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Under - Revision history
2024-03-29T09:29:17Z
Revision history for this page on the wiki
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Marshall: /* History */ J9.5 has partial structural Under support
2024-02-17T13:35:21Z
<p><span dir="auto"><span class="autocomment">History: </span> J9.5 has partial structural Under support</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 13:35, 17 February 2024</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <syntaxhighlight lang=apl inline>⍢</syntaxhighlight> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <syntaxhighlight lang=apl inline>⍢</syntaxhighlight> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <syntaxhighlight lang=apl inline>⍢×</syntaxhighlight> and <syntaxhighlight lang=apl inline>⍢|</syntaxhighlight> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020. Structural Under also appears in [[Kap]] and [[Uiua]].</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <syntaxhighlight lang=apl inline>⍢×</syntaxhighlight> and <syntaxhighlight lang=apl inline>⍢|</syntaxhighlight> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020. Structural Under also appears in [[Kap]] and [[Uiua]]<ins style="font-weight: bold; text-decoration: none;">, and is implemented for a small number of cases in [[J]] version 9.5 (2023)</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td></tr>
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Marshall
https://aplwiki.com/index.php?title=Under&diff=10776&oldid=prev
Marshall: Reference Kap and Uiua
2024-01-28T21:17:51Z
<p>Reference Kap and Uiua</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:17, 28 January 2024</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l23">Line 23:</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <syntaxhighlight lang=apl inline>⍢</syntaxhighlight> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <syntaxhighlight lang=apl inline>⍢</syntaxhighlight> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <syntaxhighlight lang=apl inline>⍢×</syntaxhighlight> and <syntaxhighlight lang=apl inline>⍢|</syntaxhighlight> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <syntaxhighlight lang=apl inline>⍢×</syntaxhighlight> and <syntaxhighlight lang=apl inline>⍢|</syntaxhighlight> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020<ins style="font-weight: bold; text-decoration: none;">. Structural Under also appears in [[Kap]] and [[Uiua]]</ins>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <syntaxhighlight lang=j inline>&.</syntaxhighlight> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <syntaxhighlight lang=j inline>&.:</syntaxhighlight> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <syntaxhighlight lang=j inline>&.</syntaxhighlight> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <syntaxhighlight lang=j inline>&.:</syntaxhighlight> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://mlochbaum.github.io/BQN/doc/under.html BQN]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://mlochbaum.github.io/BQN/doc/under.html BQN]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [https://kapdemo.dhsdevelopments.com/reference.html#_structural_under Kap]</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [https://www.uiua.org/docs/under Uiua]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{APL built-ins}}[[Category:Primitive operators]][[Category:Composition operators]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{APL built-ins}}[[Category:Primitive operators]][[Category:Composition operators]]</div></td></tr>
</table>
Marshall
https://aplwiki.com/index.php?title=Under&diff=9691&oldid=prev
Adám Brudzewsky: Text replacement - "</source>" to "</syntaxhighlight>"
2022-09-11T10:50:56Z
<p>Text replacement - "</source>" to "</syntaxhighlight>"</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 10:50, 11 September 2022</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Built-in|Under|⍢}}, or '''Dual''' (<syntaxhighlight lang=apl inline>¨</<del style="font-weight: bold; text-decoration: none;">source</del>>), is a [[primitive operator|primitive]] [[dyadic operator]] which takes two [[function]] [[operand]]s. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.<ref>[https://mlochbaum.github.io/BQN/help/under.html BQN: Circle Jot (⌾)]</ref></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Built-in|Under|⍢}}, or '''Dual''' (<syntaxhighlight lang=apl inline>¨</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>>), is a [[primitive operator|primitive]] [[dyadic operator]] which takes two [[function]] [[operand]]s. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.