doc/guide/localhost/VariableArityPolymorphism

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  26 <h1>VariableArityPolymorphism</h1>
  27 </div>
  28 <div id="content">
  29 <div id="preamble">
  30 <div class="sectionbody">
  31 <div class="paragraph"><p><a href="StandardML">Standard ML</a> programmers often face the problem of how to
  32 provide a variable-arity polymorphic function.  For example, suppose
  33 one is defining a combinator library, e.g. for parsing or pickling.
  34 The signature for such a library might look something like the
  35 following.</p></div>
  36 <div class="listingblock">
  37 <div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">COMBINATOR</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  38 <span class="w">   </span><span class="k">sig</span><span class="w"></span>
  39 <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>
  40
  41 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">int</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>
  42 <span class="w">      </span><span class="k">val</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="w"> </span><span class="n">t</span><span class="w"></span>
  43 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">string</span><span class="p">:</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  44 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">unit</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>
  45 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">tuple2</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a1</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a2</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;a1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a2</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  46 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">tuple3</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a1</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a2</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a3</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;a1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a2</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a3</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  47 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">tuple4</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a1</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a2</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a3</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a4</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  48 <span class="w">                  </span><span class="p">-&gt;</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a2</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a3</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">&#39;a4</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  49 <span class="w">      </span><span class="p">...</span><span class="w"></span>
  50 <span class="w">   </span><span class="k">end</span><span class="w"></span>
  51 </pre></div></div></div>
  52 <div class="paragraph"><p>The question is how to define a variable-arity tuple combinator.
  53 Traditionally, the only way to take a variable number of arguments in
  54 SML is to put the arguments in a list (or vector) and pass that.  So,
  55 one might define a tuple combinator with the following signature.</p></div>
  56 <div class="listingblock">
  57 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">tupleN</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">list</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">list</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
  58 </pre></div></div></div>
  59 <div class="paragraph"><p>The problem with this approach is that as soon as one places values in
  60 a list, they must all have the same type.  So, programmers often take
  61 an alternative approach, and define a family of <span class="monospaced">tuple&lt;N&gt;</span> functions,
  62 as we see in the <span class="monospaced">COMBINATOR</span> signature above.</p></div>
  63 <div class="paragraph"><p>The family-of-functions approach is ugly for many reasons.  First, it
  64 clutters the signature with a number of functions when there should
  65 really only be one.  Second, it is <em>closed</em>, in that there are a fixed
  66 number of tuple combinators in the interface, and should a client need
  67 a combinator for a large tuple, he is out of luck.  Third, this
  68 approach often requires a lot of duplicate code in the implementation
  69 of the combinators.</p></div>
  70 <div class="paragraph"><p>Fortunately, using <a href="Fold01N">Fold01N</a> and <a href="ProductType">products</a>, one can
  71 provide an interface and implementation that solves all these
  72 problems.  Here is a simple pickling module that converts values to
  73 strings.</p></div>
  74 <div class="listingblock">
  75 <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Pickler</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  76 <span class="w">   </span><span class="k">struct</span><span class="w"></span>
  77 <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><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">string</span><span class="w"></span>
  78
  79 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="p">=</span><span class="w"> </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="s">&quot;&quot;</span><span class="w"></span>
  80
  81 <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="p">.</span><span class="n">toString</span><span class="w"></span>
  82
  83 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">real</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">toString</span><span class="w"></span>
  84
  85 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">id</span><span class="w"></span>
  86
  87 <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">accum</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">*</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">list</span><span class="w"></span>
  88
  89 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
  90 <span class="w">         </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
  91 <span class="w">         </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">fold</span><span class="w"></span>
  92 <span class="w">         </span><span class="p">{</span><span class="n">finish</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">ps</span><span class="w"> </span><span class="p">=&gt;</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">concat</span><span class="w"> </span><span class="p">(</span><span class="n">rev</span><span class="w"> </span><span class="p">(</span><span class="n">ps</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="p">[]))),</span><span class="w"></span>
  93 <span class="w">          </span><span class="n">start</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">p</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">l</span><span class="p">,</span><span class="w"></span>
  94 <span class="w">          </span><span class="n">zero</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">unit</span><span class="p">}</span><span class="w"></span>
