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authorMarshall Lochbaum <mwlochbaum@gmail.com>2020-07-17 12:04:34 -0400
committerMarshall Lochbaum <mwlochbaum@gmail.com>2020-07-17 12:26:59 -0400
commiteb01eb415a5304d98c55f844188bb0defac90c67 (patch)
treee36497edff6a231b8e0b6f4290c7e37748771284 /docs/depth.html
parent7051529b8ceefabbc8a64a3a74491a87a9651801 (diff)
Character entity escaping for "&<>
Diffstat (limited to 'docs/depth.html')
-rw-r--r--docs/depth.html30
1 files changed, 15 insertions, 15 deletions
diff --git a/docs/depth.html b/docs/depth.html
index a365212a..93234609 100644
--- a/docs/depth.html
+++ b/docs/depth.html
@@ -5,21 +5,21 @@
<p>To find the depth of an array, use Depth (<code><span class='Function'>≡</span></code>). For example, the depth of a list of numbers or characters is 1:</p>
<pre> <span class='Function'>≡</span> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span>
<span class='Number'>1</span>
- <span class='Function'>≡</span> <span class='String'>"a string is a list of characters"</span>
+ <span class='Function'>≡</span> <span class='String'>&quot;a string is a list of characters&quot;</span>
<span class='Number'>1</span>
</pre>
-<p>Depth is somewhat analogous to an array's rank <code><span class='Function'>≠≢</span><span class='Value'>𝕩</span></code>, and in fact rank can be "converted" to depth by splitting rows with <code><span class='Function'><</span><span class='Composition'>⎉</span><span class='Number'>1</span></code>, reducing the rank by 1 and increasing the depth. Unlike rank, Depth doesn't care at all about its argument's shape:</p>
-<pre> <span class='Function'>≡</span> <span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Function'>⥊</span><span class='String'>"characters"</span>
+<p>Depth is somewhat analogous to an array's rank <code><span class='Function'>≠≢</span><span class='Value'>𝕩</span></code>, and in fact rank can be &quot;converted&quot; to depth by splitting rows with <code><span class='Function'>&lt;</span><span class='Composition'>⎉</span><span class='Number'>1</span></code>, reducing the rank by 1 and increasing the depth. Unlike rank, Depth doesn't care at all about its argument's shape:</p>
+<pre> <span class='Function'>≡</span> <span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Function'>⥊</span><span class='String'>&quot;characters&quot;</span>
<span class='Number'>1</span>
- <span class='Function'>≡</span> <span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>+↕</span><span class='Number'>10</span><span class='Paren'>)</span><span class='Function'>⥊</span><span class='String'>"characters"</span>
+ <span class='Function'>≡</span> <span class='Paren'>(</span><span class='Number'>1</span><span class='Function'>+↕</span><span class='Number'>10</span><span class='Paren'>)</span><span class='Function'>⥊</span><span class='String'>&quot;characters&quot;</span>
<span class='Number'>1</span>
</pre>
<p>Also unlike rank, Depth <em>does</em> care about the elements of its argument: in fact, to find the depth of an array, every element must be inspected.</p>
<pre> <span class='Function'>≡</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Separator'>,</span><span class='Number'>5</span><span class='Bracket'>⟩</span>
<span class='Number'>1</span>
- <span class='Function'>≡</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Function'><</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Separator'>,</span><span class='Number'>5</span><span class='Bracket'>⟩</span>
+ <span class='Function'>≡</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Function'>&lt;</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Separator'>,</span><span class='Number'>5</span><span class='Bracket'>⟩</span>
<span class='Number'>2</span>
- <span class='Function'>≡</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Function'><</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Separator'>,</span><span class='Function'><<<</span><span class='Number'>5</span><span class='Bracket'>⟩</span>
+ <span class='Function'>≡</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Function'>&lt;</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Separator'>,</span><span class='Function'>&lt;&lt;&lt;</span><span class='Number'>5</span><span class='Bracket'>⟩</span>
<span class='Number'>4</span>
</pre>
<p>As the above expressions suggest, the depth of an array is the maximum of its elements, plus one. The base case, a non-array (including a function, modifier, or combinator), has depth 0.</p>
@@ -62,9 +62,9 @@
</tbody>
</table>
<p>Functions such as Take and Drop use a single number per axis. When the left argument is a list of numbers, they apply to initial axes. But for convenience, a single number is also accepted, and applied to the first axis only. This is equivalent to ravelling the left argument before applying the function.</p>
-<pre> <span class='Function'>≢</span><span class='Number'>2</span><span class='Function'>↑</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Function'>⥊</span><span class='String'>"abc"</span>
+<pre> <span class='Function'>≢</span><span class='Number'>2</span><span class='Function'>↑</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Function'>⥊</span><span class='String'>&quot;abc&quot;</span>
<span class='Value'>[</span> <span class='Number'>2</span> <span class='Number'>7</span> <span class='Number'>7</span> <span class='Number'>7</span> <span class='Value'>]</span>