<ref>[https://mlochbaum.github.io/BQN/help/under.html BQN: Circle Jot (⌾)]</ref></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Model ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Model ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In dialects that support user-defined operators and invertible functions, it can be defined as (or similar):</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In dialects that support user-defined operators and invertible functions, it can be defined as (or similar):</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><syntaxhighlight lang=apl> _U_ ← {⍺←{⍵ ⋄ ⍺⍺} ⋄ ⍵⍵⍣¯1⊢(⍵⍵ ⍺)⍺⍺(⍵⍵ ⍵)}</<del style="font-weight: bold; text-decoration: none;">source</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><syntaxhighlight lang=apl> _U_ ← {⍺←{⍵ ⋄ ⍺⍺} ⋄ ⍵⍵⍣¯1⊢(⍵⍵ ⍺)⍺⍺(⍵⍵ ⍵)}</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Examples ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Examples ==</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 4 3 +_U_(10⊥⌽) 5 6 4</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 4 3 +_U_(10⊥⌽) 5 6 4</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>7 0 8</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>7 0 8</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div></<del style="font-weight: bold; text-decoration: none;">source</del>></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div></<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>First, each argument is reversed, then decoded from base 10. The addition is performed. Finally, the result is encoded to base 10 and reversed.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>First, each argument is reversed, then decoded from base 10. The addition is performed. Finally, the result is encoded to base 10 and reversed.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <syntaxhighlight lang=apl inline>f¨g</<del style="font-weight: bold; text-decoration: none;">source</del>> has the overall [[function rank]] of <syntaxhighlight lang=apl inline>g</<del style="font-weight: bold; text-decoration: none;">source</del>>. This allows <syntaxhighlight lang=apl inline>¨></<del style="font-weight: bold; text-decoration: none;">source</del>> to serve the purpose of [[Each]] from a [[Nested array model|nested]] APL. J uses <syntaxhighlight lang=j inline>&.</<del style="font-weight: bold; text-decoration: none;">source</del>> for the close form and <syntaxhighlight lang=j inline>&.:</<del style="font-weight: bold; text-decoration: none;">source</del>> for the rankless form.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <syntaxhighlight lang=apl inline>f¨g</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> has the overall [[function rank]] of <syntaxhighlight lang=apl inline>g</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>>. This allows <syntaxhighlight lang=apl inline>¨></<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> to serve the purpose of [[Each]] from a [[Nested array model|nested]] APL. J uses <syntaxhighlight lang=j inline>&.</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> for the close form and <syntaxhighlight lang=j inline>&.:</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> for the rankless form.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <syntaxhighlight lang=apl inline>¨</<del style="font-weight: bold; text-decoration: none;">source</del>> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <syntaxhighlight lang=j inline>&.</<del style="font-weight: bold; text-decoration: none;">source</del>> like SHARP and a non-close form <syntaxhighlight lang=j inline>&.:</<del style="font-weight: bold; text-decoration: none;">source</del>>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <syntaxhighlight lang=apl inline>¨</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <syntaxhighlight lang=j inline>&.</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> like SHARP and a non-close form <syntaxhighlight lang=j inline>&.:</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <syntaxhighlight lang=apl inline>⍢</<del style="font-weight: bold; text-decoration: none;">source</del>> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <syntaxhighlight lang=apl inline>⍢</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <syntaxhighlight lang=apl inline>⍢×</<del style="font-weight: bold; text-decoration: none;">source</del>> and <syntaxhighlight lang=apl inline>⍢|</<del style="font-weight: bold; text-decoration: none;">source</del>> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <syntaxhighlight lang=apl inline>⍢×</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> and <syntaxhighlight lang=apl inline>⍢|</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <syntaxhighlight lang=j inline>&.</<del style="font-weight: bold; text-decoration: none;">source</del>> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <syntaxhighlight lang=j inline>&.:</<del style="font-weight: bold; text-decoration: none;">source</del>> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <syntaxhighlight lang=j inline>&.</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <syntaxhighlight lang=j inline>&.:</<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight</ins>> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://mlochbaum.github.io/BQN/doc/under.html BQN]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://mlochbaum.github.io/BQN/doc/under.html BQN]</div></td></tr>