  95 <span class="w">         </span><span class="n">z</span><span class="w"></span>
  96
  97 <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>
  98 <span class="w">         </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"></span>
  99 <span class="w">         </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">step1</span><span class="w"></span>
 100 <span class="w">         </span><span class="p">{</span><span class="n">combine</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">p&#39;</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">x&#39;</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="n">p&#39;</span><span class="w"> </span><span class="n">x&#39;</span><span class="w"> </span><span class="n">::</span><span class="w"> </span><span class="s">&quot;,&quot;</span><span class="w"> </span><span class="n">::</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">))}</span><span class="w"></span>
 101 <span class="w">         </span><span class="n">z</span><span class="w"></span>
 102 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 103 </pre></div></div></div>
 104 <div class="paragraph"><p>If one has <span class="monospaced">n</span> picklers of types</p></div>
 105 <div class="listingblock">
 106 <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">p1</span><span class="p">:</span><span class="w"> </span><span class="n">a1</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 107 <span class="k">val</span><span class="w"> </span><span class="n">p2</span><span class="p">:</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 108 <span class="p">...</span><span class="w"></span>
 109 <span class="k">val</span><span class="w"> </span><span class="n">pn</span><span class="p">:</span><span class="w"> </span><span class="n">an</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 110 </pre></div></div></div>
 111 <div class="paragraph"><p>then one can construct a pickler for n-ary products as follows.</p></div>
 112 <div class="listingblock">
 113 <div class="content"><div class="highlight"><pre><span class="n">tuple</span><span class="w"> </span><span class="n">`p1</span><span class="w"> </span><span class="n">`p2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">`pn</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">a1</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="n">an</span><span class="p">)</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 114 </pre></div></div></div>
 115 <div class="paragraph"><p>For example, with <span class="monospaced">Pickler</span> in scope, one can prove the following
 116 equations.</p></div>
 117 <div class="listingblock">
 118 <div class="content"><div class="highlight"><pre><span class="s">&quot;&quot;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
 119 <span class="s">&quot;1&quot;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">`int</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>
 120 <span class="s">&quot;1,2.0&quot;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">`int</span><span class="w"> </span><span class="n">`real</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="mf">2.0</span><span class="p">)</span><span class="w"></span>
 121 <span class="s">&quot;1,2.0,three&quot;</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">`int</span><span class="w"> </span><span class="n">`real</span><span class="w"> </span><span class="n">`string</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="n">&amp;</span><span class="w"> </span><span class="s">&quot;three&quot;</span><span class="p">)</span><span class="w"></span>
 122 </pre></div></div></div>
 123 <div class="paragraph"><p>Here is the signature for <span class="monospaced">Pickler</span>.  It shows why the <span class="monospaced">accum</span> type is
 124 useful.</p></div>
 125 <div class="listingblock">
 126 <div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">PICKLER</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
 127 <span class="w">   </span><span class="k">sig</span><span class="w"></span>
 128 <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>
 129
 130 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">int</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>
 131 <span class="w">      </span><span class="k">val</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="w"> </span><span class="n">t</span><span class="w"></span>
 132 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">string</span><span class="p">:</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="w"></span>
 133 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">unit</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>
 134
 135 <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">accum</span><span class="w"></span>
 136 <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="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">accum</span><span class="p">,</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">(</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">prod</span><span class="w"> </span><span class="n">accum</span><span class="p">,</span><span class="w"></span>
 137 <span class="w">               </span><span class="n">&#39;z1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z2</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z3</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z4</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z5</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z6</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z7</span><span class="p">)</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">step1</span><span class="w"></span>
 138 <span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">tuple</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">&#39;a</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">accum</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;b</span><span class="w"> </span><span class="n">accum</span><span class="p">,</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="n">unit</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"></span>
 139 <span class="w">                  </span><span class="n">&#39;z1</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z2</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z3</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z4</span><span class="p">,</span><span class="w"> </span><span class="n">&#39;z5</span><span class="p">)</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">t</span><span class="w"></span>
 140 <span class="w">   </span><span class="k">end</span><span class="w"></span>
 141
 142 <span class="k">structure</span><span class="w"> </span><span class="n">Pickler</span><span class="p">:</span><span class="w"> </span><span class="n">PICKLER</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Pickler</span><span class="w"></span>
 143 </pre></div></div></div>
 144 </div>
 145 </div>
 146 </div>
 147 <div id="footnotes"><hr></div>
 148 <div id="footer">
 149 <div id="footer-text">
 150 </div>
 151 <div id="footer-badges">
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