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UniversalType
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A universal type is a type into which all other types can be embedded. Here's a <a href="StandardML">Standard ML</a> signature for a universal type.
<pre class=code>
<B><FONT COLOR="#0000FF">signature</FONT></B> UNIVERSAL_TYPE =
<B><FONT COLOR="#0000FF">sig</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> t
</FONT></B><B><FONT COLOR="#A020F0">val</FONT></B> embed: unit -> ('a -> t) * (t -> 'a option)
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
The idea is that <tt>type t</tt> is the universal type and that each call to <tt>embed</tt> returns a new pair of functions <tt>(inject, project)</tt>, where <tt>inject</tt> embeds a value into the universal type and <tt>project</tt> extracts the value from the universal type. A pair <tt>(inject, project)</tt> returned by <tt>embed</tt> works together in that <tt>project u</tt> will return <tt>SOME v</tt> if and only if <tt>u</tt> was created by <tt>inject v</tt>. If <tt>u</tt> was created by a different function <tt>inject'</tt>, then <tt>project</tt> returns <tt>NONE</tt>.
</p>
<p>
Here's an example embedding integers and reals into a universal type.
<pre class=code>
<B><FONT COLOR="#0000FF">functor</FONT></B> Test (U: UNIVERSAL_TYPE): <B><FONT COLOR="#0000FF">sig</FONT></B> <B><FONT COLOR="#0000FF">end</FONT></B> =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> (intIn: int -> U.t, intOut) = U.embed ()
<B><FONT COLOR="#A020F0">val</FONT></B> r: U.t ref = ref (intIn <B><FONT COLOR="#5F9EA0">13</FONT></B>)
<B><FONT COLOR="#A020F0">val</FONT></B> s1 =
<B><FONT COLOR="#A020F0">case</FONT></B> intOut (!r) <B><FONT COLOR="#A020F0">of</FONT></B>
NONE => <B><FONT COLOR="#BC8F8F">"NONE"</FONT></B>
| SOME i => Int.toString i
<B><FONT COLOR="#A020F0">val</FONT></B> (realIn: real -> U.t, realOut) = U.embed ()
<B><FONT COLOR="#A020F0">val</FONT></B> () = r := realIn <B><FONT COLOR="#5F9EA0">13.0</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> s2 =
<B><FONT COLOR="#A020F0">case</FONT></B> intOut (!r) <B><FONT COLOR="#A020F0">of</FONT></B>
NONE => <B><FONT COLOR="#BC8F8F">"NONE"</FONT></B>
| SOME i => Int.toString i
<B><FONT COLOR="#A020F0">val</FONT></B> s3 =
<B><FONT COLOR="#A020F0">case</FONT></B> realOut (!r) <B><FONT COLOR="#A020F0">of</FONT></B>
NONE => <B><FONT COLOR="#BC8F8F">"NONE"</FONT></B>
| SOME x => Real.toString x
<B><FONT COLOR="#A020F0">val</FONT></B> () = print (concat [s1, <B><FONT COLOR="#BC8F8F">" "</FONT></B>, s2, <B><FONT COLOR="#BC8F8F">" "</FONT></B>, s3, <B><FONT COLOR="#BC8F8F">"\n"</FONT></B>])
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
</p>
<p>
Applying <tt>Test</tt> to an appropriate implementation will print
<pre>13 NONE 13.0
</pre>
</p>
<p>
Note that two different calls to embed on the same type return different embeddings.
</p>
<p>
Standard ML does not have explicit support for universal types; however, there are at least two ways to implement them.
