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<h1>UniversalType</h1>
</div>
<div id="content">
<div id="preamble">
<div class="sectionbody">
<div class="paragraph"><p>A universal type is a type into which all other types can be embedded.
Here&#8217;s a <a href="StandardML">Standard ML</a> signature for a universal type.</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">UNIVERSAL_TYPE</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">   </span><span class="k">sig</span><span class="w"></span>
<span class="w">      </span><span class="k">type</span><span class="w"> </span><span class="n">t</span><span class="w"></span>

<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">embed</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="p">-&gt;</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">t</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </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><span class="n">option</span><span class="p">)</span><span class="w"></span>
<span class="w">   </span><span class="k">end</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>The idea is that <span class="monospaced">type t</span> is the universal type and that each call to
<span class="monospaced">embed</span> returns a new pair of functions <span class="monospaced">(inject, project)</span>, where
<span class="monospaced">inject</span> embeds a value into the universal type and <span class="monospaced">project</span> extracts
the value from the universal type.  A pair <span class="monospaced">(inject, project)</span>
returned by <span class="monospaced">embed</span> works together in that <span class="monospaced">project u</span> will return
<span class="monospaced">SOME v</span> if and only if <span class="monospaced">u</span> was created by <span class="monospaced">inject v</span>.  If <span class="monospaced">u</span> was
created by a different function <span class="monospaced">inject'</span>, then <span class="monospaced">project</span> returns
<span class="monospaced">NONE</span>.</p></div>
<div class="paragraph"><p>Here&#8217;s an example embedding integers and reals into a universal type.</p></div>
<div class="listingblock">
<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">U</span><span class="p">:</span><span class="w"> </span><span class="n">UNIVERSAL_TYPE</span><span class="p">):</span><span class="w"> </span><span class="k">sig</span><span class="w"> </span><span class="k">end</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">   </span><span class="k">struct</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="p">(</span><span class="n">intIn</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">U</span><span class="p">.</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">intOut</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">U</span><span class="p">.</span><span class="n">embed</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">U</span><span class="p">.</span><span class="n">t</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">(</span><span class="n">intIn</span><span class="w"> </span><span class="mi">13</span><span class="p">)</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">s1</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">         </span><span class="k">case</span><span class="w"> </span><span class="n">intOut</span><span class="w"> </span><span class="p">(</span><span class="n">!r</span><span class="p">)</span><span class="w"> </span><span class="k">of</span><span class="w"></span>
<span class="w">            </span><span class="n">NONE</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="s">&quot;NONE&quot;</span><span class="w"></span>
<span class="w">          </span><span class="p">|</span><span class="w"> </span><span class="n">SOME</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">Int</span><span class="p">.</span><span class="n">toString</span><span class="w"> </span><span class="n">i</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="p">(</span><span class="n">realIn</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">U</span><span class="p">.</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">realOut</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">U</span><span class="p">.</span><span class="n">embed</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">realIn</span><span class="w"> </span><span class="mf">13.0</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">s2</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">         </span><span class="k">case</span><span class="w"> </span><span class="n">intOut</span><span class="w"> </span><span class="p">(</span><span class="n">!r</span><span class="p">)</span><span class="w"> </span><span class="k">of</span><span class="w"></span>
<span class="w">            </span><span class="n">NONE</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="s">&quot;NONE&quot;</span><span class="w"></span>
<span class="w">          </span><span class="p">|</span><span class="w"> </span><span class="n">SOME</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">Int</span><span class="p">.</span><span class="n">toString</span><span class="w"> </span><span class="n">i</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="n">s3</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">         </span><span class="k">case</span><span class="w"> </span><span class="n">realOut</span><span class="w"> </span><span class="p">(</span><span class="n">!r</span><span class="p">)</span><span class="w"> </span><span class="k">of</span><span class="w"></span>
<span class="w">            </span><span class="n">NONE</span><span class="w"> </span><span class="p">=&gt;</span><span class="w"> </span><span class="s">&quot;NONE&quot;</span><span class="w"></span>