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</table>
Adám Brudzewsky
https://aplwiki.com/index.php?title=Under&diff=9674&oldid=prev
Adám Brudzewsky: Text replacement - "<source" to "<syntaxhighlight"
2022-09-11T10:41:12Z
<p>Text replacement - "<source" to "<syntaxhighlight"</p>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Built-in|Under|⍢}}, or '''Dual''' (<<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>¨</source>), is a [[primitive operator|primitive]] [[dyadic operator]] which takes two [[function]] [[operand]]s. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.<ref>[https://mlochbaum.github.io/BQN/help/under.html BQN: Circle Jot (⌾)]</ref></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Built-in|Under|⍢}}, or '''Dual''' (<<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>¨</source>), is a [[primitive operator|primitive]] [[dyadic operator]] which takes two [[function]] [[operand]]s. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.<ref>[https://mlochbaum.github.io/BQN/help/under.html BQN: Circle Jot (⌾)]</ref></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Model ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Model ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In dialects that support user-defined operators and invertible functions, it can be defined as (or similar):</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In dialects that support user-defined operators and invertible functions, it can be defined as (or similar):</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl> _U_ ← {⍺←{⍵ ⋄ ⍺⍺} ⋄ ⍵⍵⍣¯1⊢(⍵⍵ ⍺)⍺⍺(⍵⍵ ⍵)}</source></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl> _U_ ← {⍺←{⍵ ⋄ ⍺⍺} ⋄ ⍵⍵⍣¯1⊢(⍵⍵ ⍺)⍺⍺(⍵⍵ ⍵)}</source></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Examples ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Examples ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following example solves [https://leetcode.com/problems/add-two-numbers/ a LeetCode problem] which asks to add two numbers given as reversed digit lists:</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The following example solves [https://leetcode.com/problems/add-two-numbers/ a LeetCode problem] which asks to add two numbers given as reversed digit lists:</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 4 3 +_U_(10⊥⌽) 5 6 4</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div> 2 4 3 +_U_(10⊥⌽) 5 6 4</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l15">Line 15:</td>
<td colspan="2" class="diff-lineno">Line 15:</td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>f¨g</source> has the overall [[function rank]] of <<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>g</source>. This allows <<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>¨></source> to serve the purpose of [[Each]] from a [[Nested array model|nested]] APL. J uses <<del style="font-weight: bold; text-decoration: none;">source </del>lang=j inline>&.</source> for the close form and <<del style="font-weight: bold; text-decoration: none;">source </del>lang=j inline>&.:</source> for the rankless form.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>f¨g</source> has the overall [[function rank]] of <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>g</source>. This allows <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>¨></source> to serve the purpose of [[Each]] from a [[Nested array model|nested]] APL. J uses <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=j inline>&.</source> for the close form and <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=j inline>&.:</source> for the rankless form.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>¨</source> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <<del style="font-weight: bold; text-decoration: none;">source </del>lang=j inline>&.</source> like SHARP and a non-close form <<del style="font-weight: bold; text-decoration: none;">source </del>lang=j inline>&.:</source>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>¨</source> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=j inline>&.</source> like SHARP and a non-close form <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=j inline>&.:</source>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>⍢</source> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>⍢</source> in his 1978 paper [[Operators and Functions]],<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>⍢×</source> and <<del style="font-weight: bold; text-decoration: none;">source </del>lang=apl inline>⍢|</source> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>⍢×</source> and <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=apl inline>⍢|</source> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== External links ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <<del style="font-weight: bold; text-decoration: none;">source </del>lang=j inline>&.