</p>
<h2 id="head-5ab5e9f7a9267673649cd34bf0a7cf4b914d2288">Implementation Using Exceptions</h2>
<p>
While the intended use of SML exceptions is for exception handling, an accidental feature of their design is that the <tt>exn</tt> 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>
<pre class=code>
<B><FONT COLOR="#0000FF">structure</FONT></B> U:> UNIVERSAL_TYPE =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> t </FONT></B>=<B><FONT COLOR="#228B22"> exn
</FONT></B><B><FONT COLOR="#A020F0">fun</FONT></B> <B><FONT COLOR="#228B22">'a</FONT></B> embed () =
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">exception</FONT></B><B><FONT COLOR="#228B22"> <FONT COLOR="#B8860B">E</FONT> <B><FONT COLOR="#A020F0">of</FONT></B> 'a
</FONT></B><B><FONT COLOR="#A020F0">fun</FONT></B> project (e: t): 'a option =
<B><FONT COLOR="#A020F0">case</FONT></B> e <B><FONT COLOR="#A020F0">of</FONT></B>
E a => SOME a
| _ => NONE
<B><FONT COLOR="#A020F0">in</FONT></B>
(E, project)
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<h2 id="head-ce424b82430977939b90a4c1cb21f7a8e6f6301c">Implementation Using Functions and References</h2>
<pre class=code>
<B><FONT COLOR="#0000FF">structure</FONT></B> U:> UNIVERSAL_TYPE =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">datatype</FONT></B><B><FONT COLOR="#228B22"> t </FONT></B>=<B><FONT COLOR="#228B22"> <FONT COLOR="#B8860B">T</FONT> <B><FONT COLOR="#A020F0">of</FONT></B> {clear: unit -> unit,
store: unit -> unit}
</FONT></B><B><FONT COLOR="#A020F0">fun</FONT></B> <B><FONT COLOR="#228B22">'a</FONT></B> embed () =
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> r: 'a option ref = ref NONE
<B><FONT COLOR="#A020F0">fun</FONT></B> inject (a: 'a): t =
T {clear = <B><FONT COLOR="#A020F0">fn</FONT></B> () => r := NONE,
store = <B><FONT COLOR="#A020F0">fn</FONT></B> () => r := SOME a}
<B><FONT COLOR="#A020F0">fun</FONT></B> project (T {clear, store}): 'a option =
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> () = store ()
<B><FONT COLOR="#A020F0">val</FONT></B> res = !r
<B><FONT COLOR="#A020F0">val</FONT></B> () = clear ()
<B><FONT COLOR="#A020F0">in</FONT></B>
res
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#A020F0">in</FONT></B>
(inject, project)
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
Note that due to the use of a shared ref cell, the above implementation is not thread safe.
</p>
<p>
One could try to simplify the above implementation by eliminating the <tt>clear</tt> function, making <tt>type t = unit -> unit</tt>.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">structure</FONT></B> U:> UNIVERSAL_TYPE =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> t </FONT></B>=<B><FONT COLOR="#228B22"> unit -> unit
</FONT></B><B><FONT COLOR="#A020F0">fun</FONT></B> <B><FONT COLOR="#228B22">'a</FONT></B> embed () =
<B><FONT COLOR="#A020F0">let</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> r: 'a option ref = ref NONE
<B><FONT COLOR="#A020F0">fun</FONT></B> inject (a: 'a): t = <B><FONT COLOR="#A020F0">fn</FONT></B> () => r := SOME a
<B><FONT COLOR="#A020F0">fun</FONT></B> project (f: t): 'a option = (r := NONE; f (); !r)
<B><FONT COLOR="#A020F0">in</FONT></B>
(inject, project)
<B><FONT COLOR="#A020F0">end</FONT></B>
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<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 <tt>project</tt>, where the call to <tt>f</tt> stores some value in some ref cell <tt>r'</tt>. Perhaps <tt>r'</tt> is the same ref cell as <tt>r</tt>, but perhaps not. If we do not clear <tt>r'</tt> before returning from <tt>project</tt>, then <tt>r'</tt> will keep the value alive, even though it is useless.
</p>
<h2 id="head-a4bc8bf5caf54b18cea9f58e83dd4acb488deb17">Also see</h2>
<ul>
<li>
<p>
<a href="PropertyList">PropertyList</a>: Lisp-style property lists implemented with a universal type.
</p>
</li>
</ul>
</div>
<p>
<hr>
Last edited on 2005-05-29 03:04:34 by <span title="cs78147114.pp.htv.fi"><a href="VesaKarvonen">VesaKarvonen</a></span>.
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