- <span class='Function'>≢</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Function'>↑</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Function'>⥊</span><span class='String'>"abc"</span>
+ <span class='Function'>≢</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Function'>↑</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Ligature'>‿</span><span class='Number'>7</span><span class='Function'>⥊</span><span class='String'>&quot;abc&quot;</span>
<span class='Value'>[</span> <span class='Number'>2</span> <span class='Number'>1</span> <span class='Number'>1</span> <span class='Number'>7</span> <span class='Value'>]</span>
</pre>
<p>In these cases the flexibility seems trivial because the left argument has depth 1 or 0: it is an array or isn't, and it's obvious what a plain number should do. But for the second row in the table, the left argument is always an array. The general case is that the left argument is a vector and its elements correspond to right argument axes:</p>
@@ -75,21 +75,21 @@
<span class='Value'>┘</span>
</pre>
<p>This means the left argument is homogeneous of depth 2. What should an argument of depth 1, or an argument that contains non-arrays, do? One option is to continue to require the left argument to be a vector, and convert any non-array argument into an array by boxing it:</p>
-<pre> <span class='Bracket'>⟨</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Number'>1</span><span class='Bracket'>⟩</span> <span class='Function'><</span><span class='Composition'>⍟</span><span class='Paren'>(</span><span class='Number'>0</span><span class='Function'>=≡</span><span class='Paren'>)</span><span class='Modifier'>¨</span><span class='Composition'>⊸</span><span class='Function'>⊏</span> <span class='Function'>↕</span><span class='Number'>6</span><span class='Ligature'>‿</span><span class='Number'>7</span>
+<pre> <span class='Bracket'>⟨</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Number'>1</span><span class='Bracket'>⟩</span> <span class='Function'>&lt;</span><span class='Composition'>⍟</span><span class='Paren'>(</span><span class='Number'>0</span><span class='Function'>=≡</span><span class='Paren'>)</span><span class='Modifier'>¨</span><span class='Composition'>⊸</span><span class='Function'>⊏</span> <span class='Function'>↕</span><span class='Number'>6</span><span class='Ligature'>‿</span><span class='Number'>7</span>
<span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>3</span> <span class='Number'>1</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>2</span> <span class='Number'>1</span> <span class='Value'>]</span> <span class='Value'>]</span>
</pre>
<p>While very consistent, this extension represents a small convenience and makes it difficult to act on a single axis, which for Replicate and <a href="group.html">Group</a> is probably the most common way the primitive is used:</p>
-<pre> <span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span> <span class='Function'>/</span> <span class='String'>"abcde"</span>
+<pre> <span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span> <span class='Function'>/</span> <span class='String'>&quot;abcde&quot;</span>
<span class='Value'>[</span> <span class='Value'>aaabbcddeee</span> <span class='Value'>]</span>
</pre>
-<p>With the extension above, every case like this would have to use <code><span class='Function'><</span><span class='Composition'>⊸</span><span class='Function'>/</span></code> instead of just <code><span class='Function'>/</span></code>. BQN avoids this difficulty by testing the left argument's depth. A depth-1 argument applies to the first axis only, giving the behavior above.</p>
-<p>For Select, the depth-1 case is still quite useful, but it may also be desirable to choose a single cell using a list of numbers. In this case the left argument depth can be increased from the bottom using <code><span class='Function'><</span><span class='Modifier'>¨</span></code>.</p>
-<pre> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>4</span> <span class='Function'><</span><span class='Modifier'>¨</span><span class='Composition'>⊸</span><span class='Function'>⊏</span> <span class='Function'>↕</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>5</span><span class='Ligature'>‿</span><span class='Number'>2</span>
+<p>With the extension above, every case like this would have to use <code><span class='Function'>&lt;</span><span class='Composition'>⊸</span><span class='Function'>/</span></code> instead of just <code><span class='Function'>/</span></code>. BQN avoids this difficulty by testing the left argument's depth. A depth-1 argument applies to the first axis only, giving the behavior above.</p>
+<p>For Select, the depth-1 case is still quite useful, but it may also be desirable to choose a single cell using a list of numbers. In this case the left argument depth can be increased from the bottom using <code><span class='Function'>&lt;</span><span class='Modifier'>¨</span></code>.</p>
+<pre> <span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>1</span><span class='Ligature'>‿</span><span class='Number'>4</span> <span class='Function'>&lt;</span><span class='Modifier'>¨</span><span class='Composition'>⊸</span><span class='Function'>⊏</span> <span class='Function'>↕</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>5</span><span class='Ligature'>‿</span><span class='Number'>2</span>
