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Simon Forman
2018-06-06 11:20:31 -07:00
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<p>Consider the <code>x</code> combinator:</p>
<pre><code>x == dup i
<pre><code>x == dup i</code></pre>
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<p>We can apply it to a quoted program consisting of some value <code>a</code> and a function <code>B</code>:</p>
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<p>We can apply it to a quoted program consisting of some value <code>a</code> and some function <code>B</code>:</p>
<pre><code>[a B] x
[a B] a B</code></pre>
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<p>Putting it together, this is the definition of <code>B</code>:</p>
<p>Altogether, this is the definition of <code>B</code>:</p>
<pre><code>B == swap [C] dip rest cons</code></pre>
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<p>We can create a quoted program that generates the Natural numbers (0, 1, 2, ...) by using <code>0</code> for <code>a</code> and <code>[dup ++]</code> for <code>[C]</code>:</p>
<p>We can make a generator for the Natural numbers (0, 1, 2, ...) by using <code>0</code> for <code>a</code> and <code>[dup ++]</code> for <code>[C]</code>:</p>
<pre><code>[0 swap [dup ++] dip rest cons]
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<h3 id="direco"><code>direco</code><a class="anchor-link" href="#direco">&#182;</a></h3>
<h2 id="direco"><code>direco</code><a class="anchor-link" href="#direco">&#182;</a></h2>
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<h1 id="Making-Generators">Making Generators<a class="anchor-link" href="#Making-Generators">&#182;</a></h1><p>We want to define a function that accepts <code>a</code> and <code>[C]</code> and builds our quoted program:</p>
<h2 id="Making-Generators">Making Generators<a class="anchor-link" href="#Making-Generators">&#182;</a></h2><p>We want to define a function that accepts <code>a</code> and <code>[C]</code> and builds our quoted program:</p>
<pre><code> a [C] G
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<h1 id="Generating-Multiples-of-Three-and-Five">Generating Multiples of Three and Five<a class="anchor-link" href="#Generating-Multiples-of-Three-and-Five">&#182;</a></h1><p>Look at the treatment of the Project Euler Problem One in <a href="./Developing a Program.ipynb">Developing a Program.ipynb</a> and you'll see that we might be interested in generating an endless cycle of:</p>
<h2 id="Generating-Multiples-of-Three-and-Five">Generating Multiples of Three and Five<a class="anchor-link" href="#Generating-Multiples-of-Three-and-Five">&#182;</a></h2><p>Look at the treatment of the Project Euler Problem One in <a href="./Developing a Program.ipynb">Developing a Program.ipynb</a> and you'll see that we might be interested in generating an endless cycle of:</p>
<pre><code>3 2 1 3 1 2 3
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<h1 id="Project-Euler-Problem-One">Project Euler Problem One<a class="anchor-link" href="#Project-Euler-Problem-One">&#182;</a></h1>
<h2 id="Project-Euler-Problem-One">Project Euler Problem One<a class="anchor-link" href="#Project-Euler-Problem-One">&#182;</a></h2>
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<h1 id="A-generator-for-the-Fibonacci-Sequence.">A generator for the Fibonacci Sequence.<a class="anchor-link" href="#A-generator-for-the-Fibonacci-Sequence.">&#182;</a></h1><p>Consider:</p>
<h2 id="A-generator-for-the-Fibonacci-Sequence.">A generator for the Fibonacci Sequence.<a class="anchor-link" href="#A-generator-for-the-Fibonacci-Sequence.">&#182;</a></h2><p>Consider:</p>
<pre><code>[b a F] x
[b a F] b a F</code></pre>
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<h3 id="Project-Euler-Problem-Two">Project Euler Problem Two<a class="anchor-link" href="#Project-Euler-Problem-Two">&#182;</a></h3>
<h2 id="Project-Euler-Problem-Two">Project Euler Problem Two<a class="anchor-link" href="#Project-Euler-Problem-Two">&#182;</a></h2>
<pre><code>By considering the terms in the Fibonacci sequence whose values do not exceed four million,
find the sum of the even-valued terms.
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<h1 id="How-to-compile-these?">How to compile these?<a class="anchor-link" href="#How-to-compile-these?">&#182;</a></h1><p>You would probably start with a special version of <code>G</code>, and perhaps modifications to the default <code>x</code>?</p>
<h2 id="How-to-compile-these?">How to compile these?<a class="anchor-link" href="#How-to-compile-these?">&#182;</a></h2><p>You would probably start with a special version of <code>G</code>, and perhaps modifications to the default <code>x</code>?</p>
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<h1 id="An-Interesting-Variation">An Interesting Variation<a class="anchor-link" href="#An-Interesting-Variation">&#182;</a></h1>
<h2 id="An-Interesting-Variation">An Interesting Variation<a class="anchor-link" href="#An-Interesting-Variation">&#182;</a></h2>
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