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<a href="StandardML">Standard ML</a> has special syntax for products (tuples). A product type is written as
<pre>t1 * t2 * ... * tN
</pre> and a product pattern is written as
<pre>(p1, p2, ..., pN)
</pre><p>
In most situations the syntax is quite convenient. However, there are situations where the syntax is cumbersome. There are also situations in which it is useful to construct and destruct n-ary products inductively, especially when using <a href="Fold">Fold</a>.
</p>
<p>
In such situations, it is useful to have a binary product datatype with an infix constructor defined as follows.
<pre class=code>
<B><FONT COLOR="#A020F0">datatype</FONT></B><B><FONT COLOR="#228B22"> ('a, 'b) product </FONT></B>=<B><FONT COLOR="#228B22"> <FONT COLOR="#B8860B">&</FONT> <B><FONT COLOR="#A020F0">of</FONT></B> 'a * 'b
</FONT></B><B><FONT COLOR="#A020F0">infix</FONT></B> &
</PRE>
</p>
<p>
With these definitions, one can write an n-ary product as a nested binary product quite conveniently.
<pre>x1 & x2 & ... & xn
</pre>
</p>
<p>
Because of left associativity, this is the same as
<pre>(((x1 & x2) & ...) & xn)
</pre>
</p>
<p>
Because <tt>&</tt> is a constructor, the syntax can also be used for patterns.
</p>
<p>
The symbol <tt>&</tt> is inspired by the Curry-Howard isomorphism: the proof of a conjunction <tt>(A & B)</tt> is a pair of proofs <tt>(a, b)</tt>.
</p>
<h2 id="head-058073812447f89d85b57e47ba6958fee7a17440">Example: parser combinators</h2>
<p>
A typical parser combinator library provides a combinator that has a type of the form.
</p>
<pre>'a parser * 'b parser -> ('a * 'b) parser
</pre><p>
and produces a parser for the concatenation of two parsers. When more than two parsers are concatenated, the result of the resulting parser is a nested structure of pairs
</p>
<pre>(...((p1, p2), p3)..., pN)
</pre><p>
which is somewhat cumbersome.
</p>
<p>
By using a product type, the type of the concatenation combinator then becomes
</p>
<pre>'a parser * 'b parser -> ('a, 'b) product parser
</pre><p>
While this doesn't stop the nesting, it makes the pattern significantly easier to write. Instead of
</p>
<pre>(...((p1, p2), p3)..., pN)
</pre>the pattern is written as
<pre>p1 & p2 & p3 & ... & pN
</pre>which is considerably more concise. <h2 id="head-a4bc8bf5caf54b18cea9f58e83dd4acb488deb17">Also see</h2>
<ul>
<li>
<p>
<a href="VariableArityPolymorphism">VariableArityPolymorphism</a>
</p>
</li>
<li>
<p>
<a href="Utilities">Utilities</a>
</p>
</li>
</ul>
</div>
<p>
<hr>
Last edited on 2007-08-26 19:59:19 by <span title="c-71-57-91-146.hsd1.il.comcast.net"><a href="MatthewFluet">MatthewFluet</a></span>.
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