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Simon Forman
2018-06-05 17:41:22 -07:00
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<p>Consider the <code>x</code> combinator <code>x == dup i</code>:</p>
<p>Consider the <code>x</code> combinator:</p>
<pre><code>[a B] x
[a B] a B
<pre><code>x == dup i
</code></pre>
<p>Let <code>B</code> <code>swap</code> the <code>a</code> with the quote and run some function <code>[C]</code> on it.</p>
<p>We can apply it to a quoted program consisting of some value <code>a</code> and a function <code>B</code>:</p>
<pre><code>[a B] a B
<pre><code>[a B] x
[a B] a B</code></pre>
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<p>Let <code>B</code> function <code>swap</code> the <code>a</code> with the quote and run some function <code>C</code> on it to generate a new value <code>b</code>:</p>
<pre><code>B == swap [C] dip
[a B] a B
[a B] a swap [C] dip
a [a B] [C] dip
a C [a B]
b [a B]</code></pre>
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<p>Now discard the quoted <code>a</code> with <code>rest</code> and <code>cons</code> the result of <code>C</code> on <code>a</code> whatever that is:</p>
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<p>Now discard the quoted <code>a</code> with <code>rest</code> then <code>cons</code> <code>b</code>:</p>
<pre><code>aC [a B] rest cons
aC [B] cons
[aC B]
<pre><code>b [a B] rest cons
b [B] cons
[b B]</code></pre>
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<p>Altogether, this is the definition of <code>B</code>:</p>
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<p>Putting it together, this is the definition of <code>B</code>:</p>
<pre><code>B == swap [C] dip rest cons
<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 (integers 0, 1, 2, ...) by using <code>0</code> for <code>a</code> and <code>[dup ++]</code> for <code>[C]</code>:</p>
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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>
<pre><code>[0 swap [dup ++] dip rest cons]
@@ -11983,16 +12015,19 @@ aC [B] cons
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<h1 id="Generating-Generators">Generating Generators<a class="anchor-link" href="#Generating-Generators">&#182;</a></h1><p>We want to go from:</p>
<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>
<pre><code>a [C] G
<pre><code> a [C] G
-------------------------
[a swap [C] direco]</code></pre>
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<p>to:</p>
<pre><code>[a swap [C] direco]
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<p>Working in reverse:</p>
<pre><code>[a swap [C] direco] cons
@@ -12005,12 +12040,7 @@ a [C] [direco] cons [swap]
<p>Reading from the bottom up:</p>
<pre><code>G == [direco] cons [swap] swap concat cons
G == [direco] cons [swap] swoncat cons
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<p>We can try it out:</p>
<pre><code>0 [dup ++] G</code></pre>
G == [direco] cons [swap] swoncat cons</code></pre>
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<p>Let's try it out:</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;0 [dup ++] G&#39;</span><span class="p">)</span>
<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;0 [dup ++] G&#39;</span><span class="p">)</span>
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<pre> . 0 [dup ++] G
0 . [dup ++] G
0 [dup ++] . G
0 [dup ++] . [direco] cons [swap] swoncat cons
0 [dup ++] [direco] . cons [swap] swoncat cons
0 [[dup ++] direco] . [swap] swoncat cons
0 [[dup ++] direco] [swap] . swoncat cons
0 [[dup ++] direco] [swap] . swap concat cons
0 [swap] [[dup ++] direco] . concat cons
0 [swap [dup ++] direco] . cons
[0 swap [dup ++] direco] .
<pre>[0 swap [dup ++] direco]
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;0 [dup ++] G x&#39;</span><span class="p">)</span>
<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;0 [dup ++] G x x x pop&#39;</span><span class="p">)</span>
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<pre> . 0 [dup ++] G x
0 . [dup ++] G x
0 [dup ++] . G x
0 [dup ++] . [direco] cons [swap] swoncat cons x
0 [dup ++] [direco] . cons [swap] swoncat cons x
0 [[dup ++] direco] . [swap] swoncat cons x
0 [[dup ++] direco] [swap] . swoncat cons x
0 [[dup ++] direco] [swap] . swap concat cons x
0 [swap] [[dup ++] direco] . concat cons x
0 [swap [dup ++] direco] . cons x
[0 swap [dup ++] direco] . x
[0 swap [dup ++] direco] . 0 swap [dup ++] direco
[0 swap [dup ++] direco] 0 . swap [dup ++] direco
0 [0 swap [dup ++] direco] . [dup ++] direco
0 [0 swap [dup ++] direco] [dup ++] . direco
0 [0 swap [dup ++] direco] [dup ++] . dip rest cons
0 . dup ++ [0 swap [dup ++] direco] rest cons
0 0 . ++ [0 swap [dup ++] direco] rest cons
0 1 . [0 swap [dup ++] direco] rest cons
0 1 [0 swap [dup ++] direco] . rest cons
0 1 [swap [dup ++] direco] . cons
0 [1 swap [dup ++] direco] .
