary Sums</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>Consider the sum type as defined below:</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>While a generic sum type such as defined above is very useful, it has\r
a number of limitations.  As an example, we could write the function\r
<span class="monospaced">out</span> to extract the value from a sum as follows:</p></div>\r
<div class="listingblock">\r
<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>\r
<span class="w">    </span><span class="k">case</span><span class="w"> </span><span class="n">s</span><span class="w"></span>\r
<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>\r
<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>\r
</pre></div></div></div>\r
<div class="paragraph"><p>As can be seen from the type of <span class="monospaced">out</span>, it is limited in the sense that\r
it requires both variants of the sum to have the same type.  So, <span class="monospaced">out</span>\r
cannot be used to extract the value of a sum of two different types,\r
such as the type <span class="monospaced">(int, real) Sum.t</span>.  As another example of a\r
limitation, consider the following attempt at a <span class="monospaced">succ</span> function:</p></div>\r
<div class="listingblock">\r
<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>\r
<span class="w">    </span><span class="k">case</span><span class="w"> </span><span class="n">s</span><span class="w"></span>\r
<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>\r
<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>\r
</pre></div></div></div>\r
<div class="paragraph"><p>The above definition of <span class="monospaced">succ</span> cannot be typed, because there is no\r
type for the codomain within SML.</p></div>\r
</div>\r
</div>\r
<div class="sect1">\r
<h2 id="_static_sums">Static Sums</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>Interestingly, it is possible to define values <span class="monospaced">inL</span>, <span class="monospaced">inR</span>, and\r
<span class="monospaced">match</span> that satisfy the laws</p></div>\r
<div class="listingblock">\r
<div class="content monospaced">\r
<pre>match (inL x) (f, g) = f x\r
match (inR x) (f, g) = g x</pre>\r
</div></div>\r
<div class="paragraph"><p>and do not suffer from the same limitions.  The definitions are\r
actually quite trivial:</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>Now, given the <span class="monospaced">succ</span> function defined as</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<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>\r
<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>\r
</pre></div></div></div>\r
<div class="paragraph"><p>we get</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
</pre></div></div></div>\r
<div class="paragraph"><p>To better understand how this works, consider the following signature\r
for static sums:</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>Above, <span class="monospaced">'d</span> stands for domain and <span class="monospaced">'c</span> for codomain.  The key\r
difference between an ordinary sum type, like <span class="monospaced">(int, real) Sum.t</span>, and\r
a static sum type, like <span class="monospaced">(int, real, real, int, real) StaticSum.t</span>, is\r
that the ordinary sum type says nothing about the type of the result\r
of deconstructing a sum while the static sum type specifies the type.</p></div>\r
<div class="paragraph"><p>With the sealed static sum module, we get the type</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="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>\r
</pre></div></div></div>\r
<div class="paragraph"><p>for the previously defined <span class="monospaced">succ</span> function.  The type specifies that\r
<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>.\r
For example, the type of <span class="monospaced">StaticSum.inL 1</span> is\r
<span class="monospaced">(int, 'cL, 'dR, 'cR, 'cL) StaticSum.t</span>.  Unifying this with the\r
argument type of <span class="monospaced">succ</span> gives the type <span class="monospaced">(int, int, real, real, int)\r
StaticSum.t -&gt; int</span>.</p></div>\r
<div class="paragraph"><p>The <span class="monospaced">out</span> function is quite useful on its own.  Here is how it can be\r
defined:</p></div>\r
<div class="listingblock">\r
<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>\r
<span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>\r
<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>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>Due to the value restriction, lack of first class polymorphism and\r
polymorphic recursion, the usefulness and convenience of static sums\r
is somewhat limited in SML.  So, don&#8217;t throw away the ordinary sum\r
type just yet.  Static sums can nevertheless be quite useful.</p></div>\r
<div class="sect2">\r
<h3 id="_example_send_and_receive_with_argument_type_dependent_result_types">Example: Send and Receive with Argument Type Dependent Result Types</h3>\r
<div class="paragraph"><p>In some situations it would seem useful to define functions whose\r
result type would depend on some of the arguments.  Traditionally such\r
functions have been thought to be impossible in SML and the solution\r
has been to define multiple functions.  For example, the\r
<a href="http://www.standardml.org/Basis/socket.html"><span class="monospaced">Socket</span> structure</a> of the\r
Basis library defines 16 <span class="monospaced">send</span> and 16 <span class="monospaced">recv</span> functions.  In contrast,\r
the Net structure\r
(<a href="https://github.com/MLton/mltonlib/blob/master/com/sweeks/basic/unstable/net.sig"><span class="monospaced">net.sig</span></a>) of the\r
Basic library designed by Stephen Weeks defines only a single <span class="monospaced">send</span>\r
and a single <span class="monospaced">receive</span> and the result types of the functions depend on\r
their arguments.  The implementation\r