</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <<del style="font-weight: bold; text-decoration: none;">source </del>lang=j inline>&.:</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=j inline>&.</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <<ins style="font-weight: bold; text-decoration: none;">syntaxhighlight </ins>lang=j inline>&.:</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://mlochbaum.github.io/BQN/doc/under.html BQN]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [https://mlochbaum.github.io/BQN/doc/under.html BQN]</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
</table>
Adám Brudzewsky
https://aplwiki.com/index.php?title=Under&diff=8640&oldid=prev
Marshall: /* External links */ BQN
2022-05-28T02:38:10Z
<p><span dir="auto"><span class="autocomment">External links: </span> BQN</span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 02:38, 28 May 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <source lang=j inline>&.</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <source lang=j inline>&.:</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* [[J]]: <source lang=j inline>&.</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdot NuVoc], [https://www.jsoftware.com/help/dictionary/d631.htm dictionary]; <source lang=j inline>&.:</source> [https://code.jsoftware.com/wiki/Vocabulary/ampdotco NuVoc], [https://www.jsoftware.com/help/dictionary/d631c.htm dictionary]</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">* [https://mlochbaum.github.io/BQN/doc/under.html BQN]</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== References ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div><references/></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{APL built-ins}}[[Category:Primitive operators]][[Category:Composition operators]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>{{APL built-ins}}[[Category:Primitive operators]][[Category:Composition operators]]</div></td></tr>
</table>
Marshall
https://aplwiki.com/index.php?title=Under&diff=8475&oldid=prev
Marshall: Text replacement - ""Operators and Functions"" to "Operators and Functions"
2022-04-29T15:32:21Z
<p>Text replacement - ""Operators and Functions"" to "<a href="/wiki/Operators_and_Functions" title="Operators and Functions">Operators and Functions</a>"</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 15:32, 29 April 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <source lang=apl inline>¨</source> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <source lang=j inline>&.</source> like SHARP and a non-close form <source lang=j inline>&.:</source>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <source lang=apl inline>¨</source> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <source lang=j inline>&.</source> like SHARP and a non-close form <source lang=j inline>&.:</source>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <source lang=apl inline>⍢</source> in his 1978 paper <del style="font-weight: bold; text-decoration: none;">"</del>Operators and Functions<del style="font-weight: bold; text-decoration: none;">"</del>,<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <source lang=apl inline>⍢</source> in his 1978 paper <ins style="font-weight: bold; text-decoration: none;">[[</ins>Operators and Functions<ins style="font-weight: bold; text-decoration: none;">]]</ins>,<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <source lang=apl inline>⍢×</source> and <source lang=apl inline>⍢|</source> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <source lang=apl inline>⍢×</source> and <source lang=apl inline>⍢|</source> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td></tr>
</table>
Marshall
https://aplwiki.com/index.php?title=Under&diff=8441&oldid=prev
Adám Brudzewsky at 23:10, 26 April 2022
2022-04-26T23:10:29Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 23:10, 26 April 2022</td>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>{{Built-in|Under|⍢}}, or '''Dual''' (<<del style="font-weight: bold; text-decoration: none;">code</del>>¨</<del style="font-weight: bold; text-decoration: none;">code</del>>), is a [[primitive operator|primitive]] [[dyadic operator]] which takes two [[function]] [[operand]]s. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.<ref>[https://mlochbaum.github.io/BQN/help/under.html BQN: Circle Jot (⌾)]</ref></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>{{Built-in|Under|⍢}}, or '''Dual''' (<<ins style="font-weight: bold; text-decoration: none;">source lang=apl inline</ins>>¨</<ins style="font-weight: bold; text-decoration: none;">source</ins>>), is a [[primitive operator|primitive]] [[dyadic operator]] which takes two [[function]] [[operand]]s. