WikiSysop at 15:57, 9 February 2024

2024-02-09T15:57:12Z

← Older revision		Revision as of 11:57, 9 February 2024
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			==Notation==

			The symbol chosen for this operator is the Del overstruck with Tilde (<apl>⍫</apl>), entered from the keyboard as Alt-’K’ or Alt-Shift-'k' (U+236B).

	==Introduction==		==Introduction==
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	P←1 0 0 1 0 1 0 0 0		P←1 0 0 1 0 1 0 0 0
	R←1 2 3 4 5 6 7 8 9		R←1 2 3 4 5 6 7 8 9
	+\R-P\¯2-\P/+\¯1↓0,R		+\R- P\ ¯2-\ P/ +\ ¯1↓0,R
	1 3 6 4 9 6 13 21 30		1 3 6 4 9 6 13 21 30
	</pre></apll>		</pre></apll>
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	<apll>f<sup>-1</sup> g<sup>-1</sup> f g</apll>		<apll>f<sup>-1</sup> g<sup>-1</sup> f g</apll>

	<p>where <apll>g ←→ +\</apll> <apll>g⍣¯1 ←→ ¯2∘(-\)</apll> <apll>f ←→ P∘/ ~~f⍣¯1~~ ←→ P∘\</apll></p>		<p>where <apll>f⍣¯1 ←→ P∘\</apll>     <apll>g⍣¯1 ←→ ¯2∘(-\)</apll>     <apll>f ←→ P∘/</apll>     <apll>g ←→ +\</apll></p>

	<p>which can be re-written as</p>		<p>which can be re-written as</p>

	<apll>+\R-P∘/⍫(+\)¯1↓0,R</apll>		<apll><pre> +\R- P∘/⍫(+\) ¯1↓0,R
			1 3 6 4 9 6 13 21 30
			</pre></apll>

			<p>For more information about inverses, see the [[Power#Inverses\|Power Operator]].

	==Implementation==		==Implementation==

WikiSysop: Created page with "

Z←{L} f⍫g R applies the function g then f then g⍣¯1 and then f⍣¯1 to or between the arguments.

<..."
2024-02-07T14:31:52Z

Created page with "<table border="1" cellpadding="5" cellspacing="0" rules="none" summary=""> <tr> <td> <table border="0" cellpadding="5" cellspacing="0" summary=""> <tr> <td valign="top"><apll>Z←{L} f⍫g R</apll></td> <td></td> <td></td> <td>applies the function <apll>g</apll> then <apll>f</apll> then <apll>g⍣¯1</apll> and then <apll>f⍣¯1</apll> to or between the arguments.</td> </tr> </table> <..."

New page
<table border="1" cellpadding="5" cellspacing="0" rules="none" summary="">
<tr>
<td>
<table border="0" cellpadding="5" cellspacing="0" summary="">
<tr>
<td valign="top"><apll>Z←{L} f⍫g R</apll></td>
<td></td>
<td></td>
<td>applies the function <apll>g</apll> then <apll>f</apll> then <apll>g⍣¯1</apll> and then <apll>f⍣¯1</apll> to or between the arguments.</td>
</tr>
</table>
</td>
</tr>
<tr>
<td><apll>L</apll> is an optional array.</td>
</tr>
<tr>
<td><apll>R</apll> is an array.</td>
</tr>
<tr>
<td><apll>f</apll> is an arbitrary ambivalent function with an inverse.</td>
</tr>
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<td><apll>g</apll> is an arbitrary monadic function with an inverse.</td>
</tr>
</table>

==Introduction==

The concept of the Commutator operator is taken from mathematical [https://en.wikipedia.org/wiki/Commutator Group Theory] from the 1830s.

For example, the algorithm for [https://dl.acm.org/doi/pdf/10.1145/800136.804488 Partitioned Plus Scan] where <apll>P</apll> is the Boolean partition vector (always with a leading 1), in modern notation looks like this:

<apll><pre>
P←1 0 0 1 0 1 0 0 0
R←1 2 3 4 5 6 7 8 9
+\R-P\¯2-\P/+\¯1↓0,R
1 3 6 4 9 6 13 21 30
</pre></apll>

An interesting observation on this one-liner is that it contains two interleaved pairs of inverses: <apll>P\</apll> and <apll>P/</apll> as one and <apll>¯2-\</apll> and <apll>+\</apll> as the other. This interleaving of inverses exactly matches the template for the Commutator Operator:

<apll>f-1 g-1 f g</apll>

where <apll>g ←→ +\</apll> <apll>g⍣¯1 ←→ ¯2∘(-\)</apll> <apll>f ←→ P∘/ f⍣¯1 ←→ P∘\</apll>

which can be re-written as

<apll>+\R-P∘/⍫(+\)¯1↓0,R</apll>

==Implementation==

This operator is implemented via the anonymous function:

<apll><pre>
{⍺←⍣0 ⋄ (⍵⍵ ⍺) ⍺⍺⍣¯1 ⍵⍵⍣¯1 (⍵⍵ ⍺) ⍺⍺ ⍵⍵ ⍵}
</pre></apll>