<pre>0 1 2
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;1 [dup 1 &lt;&lt;] G x x x x x x x x x&#39;</span><span class="p">)</span>
<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;1 [dup 1 &lt;&lt;] G x x x x x x x x x pop&#39;</span><span class="p">)</span>
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<pre>1 2 4 8 16 32 64 128 256 [512 swap [dup 1 &lt;&lt;] direco]
<pre>1 2 4 8 16 32 64 128 256
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<h1 id="n-[x]-times"><code>n [x] times</code><a class="anchor-link" href="#n-[x]-times">&#182;</a></h1><p>If we have one of these quoted programs we can drive it using <code>times</code> with the <code>x</code> combinator.</p>
<p>Let's define a word <code>n_range</code> that takes a starting integer and a count and leaves that many consecutive integers on the stack. For example:</p>
<h3 id="[x]-times"><code>[x] times</code><a class="anchor-link" href="#[x]-times">&#182;</a></h3><p>If we have one of these quoted programs we can drive it using <code>times</code> with the <code>x</code> combinator.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;23 [dup ++] G 5 [x] times pop&#39;</span><span class="p">)</span>
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<pre>23 24 25 26 27
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<p>We can use <code>dip</code> to untangle <code>[dup ++] G</code> from the arguments.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;23 5 [[dup ++] G] dip [x] times pop&#39;</span><span class="p">)</span>
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<pre>23 24 25 26 27
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<p>Now that the givens (arguments) are on the left we have the definition we're looking for:</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;n_range == [[dup ++] G] dip [x] times pop&#39;</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;450 10 n_range&#39;</span><span class="p">)</span>
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<pre>450 451 452 453 454 455 456 457 458 459
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<p>This is better just using the <code>times</code> combinator though...</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;450 9 [dup ++] times&#39;</span><span class="p">)</span>
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<pre>450 451 452 453 454 455 456 457 458 459
<pre>23 24 25 26 27 [28 swap [dup ++] direco]
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;PE1.1 == dup [3 &amp;] dip 2 &gt;&gt;&#39;</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;14811 PE1.1&#39;</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;14811 [PE1.1] G&#39;</span><span class="p">)</span>
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<pre>3 [3702 swap [PE1.1] direco]
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;PE1.1.check == dup [pop 14811] [] branch&#39;</span><span class="p">)</span>
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<pre>[14811 swap [PE1.1.check PE1.1] direco]
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<p>(It would be more efficient to reset the int every seven cycles but that's a little beyond the scope of this article. This solution does extra work, but not much, and we're not using it "in production" as they say.)</p>
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<pre>3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 [57 swap [PE1.1.check PE1.1] dip rest cons]
<pre>3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 1 2 3 3 2 1 3 [57 swap [PE1.1.check PE1.1] direco]
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@@ -12654,10 +12537,10 @@ G == [direco] cons [swap] swoncat cons
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;[14811 swap [PE1.1.check PE1.1] dip rest cons] 466 [x] times pop enstacken sum&#39;</span><span class="p">)</span>
<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;[14811 swap [PE1.1.check PE1.1] direco] 466 [x] times pop enstacken sum&#39;</span><span class="p">)</span>
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@@ -12693,7 +12576,7 @@ G == [direco] cons [swap] swoncat cons
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;PE1.2 == + dup [+] dip&#39;</span><span class="p">)</span>
@@ -12708,54 +12591,14 @@ G == [direco] cons [swap] swoncat cons
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<p>Now we can add <code>PE1.2</code> to the quoted program given to <code>times</code>.</p>
<p>Now we can add <code>PE1.2</code> to the quoted program given to <code>G</code>.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;0 0 [0 swap [PE1.1.check PE1.1] direco] 466 [x [PE1.2] dip] times popop&#39;</span><span class="p">)</span>
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<pre>233168
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<p>Or using <code>G</code> we can write:</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;0 0 0 [PE1.1.check PE1.1] G 466 [x [PE1.2] dip] times popop&#39;</span><span class="p">)</span>