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.<ref>[https://mlochbaum.github.io/BQN/help/under.html BQN: Circle Jot (⌾)]</ref></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Model ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Model ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In dialects that support user-defined operators and invertible functions, it can be defined as (or similar):</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>In dialects that support user-defined operators and invertible functions, it can be defined as (or similar):</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> </del>_U_ ← {⍺←{⍵ ⋄ ⍺⍺} ⋄ ⍵⍵⍣¯1⊢(⍵⍵ ⍺)⍺⍺(⍵⍵ ⍵)}</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><source lang=apl> </ins>_U_ ← {⍺←{⍵ ⋄ ⍺⍺} ⋄ ⍵⍵⍣¯1⊢(⍵⍵ ⍺)⍺⍺(⍵⍵ ⍵)}<ins style="font-weight: bold; text-decoration: none;"></source></ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Examples ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Examples ==</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> ⍝ </del>https://leetcode.com/problems/add-two-numbers/</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">The following example solves [</ins>https://leetcode.com/problems/add-two-numbers/ <ins style="font-weight: bold; text-decoration: none;">a LeetCode problem] which asks to add two numbers given as reversed digit lists:</ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> 2 4 3 +_U_(10⊥⌽) 5 6 4</del></div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"><source lang=apl></ins></div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div><del style="font-weight: bold; text-decoration: none;"> 7 0 8</del></div></td><td colspan="2" class="diff-side-added"></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"> 2 4 3 +_U_(10⊥⌽) 5 6 4</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">7 0 8</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></source></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">First, each argument is reversed, then decoded from base 10. The addition is performed. Finally, the result is encoded to base 10 and reversed.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <source lang=apl inline>f¨g</source> has the overall [[function rank]] of <source lang=apl inline>g</source>. This allows <<del style="font-weight: bold; text-decoration: none;">code</del>>¨></<del style="font-weight: bold; text-decoration: none;">code</del>> to serve the purpose of [[Each]] from a [[Nested array model|nested]] APL. J uses <<del style="font-weight: bold; text-decoration: none;">code</del>>&.</<del style="font-weight: bold; text-decoration: none;">code</del>> for the close form and <<del style="font-weight: bold; text-decoration: none;">code</del>>&.:</<del style="font-weight: bold; text-decoration: none;">code</del>> for the rankless form.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <source lang=apl inline>f¨g</source> has the overall [[function rank]] of <source lang=apl inline>g</source>. This allows <<ins style="font-weight: bold; text-decoration: none;">source lang=apl inline</ins>>¨></<ins style="font-weight: bold; text-decoration: none;">source</ins>> to serve the purpose of [[Each]] from a [[Nested array model|nested]] APL. J uses <<ins style="font-weight: bold; text-decoration: none;">source lang=j inline</ins>>&.</<ins style="font-weight: bold; text-decoration: none;">source</ins>> for the close form and <<ins style="font-weight: bold; text-decoration: none;">source lang=j inline</ins>>&.:</<ins style="font-weight: bold; text-decoration: none;">source</ins>> for the rankless form.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <<del style="font-weight: bold; text-decoration: none;">code</del>>¨</<del style="font-weight: bold; text-decoration: none;">code</del>> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <source lang=j inline>&.</source> like SHARP and a non-close form <source lang=j inline>&.:</source>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The most widely known form of Under for much of APL's history was [[SHARP APL]]'s Dual operator, written <<ins style="font-weight: bold; text-decoration: none;">source lang=apl inline</ins>>¨</<ins style="font-weight: bold; text-decoration: none;">source</ins>> and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980,<ref>[[Bob Bernecky]] and [[Ken Iverson]]. [https://www.jsoftware.com/papers/opea.htm Operators and Enclosed Arrays] at [[IPSA '80]]</ref> and implemented for only a small number of right operands in 1981.<ref>[https://www.jsoftware.com/papers/satn41.htm "Composition and Enclosure"]. SATN-41, 1981-06-20.</ref> The name "Dual" was used in [[Rationalized APL]] in 1983, and the full SHARP implementation using [[function rank]] was completed just after.<ref>[https://www.jsoftware.com/papers/satn45.htm "Language Extensions of May 1983"]. SATN-45, 1983-05-02.</ref> Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <source lang=j inline>&.</source> like SHARP and a non-close form <source lang=j inline>&.:</source>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <source lang=apl inline>⍢</source> in his 1978 paper "Operators and Functions",<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <source lang=apl inline>⍢</source> in his 1978 paper "Operators and Functions",<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981, preceding SHARP's limited implementation. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td></tr>