<span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>2</span> <span class='Number'>1</span> <span class='Number'>4</span> <span class='Number'>0</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>2</span> <span class='Number'>1</span> <span class='Number'>4</span> <span class='Number'>1</span> <span class='Value'>]</span> <span class='Value'>]</span>
</pre>
<h2 id="the-depth-composition">The Depth composition</h2>
<p>The Depth composition (<code><span class='Composition'>⚇</span></code>) is a generalization of Each that allows diving deeper into an array. To illustrate it we'll use a shape <code><span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>3</span></code> array of lists of lists.</p>
-<pre> <span class='Function'>⊢</span> <span class='Value'>n</span> <span class='Gets'>←</span> <span class='Function'><</span><span class='Composition'>⎉</span><span class='Number'>1</span><span class='Composition'>⍟</span><span class='Number'>2</span> <span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Function'>⥊↕</span><span class='Number'>48</span>
+<pre> <span class='Function'>⊢</span> <span class='Value'>n</span> <span class='Gets'>←</span> <span class='Function'>&lt;</span><span class='Composition'>⎉</span><span class='Number'>1</span><span class='Composition'>⍟</span><span class='Number'>2</span> <span class='Number'>4</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>2</span><span class='Function'>⥊↕</span><span class='Number'>48</span>
<span class='Value'>┌</span>
<span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>0</span> <span class='Number'>1</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>2</span> <span class='Number'>3</span> <span class='Value'>]</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>4</span> <span class='Number'>5</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>6</span> <span class='Number'>7</span> <span class='Value'>]</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>8</span> <span class='Number'>9</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>10</span> <span class='Number'>11</span> <span class='Value'>]</span> <span class='Value'>]</span>
<span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>12</span> <span class='Number'>13</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>14</span> <span class='Number'>15</span> <span class='Value'>]</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>16</span> <span class='Number'>17</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>18</span> <span class='Number'>19</span> <span class='Value'>]</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Value'>[</span> <span class='Number'>20</span> <span class='Number'>21</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Number'>22</span> <span class='Number'>23</span> <span class='Value'>]</span> <span class='Value'>]</span>
@@ -125,7 +125,7 @@
<span class='Value'>┘</span>
</pre>
<p>While a negative depth tells how many levels to go down, a non-negative depth gives the maximum depth of the argument before applying the left operand. On a depth-3 array like above, depth <code><span class='Number'>2</span></code> is equivalent to <code><span class='Number'>¯1</span></code> and depth <code><span class='Number'>1</span></code> is equivalent to <code><span class='Number'>¯2</span></code>. A depth of <code><span class='Number'>0</span></code> means to loop until non-arrays are reached, that is, apply <a href="https://aplwiki.com/wiki/Pervasion">pervasively</a>, like a scalar function.</p>
-<pre> <span class='Bracket'>⟨</span><span class='String'>'a'</span><span class='Separator'>,</span><span class='String'>"bc"</span><span class='Bracket'>⟩</span> <span class='Function'>≍</span><span class='Composition'>⚇</span><span class='Number'>0</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Bracket'>⟩</span>
+<pre> <span class='Bracket'>⟨</span><span class='String'>'a'</span><span class='Separator'>,</span><span class='String'>&quot;bc&quot;</span><span class='Bracket'>⟩</span> <span class='Function'>≍</span><span class='Composition'>⚇</span><span class='Number'>0</span> <span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Ligature'>‿</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Bracket'>⟩</span>
<span class='Value'>[</span> <span class='Value'>[</span> <span class='Value'>[</span> <span class='Value'>a</span> <span class='Number'>2</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Value'>a</span> <span class='Number'>3</span> <span class='Value'>]</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Value'>[</span> <span class='Value'>b</span> <span class='Number'>4</span> <span class='Value'>]</span> <span class='Value'>[</span> <span class='Value'>c</span> <span class='Number'>4</span> <span class='Value'>]</span> <span class='Value'>]</span> <span class='Value'>]</span>
</pre>
<p>With a positive operand, Depth doesn't have to use the same depth everywhere. Here, Length is applied as soon as the depth for a particular element is 1 or less, including if the argument has depth 0. For example, it maps over <code><span class='Bracket'>⟨</span><span class='Number'>2</span><span class='Separator'>,</span><span class='Bracket'>⟨</span><span class='Number'>3</span><span class='Separator'>,</span><span class='Number'>4</span><span class='Bracket'>⟩⟩</span></code>, but not over <code><span class='Bracket'>⟨</span><span class='Number'>11</span><span class='Separator'>,</span><span class='Number'>12</span><span class='Bracket'>⟩</span></code>, even though these are elements of the same array.</p>