(<a href="https://github.com/MLton/mltonlib/blob/master/com/sweeks/basic/unstable/net.sml"><span class="monospaced">net.sml</span></a>) uses\r
static sums (with a slighly different signature:\r
<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>\r
</div>\r
<div class="sect2">\r
<h3 id="_example_picking_monad_results">Example: Picking Monad Results</h3>\r
<div class="paragraph"><p>Suppose that we need to write a parser that accepts a pair of integers\r
and returns their sum given a monadic parsing combinator library.  A\r
part of the signature of such library could look like this</p></div>\r
<div class="listingblock">\r
<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>\r
<span class="w">   </span><span class="k">include</span><span class="w"> </span><span class="n">MONAD</span><span class="w"></span>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">   </span><span class="cm">(* ... *)</span><span class="w"></span>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>where the <span class="monospaced">MONAD</span> signature could be defined as</p></div>\r
<div class="listingblock">\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="k">end</span><span class="w"></span>\r
<span class="k">infix</span><span class="w"> </span><span class="n">&gt;&gt;=</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>The straightforward, but tedious, way to write the desired parser is:</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="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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
</pre></div></div></div>\r
<div class="paragraph"><p>In Haskell, the parser could be written using the <span class="monospaced">do</span> notation\r
considerably less verbosely as:</p></div>\r
<div class="listingblock">\r
<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>\r
</pre></div></div></div>\r
<div class="paragraph"><p>SML doesn&#8217;t provide a <span class="monospaced">do</span> notation, so we need another solution.</p></div>\r
<div class="paragraph"><p>Suppose we would have a "pick" notation for monads that would allows\r
us to write the parser 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="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>\r
</pre></div></div></div>\r
<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>\r
<div class="paragraph"><p>Roughly speaking</p></div>\r
<div class="ulist"><ul>\r
<li>\r
<p>\r
<span class="monospaced">&grave;p</span> means that the result of <span class="monospaced">p</span> is dropped,\r
</p>\r
</li>\r
<li>\r
<p>\r
<span class="monospaced">\p</span> means that the result of <span class="monospaced">p</span> is taken,\r
</p>\r
</li>\r
<li>\r
<p>\r
<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\r
</p>\r
</li>\r
<li>\r
<p>\r
<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.\r
</p>\r
</li>\r
</ul></div>\r
<div class="paragraph"><p>The difficulty is in implementing the concatenation combinator <span class="monospaced">^</span>.\r
The type of the result of the concatenation depends on the types of\r
the arguments.</p></div>\r
<div class="paragraph"><p>Using static sums and the <a href="ProductType">product type</a>, the pick\r
notation for monads can be implemented as follows:</p></div>\r
<div class="listingblock">\r
<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>\r
<span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>\r
<span class="k">in</span><span class="w"></span>\r
<span class="w">   </span><span class="k">struct</span><span class="w"></span>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">   </span><span class="k">end</span><span class="w"></span>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>The above implementation is inefficient, however.  It uses many more\r
bind operations, <span class="monospaced">&gt;&gt;=</span>, than necessary.  That can be solved with an\r
additional level of abstraction:</p></div>\r
<div class="listingblock">\r
<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>\r
<span class="w">   </span><span class="k">open</span><span class="w"> </span><span class="n">StaticSum</span><span class="w"></span>\r
<span class="k">in</span><span class="w"></span>\r
<span class="w">   </span><span class="k">struct</span><span class="w"></span>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<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>\r
<span class="w">   </span><span class="k">end</span><span class="w"></span>\r
<span class="k">end</span><span class="w"></span>\r
</pre></div></div></div>\r
<div class="paragraph"><p>After instantiating and opening either of the above monad pick\r
implementations, the previously given definition of <span class="monospaced">p</span> can be\r
compiled and results in a parser whose result is of type <span class="monospaced">int</span>.  Here\r
is a functor to test the theory:</p></div>\r
<div class="listingblock">\r
<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>\r
<span class="w">   </span><span class="k">local</span><span class="w"></span>\r
<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>\r
<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>\r
<span class="w">   </span><span class="k">in</span><span class="w"></span>\r
<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>\r
<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>\r
<span class="w">   </span><span class="k">end</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>\r
<div class="sect1">\r
<h2 id="_also_see">Also see</h2>\r
<div class="sectionbody">\r
<div class="paragraph"><p>There are a number of related techniques.  Here are some of them.</p></div>\r
<div class="ulist"><ul>\r
<li>\r
<p>\r
<a href="Fold">Fold</a>\r
</p>\r
</li>\r
<li>\r
<p>\r
<a href="TypeIndexedValues">TypeIndexedValues</a>\r
</p>\r
</li>\r
</ul></div>\r
</div>\r
</div>\r
</div>\r
<div id="footnotes"><hr></div>\r
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<div id="footer-text">\r
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