</table>
Adám Brudzewsky
https://aplwiki.com/index.php?title=Under&diff=8438&oldid=prev
Marshall: /* Close composition */ Move ¨> description from history section
2022-04-26T14:57:52Z
<p><span dir="auto"><span class="autocomment">Close composition: </span> Move ¨> description from history section</span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<col class="diff-marker" />
<col class="diff-content" />
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:57, 26 April 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== Close composition ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <source lang=apl inline>f¨g</source> has the overall [[function rank]] of <source lang=apl inline>g</source>. J uses <code>&.</code> for the close form and <code>&.:</code> for the rankless form.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <source lang=apl inline>f¨g</source> has the overall [[function rank]] of <source lang=apl inline>g</source><ins style="font-weight: bold; text-decoration: none;">. This allows <code>¨></code> to serve the purpose of [[Each]] from a [[Nested array model|nested]] APL</ins>. J uses <code>&.</code> for the close form and <code>&.:</code> for the rankless form.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td></tr>
</table>
Marshall
https://aplwiki.com/index.php?title=Under&diff=8437&oldid=prev
Marshall: /* History */ Include earlier SHARP publications
2022-04-26T14:56:36Z
<p><span dir="auto"><span class="autocomment">History: </span> Include earlier SHARP publications</span></p>
<table style="background-color: #fff; color: #202122;" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 14:56, 26 April 2022</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>== History ==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>The Dual operator, written <code>¨</code>, <del style="font-weight: bold; text-decoration: none;">appeared in 1983 both in </del>[[<del style="font-weight: bold; text-decoration: none;">Rationalized APL</del>]] and [[<del style="font-weight: bold; text-decoration: none;">SHARP APL</del>]]<ref>[https://www.jsoftware.com/papers/<del style="font-weight: bold; text-decoration: none;">satn45</del>.htm "<del style="font-weight: bold; text-decoration: none;">Language Extensions of May 1983</del>"]. SATN-<del style="font-weight: bold; text-decoration: none;">45</del>, <del style="font-weight: bold; text-decoration: none;">1983</del>-<del style="font-weight: bold; text-decoration: none;">05</del>-<del style="font-weight: bold; text-decoration: none;">02</del>.</ref> <del style="font-weight: bold; text-decoration: none;">just after. In both cases it </del>was <del style="font-weight: bold; text-decoration: none;">defined to be a </del>[[<del style="font-weight: bold; text-decoration: none;">close composition</del>]], <del style="font-weight: bold; text-decoration: none;">so that </del>the <del style="font-weight: bold; text-decoration: none;">overall </del>function <del style="font-weight: bold; text-decoration: none;">has the </del>rank <del style="font-weight: bold; text-decoration: none;">of the right operand; this allows </del><<del style="font-weight: bold; text-decoration: none;">code</del>><del style="font-weight: bold; text-decoration: none;">¨><</del>/<del style="font-weight: bold; text-decoration: none;">code> to serve the purpose </del>of <del style="font-weight: bold; text-decoration: none;">[[Each</del>]<del style="font-weight: bold; text-decoration: none;">] from a [[Nested array model|nested]] APL</del>. Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <source lang=j inline>&.</source> like SHARP and a non-close form <source lang=j inline>&.:</source>.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>The <ins style="font-weight: bold; text-decoration: none;">most widely known form of Under for much of APL's history was [[SHARP APL]]'s </ins>Dual operator, written <code>¨</code> <ins style="font-weight: bold; text-decoration: none;">and defined as a [[close composition]]. It was presented by [[Ken Iverson]] and [[Bob Bernecky]] with the name "with" in 1980</ins>,<ins style="font-weight: bold; text-decoration: none;"><ref>[[Bob Bernecky]] and </ins>[[<ins style="font-weight: bold; text-decoration: none;">Ken Iverson</ins>]]<ins style="font-weight: bold; text-decoration: none;">. [https://www.jsoftware.com/papers/opea.htm Operators </ins>and <ins style="font-weight: bold; text-decoration: none;">Enclosed Arrays] at </ins>[[<ins style="font-weight: bold; text-decoration: none;">IPSA '80</ins>]]<ins style="font-weight: bold; text-decoration: none;"></ref> and implemented for only a small number of right operands in 1981.