@@ -12791,73 +12634,135 @@ G == [direco] cons [swap] swoncat cons
<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>
<pre><code>[b a F] x
[b a F] b a F
[b a F] b a F</code></pre>
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<p>The obvious first thing to do is just add <code>b</code> and <code>a</code>:</p>
<pre><code>[b a F] b a +
[b a F] b+a
[b a F] b+a</code></pre>
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<p>From here we want to arrive at:</p>
<pre><code>b [b+a b F]
<pre><code>b [b+a b F]</code></pre>
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<p>Let's start with <code>swons</code>:</p>
<pre><code>[b a F] b+a swons
[b+a b a F]
[b+a b a F]</code></pre>
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<p>Considering this quote as a stack:</p>
<pre><code>F a b b+a
<pre><code>F a b b+a</code></pre>
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<p>We want to get it to:</p>
<pre><code>F b b+a b
<pre><code>F b b+a b</code></pre>
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<p>So:</p>
<pre><code>F a b b+a popdd over
F b b+a b
F b b+a b</code></pre>
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<p>And therefore:</p>
<pre><code>[b+a b a F] [popdd over] infra
[b b+a b F]
[b b+a b F]</code></pre>
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<p>And lastly:</p>
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<p>But we can just use <code>cons</code> to carry <code>b+a</code> into the quote:</p>
<pre><code>[b a F] b+a [popdd over] cons infra
[b a F] [b+a popdd over] infra
[b b+a b F]</code></pre>
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<p>Lastly:</p>
<pre><code>[b b+a b F] uncons
b [b+a b F]
b [b+a b F]</code></pre>
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<p>Done.</p>
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<p>Putting it all together:</p>
<pre><code>F == + swons [popdd over] infra uncons
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<p>And:</p>
<pre><code>fib_gen == [1 1 F]</code></pre>
<pre><code>F == + [popdd over] cons infra uncons
fib_gen == [1 1 F]</code></pre>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;fib == + swons [popdd over] infra uncons&#39;</span><span class="p">)</span>
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@@ -12867,7 +12772,7 @@ b [b+a b F]
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@@ -12880,7 +12785,7 @@ b [b+a b F]
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@@ -12914,7 +12819,8 @@ b [b+a b F]
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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>
<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.
<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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<p>Now that we have a generator for the Fibonacci sequence, we need a function that adds a term in the sequence to a sum if it is even, and <code>pop</code>s it otherwise.</p>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;PE2 == 0 fib_gen x [pop &gt;4M] [popop] [[PE2.1] dip x] primrec&#39;</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;PE2.2 == x x x [popop] dipd&#39;</span><span class="p">)</span>
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@@ -13213,6 +13119,115 @@ o + e = o
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<h1 id="An-Interesting-Variation">An Interesting Variation<a class="anchor-link" href="#An-Interesting-Variation">&#182;</a></h1>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">define</span><span class="p">(</span><span class="s1">&#39;codireco == cons dip rest cons&#39;</span><span class="p">)</span>
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">V</span><span class="p">(</span><span class="s1">&#39;[0 [dup ++] codireco] x&#39;</span><span class="p">)</span>
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<pre> . [0 [dup ++] codireco] x
[0 [dup ++] codireco] . x
[0 [dup ++] codireco] . 0 [dup ++] codireco
[0 [dup ++] codireco] 0 . [dup ++] codireco
[0 [dup ++] codireco] 0 [dup ++] . codireco
[0 [dup ++] codireco] 0 [dup ++] . cons dip rest cons
[0 [dup ++] codireco] [0 dup ++] . dip rest cons
. 0 dup ++ [0 [dup ++] codireco] rest cons
0 . dup ++ [0 [dup ++] codireco] rest cons
0 0 . ++ [0 [dup ++] codireco] rest cons
0 1 . [0 [dup ++] codireco] rest cons
0 1 [0 [dup ++] codireco] . rest cons
0 1 [[dup ++] codireco] . cons
0 [1 [dup ++] codireco] .
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<div class=" highlight hl-ipython2"><pre><span></span><span class="n">J</span><span class="p">(</span><span class="s1">&#39;230 [dup ++] G 5 [x] times pop&#39;</span><span class="p">)</span>
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<pre>230 231 232 233 234
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