<span class="w">          </span><span class="p">|</span><span class="w"> </span><span class="n">SOME</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">Real</span><span class="p">.</span><span class="n">toString</span><span class="w"> </span><span class="n">x</span><span class="w"></span>
<span class="w">      </span><span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">print</span><span class="w"> </span><span class="p">(</span><span class="n">concat</span><span class="w"> </span><span class="p">[</span><span class="n">s1</span><span class="p">,</span><span class="w"> </span><span class="s">&quot; &quot;</span><span class="p">,</span><span class="w"> </span><span class="n">s2</span><span class="p">,</span><span class="w"> </span><span class="s">&quot; &quot;</span><span class="p">,</span><span class="w"> </span><span class="n">s3</span><span class="p">,</span><span class="w"> </span><span class="s">&quot;</span><span class="se">\n</span><span class="s">&quot;</span><span class="p">])</span><span class="w"></span>
<span class="w">   </span><span class="k">end</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>Applying <span class="monospaced">Test</span> to an appropriate implementation will print</p></div>
<div class="listingblock">
<div class="content monospaced">
<pre>13 NONE 13.0</pre>
</div></div>
<div class="paragraph"><p>Note that two different calls to embed on the same type return
different embeddings.</p></div>
<div class="paragraph"><p>Standard ML does not have explicit support for universal types;
however, there are at least two ways to implement them.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_implementation_using_exceptions">Implementation Using Exceptions</h2>
<div class="sectionbody">
<div class="paragraph"><p>While the intended use of SML exceptions is for exception handling, an
accidental feature of their design is that the <span class="monospaced">exn</span> type is a
universal type.  The implementation relies on being able to declare
exceptions locally to a function and on the fact that exceptions are
<a href="GenerativeException">generative</a>.</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">U</span><span class="p">:&gt;</span><span class="w"> </span><span class="n">UNIVERSAL_TYPE</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">   </span><span class="k">struct</span><span class="w"></span>
<span class="w">      </span><span class="k">type</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">exn</span><span class="w"></span>

<span class="w">      </span><span class="k">fun</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">embed</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">         </span><span class="k">let</span><span class="w"></span>
<span class="w">            </span><span class="k">exception</span><span class="w"> </span><span class="n">E</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="w">            </span><span class="k">fun</span><span class="w"> </span><span class="n">project</span><span class="w"> </span><span class="p">(</span><span class="n">e</span><span class="p">:</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">option</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">               </span><span class="k">case</span><span class="w"> </span><span class="n">e</span><span class="w"> </span><span class="k">of</span><span class="w"></span>
<span class="w">                  </span><span class="n">E</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">SOME</span><span class="w"> </span><span class="n">a</span><span class="w"></span>
<span class="w">                </span><span class="p">|</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">NONE</span><span class="w"></span>
<span class="w">         </span><span class="k">in</span><span class="w"></span>
<span class="w">            </span><span class="p">(</span><span class="n">E</span><span class="p">,</span><span class="w"> </span><span class="n">project</span><span class="p">)</span><span class="w"></span>
<span class="w">         </span><span class="k">end</span><span class="w"></span>
<span class="w">   </span><span class="k">end</span><span class="w"></span>
</pre></div></div></div>
</div>
</div>
<div class="sect1">
<h2 id="_implementation_using_functions_and_references">Implementation Using Functions and References</h2>
<div class="sectionbody">
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">U</span><span class="p">:&gt;</span><span class="w"> </span><span class="n">UNIVERSAL_TYPE</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">   </span><span class="k">struct</span><span class="w"></span>
<span class="w">      </span><span class="k">datatype</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">T</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="p">{</span><span class="n">clear</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">unit</span><span class="p">,</span><span class="w"></span>
<span class="w">                         </span><span class="n">store</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">unit</span><span class="p">}</span><span class="w"></span>

<span class="w">      </span><span class="k">fun</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">embed</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">         </span><span class="k">let</span><span class="w"></span>
<span class="w">            </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="n">NONE</span><span class="w"></span>
<span class="w">            </span><span class="k">fun</span><span class="w"> </span><span class="n">inject</span><span class="w"> </span><span class="p">(</span><span class="n">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">t</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">               </span><span class="n">T</span><span class="w"> </span><span class="p">{</span><span class="n">clear</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="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">NONE</span><span class="p">,</span><span class="w"></span>