</ins><ref>[https://www.jsoftware.com/papers/<ins style="font-weight: bold; text-decoration: none;">satn41</ins>.htm "<ins style="font-weight: bold; text-decoration: none;">Composition and Enclosure</ins>"]. SATN-<ins style="font-weight: bold; text-decoration: none;">41</ins>, <ins style="font-weight: bold; text-decoration: none;">1981</ins>-<ins style="font-weight: bold; text-decoration: none;">06</ins>-<ins style="font-weight: bold; text-decoration: none;">20</ins>.</ref> <ins style="font-weight: bold; text-decoration: none;">The name "Dual" </ins>was <ins style="font-weight: bold; text-decoration: none;">used in </ins>[[<ins style="font-weight: bold; text-decoration: none;">Rationalized APL</ins>]] <ins style="font-weight: bold; text-decoration: none;">in 1983</ins>, <ins style="font-weight: bold; text-decoration: none;">and </ins>the <ins style="font-weight: bold; text-decoration: none;">full SHARP implementation using [[</ins>function rank<ins style="font-weight: bold; text-decoration: none;">]] was completed just after.</ins><<ins style="font-weight: bold; text-decoration: none;">ref</ins>><ins style="font-weight: bold; text-decoration: none;">[https://www.jsoftware.com/papers</ins>/<ins style="font-weight: bold; text-decoration: none;">satn45.htm "Language Extensions </ins>of <ins style="font-weight: bold; text-decoration: none;">May 1983"</ins>]. <ins style="font-weight: bold; text-decoration: none;">SATN-45, 1983-05-02.</ref> </ins>Iverson changed the name to Under for his 1987 [[A Dictionary of APL]], and [[J]] uses this name as well. It includes both a close form <source lang=j inline>&.</source> like SHARP and a non-close form <source lang=j inline>&.:</source>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>However, <del style="font-weight: bold; text-decoration: none;">[[Ken </del>Iverson<del style="font-weight: bold; text-decoration: none;">]] </del>had defined the Dual operator with glyph <source lang=apl inline>⍢</source> in his 1978 paper "Operators and Functions",<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>However, Iverson had defined the Dual operator with glyph <source lang=apl inline>⍢</source> in his 1978 paper "Operators and Functions",<ref>[[Ken Iverson]]. [https://www.jsoftware.com/papers/opfns.htm ''Operators and Functions''], §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.</ref> using a non-close form because [[function rank]] had not yet been invented. [[NARS]] featured this operator in 1981<ins style="font-weight: bold; text-decoration: none;">, preceding SHARP's limited implementation</ins>. [[Roger Hui]] proposed Under with the same definition as a potential addition to [[Dyalog APL]] at [[Dyalog '15]], and it was added to both [[Extended Dyalog APL]] and [[dzaima/APL]] in 2018.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br/></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <source lang=apl inline>⍢×</source> and <source lang=apl inline>⍢|</source> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>Structural Under was developed by [[Marshall Lochbaum]] around 2017,<ref>[[Marshall Lochbaum]]. [https://mlochbaum.github.io/BQN/commentary/history.html#structural-under BQN's development history: Structural Under].</ref> and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later [[dzaima]] would implement <source lang=apl inline>⍢×</source> and <source lang=apl inline>⍢|</source> in the same spirit, now featured in dzaima/APL and [[dzaima/BQN]]. The language [[BQN]] has included structural and computational Under, using the glyph <code>⌾</code>, since its initial design in 2020.</div></td></tr>
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Marshall
https://aplwiki.com/index.php?title=Under&diff=8417&oldid=prev
Marshall: Close composition (SHARP/J) section
2022-04-25T17:22:39Z
<p>Close composition (SHARP/J) section</p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 17:22, 25 April 2022</td>
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<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">== Close composition ==</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">In [[SHARP APL]] and [[J]], Under is implemented as a [[close composition]], meaning that (using SHARP syntax) <source lang=apl inline>f¨g</source> has the overall [[function rank]] of <source lang=apl inline>g</source>. J uses <code>&.</code> for the close form and <code>&.:</code> for the rankless form.</ins></div></td></tr>
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Marshall