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<a name="Reduce"></a>
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<a name="Reduce-1"></a>
<h4 class="subsubsection">2.7.13.3 Reduce</h4>

<a name="index-reduce"></a>
<p>In the virtual code fragment shown below, <code><var>f</var></code> should be
regarded as the virtual code for a binary operator, and <code><var>k</var></code>
is a constant.
</p>
<dl compact="compact">
<dt> <em>T24</em></dt>
<dd><p>[[<code>reduce</code>]] <code>(<var>f</var>,<var>k</var>)</code> = <code>((nil,nil),((<var>f</var>,<var>k</var>),nil))</code>
</p></dd>
</dl>

<p>By constructing a tree in the form shown, the <code>silly</code>
mnemonic <code>reduce</code> effectively constructs the code for a function
operating on lists in a particular way.
</p>
<p>The effect of evaluating an application in this form with an argument
representing a list can be broken down into several cases.
</p>
<ul>
<li> If the list is empty, then the value of <code><var>k</var></code> is the
result.
</li><li> If the list has only one item, then that item is the result.
</li><li> if the list has exactly two items, the first being <code><var>x</var></code> and the
second being <code><var>y</var></code>, then  the result is  <code><var>f</var>
(<var>x</var>,<var>y</var>)</code>.
</li><li> If the list has more than two items and an even number of them, the
result is that of evaluating the application with the list of values
obtained by partitioning the list into pairs of adjacent items, and
evaluating <code><var>f</var></code> with each pair.
</li><li> If the list has more than two items and an odd number of them, the
result is that of evaluating the application with the list of values
obtained by partitioning the list into pairs of adjacent items excluding
the last one, evaluating <code><var>f</var></code> with each pair, and then
appending the last item to the list of values.
</li></ul>

<p>In the last two cases, evaluation of the application is not necessarily
finished after just one traversal of the list, because it has to carry
on until the list is reduced to a single item.
</p>
<p>Some care has been taken to describe this operation in detail because it
differs from comparable operations common to functional programming
<a name="index-fold"></a>
languages, variously known as fold or reduce. All of these operations
could be used, for example, to compute the summation of a list of
numbers. The crucial differences are as follows.
</p>
<ul>
<li> Whereas a fold or a reduce is conventionally either of the left or
right variety, this <code>reduce</code> is neither.
</li><li> The vacuous case result <code><var>k</var></code> is never used at all unless
the argument is the empty list.
</li><li> This <code>reduce</code> is not very useful if the operator
<code><var>f</var></code> is not associative.
</li></ul>

<p>The reason for defining the semantics of <code>reduce</code> in this way
instead of the normal way is that a distributed implementation of this
<a name="index-distributed-implementation-1"></a>
one could work in logarithmic time, so it&rsquo;s worth making it easy for a
language processor to detect the pattern in case the virtual code is
ever going to be targeted to such an implementation. Of course, nothing
prevents the conventional left or right reduction semantics from being
translated to virtual code by explicit recursion.
<a name="index-recursion-2"></a>
</p>
<p>The precise semantics of this operation are as follows, where
<code><var>f</var></code> is not <code>nil</code>, <code><var>k</var></code> is unconstrained, and
<code>pairwise</code> represents a function to be explained presently.
<a name="index-iterate-1"></a>
<a name="index-pairwise"></a>
<a name="index-bu-1"></a>
<a name="index-right-2"></a>
</p>
<dl compact="compact">
<dt> <em>P29</em></dt>
<dd><p>([[<code>reduce</code>]] <code>(<var>f</var>,<var>k</var>)</code>) <code>nil</code> = <code><var>k</var></code>
</p></dd>
<dt> <em>P30</em></dt>
<dd><p>([[<code>reduce</code>]] <code>(<var>f</var>,<var>k</var>)</code>) <code>(<var>x</var>,<var>y</var>)</code> = <br>
   
[[<code>left</code>]] ([[<code>bu(iterate,right)</code>]] [[<code>pairwise</code>]] <code><var>f</var></code>) <code>(<var>x</var>,<var>y</var>)</code> 
</p></dd>
</dl>

<p>The latter property leverages off some virtual machine features and
<code>silly</code> code that has been defined already. The function
implemented by [[<code>pairwise</code>]] <code><var>f</var></code> is the one that
partitions its argument into pairs and evaluates <code><var>f</var></code> with
each pair as described above. The combination of that with
<code>bu(iterate,right)</code> results in an application that repeatedly
performs [[<code>pairwise</code>]] <code><var>f</var></code> while the resulting list
still has a tail (i.e., a <code>right</code> side), and stops when the list
has only a single item (i.e., when <code>right</code> is false). The
<code>left</code> operation then extracts the item.
</p>
<p>For the sake of completeness, it is tedious but straightforward to
give an exact definition for <code>pairwise</code>. The short version is that
it can be anything that satisfies these three equations.
</p>
<dl compact="compact">
<dt> <em>E1</em></dt>
<dd><p>([[<code>pairwise</code>]] <code><var>f</var></code>) <code>nil</code> = <code>nil</code>
</p></dd>
<dt> <em>E2</em></dt>
<dd><p>([[<code>pairwise</code>]] <code><var>f</var></code>) <code>(<var>x</var>,nil)</code> = <code>(<var>x</var>,nil)</code>
</p></dd>
<dt> <em>E3</em></dt>
<dd><p>([[<code>pairwise</code>]] <code><var>f</var></code>) <code>(<var>x</var>,(<var>y</var>,<var>z</var>))</code> =
<code>(<var>f</var> (<var>x</var>,<var>y</var>),</code>([[<code>pairwise</code>]] <code><var>f</var></code>) <code><var>z</var>)</code>
</p></dd>
</dl>

<p>For the long version, see <a href="Pairwise.html#Pairwise">Pairwise</a>.
</p>
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