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VariableArityPolymorphism
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<a href="StandardML">Standard ML</a> programmers often face the problem of how to provide a variable-arity polymorphic function. For example, suppose one is defining a combinator library, e.g. for parsing or pickling. The signature for such a library might look something like the following.
<pre class=code>
<B><FONT COLOR="#0000FF">signature</FONT></B> COMBINATOR =
<B><FONT COLOR="#0000FF">sig</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> 'a t
</FONT></B><B><FONT COLOR="#A020F0">val</FONT></B> int: int t
<B><FONT COLOR="#A020F0">val</FONT></B> real: real t
<B><FONT COLOR="#A020F0">val</FONT></B> string: string t
<B><FONT COLOR="#A020F0">val</FONT></B> unit: unit t
<B><FONT COLOR="#A020F0">val</FONT></B> tuple2: 'a1 t * 'a2 t -> ('a1 * 'a2) t
<B><FONT COLOR="#A020F0">val</FONT></B> tuple3: 'a1 t * 'a2 t * 'a3 t -> ('a1 * 'a2 * 'a3) t
<B><FONT COLOR="#A020F0">val</FONT></B> tuple4: 'a1 t * 'a2 t * 'a3 t * 'a4 t
-> ('a1 * 'a2 * 'a3 * 'a4) t
...
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
The question is how to define a variable-arity tuple combinator. Traditionally, the only way to take a variable number of arguments in SML is to put the arguments in a list (or vector) and pass that. So, one might define a tuple combinator with the following signature.
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> tupleN: 'a list -> 'a list t
</PRE>
<p>
</p>
<p>
The problem with this approach is that as soon as one places values in a list, they must all have the same type. So, programmers often take an alternative approach, and define a family of <tt>tuple<N></tt> functions, as we see in the <tt>COMBINATOR</tt> signature above.
</p>
<p>
The family-of-functions approach is ugly for many reasons. First, it clutters the signature with a number of functions when there should really only be one. Second, it is <em>closed</em>, in that there are a fixed number of tuple combinators in the interface, and should a client need a combinator for a large tuple, he is out of luck. Third, this approach often requires a lot of duplicate code in the implementation of the combinators.
</p>
<p>
Fortunately, using <a href="Fold01N">Fold01N</a> and <a href="ProductType">products</a>, one can provide an interface and implementation that solves all these problems. Here is a simple pickling module that converts values to strings.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">structure</FONT></B> Pickler =
<B><FONT COLOR="#0000FF">struct</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> 'a t </FONT></B>=<B><FONT COLOR="#228B22"> 'a -> string
</FONT></B><B><FONT COLOR="#A020F0">val</FONT></B> unit = <B><FONT COLOR="#A020F0">fn</FONT></B> () => <B><FONT COLOR="#BC8F8F">""</FONT></B>
<B><FONT COLOR="#A020F0">val</FONT></B> int = Int.toString
<B><FONT COLOR="#A020F0">val</FONT></B> real = Real.toString
<B><FONT COLOR="#A020F0">val</FONT></B> string = id
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> 'a accum </FONT></B>=<B><FONT COLOR="#228B22"> 'a * string list -> string list
</FONT></B><B><FONT COLOR="#A020F0">val</FONT></B> tuple =
<B><FONT COLOR="#A020F0">fn</FONT></B> z =>
Fold01N.fold
{finish = <B><FONT COLOR="#A020F0">fn</FONT></B> ps => <B><FONT COLOR="#A020F0">fn</FONT></B> x => concat (rev (ps (x, []))),
start = <B><FONT COLOR="#A020F0">fn</FONT></B> p => <B><FONT COLOR="#A020F0">fn</FONT></B> (x, l) => p x :: l,
zero = unit}
z
<B><FONT COLOR="#A020F0">val</FONT></B> ` =
<B><FONT COLOR="#A020F0">fn</FONT></B> z =>
Fold01N.step1
{combine = (<B><FONT COLOR="#A020F0">fn</FONT></B> (p, p') => <B><FONT COLOR="#A020F0">fn</FONT></B> (x & x', l) => p' x' :: <B><FONT COLOR="#BC8F8F">","</FONT></B> :: p (x, l))}
z
<B><FONT COLOR="#0000FF">end</FONT></B>
</PRE>
<p>
</p>
<p>
If one has <tt>n</tt> picklers of types
</p>
<pre class=code>
<B><FONT COLOR="#A020F0">val</FONT></B> p1: a1 Pickler.t
<B><FONT COLOR="#A020F0">val</FONT></B> p2: a2 Pickler.t
...
<B><FONT COLOR="#A020F0">val</FONT></B> pn: an Pickler.t
</PRE>
<p>
</p>
<p>
then one can construct a pickler for n-ary products as follows.
</p>
<pre class=code>
tuple `p1 `p2 ... `pn $ : (a1 & a2 & ... & an) Pickler.t
</PRE>
<p>
</p>
<p>
For example, with <tt>Pickler</tt> in scope, one can prove the following equations.
</p>
<pre class=code>
<B><FONT COLOR="#BC8F8F">""</FONT></B> = tuple $ ()
<B><FONT COLOR="#BC8F8F">"1"</FONT></B> = tuple `int $ <B><FONT COLOR="#5F9EA0">1</FONT></B>
<B><FONT COLOR="#BC8F8F">"1,2.0"</FONT></B> = tuple `int `real $ (<B><FONT COLOR="#5F9EA0">1</FONT></B> & <B><FONT COLOR="#5F9EA0">2.0</FONT></B>)
<B><FONT COLOR="#BC8F8F">"1,2.0,three"</FONT></B> = tuple `int `real `string $ (<B><FONT COLOR="#5F9EA0">1</FONT></B> & <B><FONT COLOR="#5F9EA0">2.0</FONT></B> & <B><FONT COLOR="#BC8F8F">"three"</FONT></B>)
</PRE>
<p>
</p>
<p>
Here is the signature for <tt>Pickler</tt>. It shows why the <tt>accum</tt> type is useful.
</p>
<pre class=code>
<B><FONT COLOR="#0000FF">signature</FONT></B> PICKLER =
<B><FONT COLOR="#0000FF">sig</FONT></B>
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> 'a t
</FONT></B><B><FONT COLOR="#A020F0">val</FONT></B> int: int t
<B><FONT COLOR="#A020F0">val</FONT></B> real: real t
<B><FONT COLOR="#A020F0">val</FONT></B> string: string t
<B><FONT COLOR="#A020F0">val</FONT></B> unit: unit t
<B><FONT COLOR="#A020F0">type</FONT></B><B><FONT COLOR="#228B22"> 'a accum
</FONT></B><B><FONT COLOR="#A020F0">val</FONT></B> ` : ('a accum, 'b t, ('a, 'b) prod accum,
'z1, 'z2, 'z3, 'z4, 'z5, 'z6, 'z7) Fold01N.step1
<B><FONT COLOR="#A020F0">val</FONT></B> tuple: ('a t, 'a accum, 'b accum, 'b t, unit t,
'z1, 'z2, 'z3, 'z4, 'z5) Fold01N.t
<B><FONT COLOR="#0000FF">end</FONT></B>
<B><FONT COLOR="#0000FF">structure</FONT></B> Pickler: PICKLER = Pickler
</PRE>
<p>
</p>
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<p>
<hr>
Last edited on 2006-03-21 22:06:02 by <span title="adsl-71-141-16-94.dsl.snfc21.sbcglobal.net"><a href="StephenWeeks">StephenWeeks</a></span>.
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