<span class="w">                  </span><span class="n">store</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="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">SOME</span><span class="w"> </span><span class="n">a</span><span class="p">}</span><span class="w"></span>
<span class="w">            </span><span class="k">fun</span><span class="w"> </span><span class="n">project</span><span class="w"> </span><span class="p">(</span><span class="n">T</span><span class="w"> </span><span class="p">{</span><span class="n">clear</span><span class="p">,</span><span class="w"> </span><span class="n">store</span><span class="p">}):</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">               </span><span class="k">let</span><span class="w"></span>
<span class="w">                  </span><span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">store</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
<span class="w">                  </span><span class="k">val</span><span class="w"> </span><span class="n">res</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">!r</span><span class="w"></span>
<span class="w">                  </span><span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">clear</span><span class="w"> </span><span class="p">()</span><span class="w"></span>
<span class="w">               </span><span class="k">in</span><span class="w"></span>
<span class="w">                  </span><span class="n">res</span><span class="w"></span>
<span class="w">               </span><span class="k">end</span><span class="w"></span>
<span class="w">         </span><span class="k">in</span><span class="w"></span>
<span class="w">            </span><span class="p">(</span><span class="n">inject</span><span class="p">,</span><span class="w"> </span><span class="n">project</span><span class="p">)</span><span class="w"></span>
<span class="w">         </span><span class="k">end</span><span class="w"></span>
<span class="w">   </span><span class="k">end</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>Note that due to the use of a shared ref cell, the above
implementation is not thread safe.</p></div>
<div class="paragraph"><p>One could try to simplify the above implementation by eliminating the
<span class="monospaced">clear</span> function, making <span class="monospaced">type t = unit -&gt; unit</span>.</p></div>
<div class="listingblock">
<div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">U</span><span class="p">:&gt;</span><span class="w"> </span><span class="n">UNIVERSAL_TYPE</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">   </span><span class="k">struct</span><span class="w"></span>
<span class="w">      </span><span class="k">type</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">unit</span><span class="w"> </span><span class="p">-&gt;</span><span class="w"> </span><span class="n">unit</span><span class="w"></span>

<span class="w">      </span><span class="k">fun</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">embed</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"></span>
<span class="w">         </span><span class="k">let</span><span class="w"></span>
<span class="w">            </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">&#39;a</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="n">NONE</span><span class="w"></span>
<span class="w">            </span><span class="k">fun</span><span class="w"> </span><span class="n">inject</span><span class="w"> </span><span class="p">(</span><span class="n">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">t</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="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">SOME</span><span class="w"> </span><span class="n">a</span><span class="w"></span>
<span class="w">            </span><span class="k">fun</span><span class="w"> </span><span class="n">project</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="p">:</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">option</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">NONE</span><span class="p">;</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">!r</span><span class="p">)</span><span class="w"></span>
<span class="w">         </span><span class="k">in</span><span class="w"></span>
<span class="w">            </span><span class="p">(</span><span class="n">inject</span><span class="p">,</span><span class="w"> </span><span class="n">project</span><span class="p">)</span><span class="w"></span>
<span class="w">         </span><span class="k">end</span><span class="w"></span>
<span class="w">   </span><span class="k">end</span><span class="w"></span>
</pre></div></div></div>
<div class="paragraph"><p>While correct, this approach keeps the contents of the ref cell alive
longer than necessary, which could cause a space leak.  The problem is
in <span class="monospaced">project</span>, where the call to <span class="monospaced">f</span> stores some value in some ref cell
<span class="monospaced">r'</span>.  Perhaps <span class="monospaced">r'</span> is the same ref cell as <span class="monospaced">r</span>, but perhaps not.  If
we do not clear <span class="monospaced">r'</span> before returning from <span class="monospaced">project</span>, then <span class="monospaced">r'</span> will
keep the value alive, even though it is useless.</p></div>
</div>
</div>
<div class="sect1">
<h2 id="_also_see">Also see</h2>
<div class="sectionbody">
<div class="ulist"><ul>
<li>
<p>
<a href="PropertyList">PropertyList</a>: Lisp-style property lists implemented with a universal type
</p>
</li>
</ul></div>
</div>
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