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{{Built-ins|Hooks|⍛|⍩}} are asymmetrical forms of [[function composition]] that first apply one of the composed [[function]]s to one [[argument]], then applies the other function to one argument and the result. [[Kap]] has the two [[operator]]s [[Beside|Compose]] (<syntaxhighlight lang=apl inline>∘</syntaxhighlight>) and [[Inverse compose]] (<syntaxhighlight lang=apl inline>⍛</syntaxhighlight>). [[BQN]] has Before (<code>⊸</code>) and After (<code>⟜</code>), which also serve the purpose of the [[Bind]] operator. In [[J]], a 2-[[train]] is a hook, while [[I]] adds the mirror image to give two functions (I has first-class functions but no operators) hook (<code>h</code>) and backhook (<code>H</code>).
Hook as an operator first appeared in [[A Dictionary of APL]] as [[Withe]] (<syntaxhighlight lang=apl inline>⍩</syntaxhighlight>). However, [[Ken Iverson|Iverson]] and [[Eugene McDonnell|McDonnell]]'s paper ''Phrasal Forms'', which introduced [[train]]s<ref>[[Ken Iverson]] and [[Eugene McDonnell]]. [http://www.jsoftware.com/papers/fork.htm Phrasal forms] at [[APL89]].</ref>, proposed hook as the meaning of a 2-train, and this was soon included in [[J]]. This definition specifies that <syntaxhighlight lang=j inline>(F G) y</syntaxhighlight> is <syntaxhighlight lang=j inline>y F G y</syntaxhighlight> and <syntaxhighlight lang=j inline>x (F G) y</syntaxhighlight> is <syntaxhighlight lang=j inline>x F G y</syntaxhighlight>. However, [[Roger Hui]] later opined that this definition was better suited to a dyadic operator (which could be denoted <syntaxhighlight lang=j inline>h.</syntaxhighlight>) than an element of syntax,<ref>[[Roger Hui]]. [https://code.jsoftware.com/wiki/Essays/Hook_Conjunction%3F Hook Conjunction?]. J Wiki essays. 2006. Accessed 2021-02-08.</ref> and defined the 2-train to represent [[Atop]] instead when he led the introduction of trains to [[Dyalog APL]]. By that time, Dyalog had long had the [[Beside]] (originally called ''Compose'') operator which is equivalent to <syntaxhighlight lang=j inline>x F h. G y</syntaxhighlight>, that is, the dyadic case. The monadic functionality can be achieved using [[Commute]] as <syntaxhighlight lang=apl inline>F∘G⍨y</syntaxhighlight> and the full [[ambivalent]] function can be written as <syntaxhighlight lang=apl inline>F∘G⍨⍨</syntaxhighlight>.
[[File:Hooks.png|thumb|right|Hook operators in [[BQN]]]]
[[File:Hooks.png|thumb|right|Hook operators in [[BQN]]]]
In order to maintain symmetry, [[Kap]] changes the meaning of <syntaxhighlight lang=apl inline>∘</syntaxhighlight> to be a proper hook, and thus aligns itself with [[I]] and [[BQN]], which also have two hooks. Kap defines Compose (<syntaxhighlight lang=apl inline>∘</syntaxhighlight>) to be the dyadic operator <syntaxhighlight lang=apl inline>{⍺←⍵ ⋄ ⍺ ⍺⍺ ⍵⍵ ⍵}</syntaxhighlight> and Inverse Compose (<syntaxhighlight lang=apl inline>⍛</syntaxhighlight>) to be <syntaxhighlight lang=apl inline>{⍺←⍵ ⋄ (⍺⍺ ⍺) ⍵⍵ ⍵}</syntaxhighlight>. In the dyadic case these operators are identical to [[Beside]] and [[Reverse Compose]] respectively, but in the monadic case, <syntaxhighlight lang=apl inline>∘</syntaxhighlight> differs from Beside because the argument is used twice: the second function application takes it as an argument directly in addition to the result of the first function application.


A '''hook''' is an asymmetrical form of function composition that first applies one of the composed functions to one argument, then applies the other function to one argument and the result. In [[J]], a 2-[[train]] is a hook, while [[I]] adds the mirror image to give two functions (I has first-class functions but no operators) hook (<code>h</code>) and backhook (<code>H</code>). [[BQN]] uses two [[operator]]s Before (<code>⊸</code>) and After (<code>⟜</code>), which also serve the purpose of the [[Bind]] operator.
The two hooks can be used together to form a [[split-compose]] construct:
 
The meaning of 2-train as hook was first proposed in [[Ken Iverson|Iverson]] and [[Eugene McDonnell|McDonnell]]'s paper ''Phrasal Forms'' introducing [[train]]s<ref>[[Ken Iverson]] and [[Eugene McDonnell]]. [http://www.jsoftware.com/papers/fork.htm Phrasal forms] at [[APL89]].</ref>, and soon included in [[J]]. This definition specifies that <source lang=j inline>(F G) y</source> is <source lang=j inline>y F G y</source> and <source lang=j inline>x (F G) y</source> is <source lang=j inline>x F G y</source>. However, [[Roger Hui]] later opined that this definition was better suited to a dyadic operator than an element of syntax,<ref>[[Roger Hui]]. [https://code.jsoftware.com/wiki/Essays/Hook_Conjunction%3F Hook Conjunction?]. J Wiki essays. 2006. Accessed 2021-02-08.</ref> and defined to 2-train to represent [[Atop]] instead when he led the introduction of trains to [[Dyalog APL]].
 
In [[I]] and [[BQN]], there are two hooks in order to maintain symmetry: for example, BQN defines Before (<code>⊸</code>) to be the dyadic operator <code>{(𝔽𝕨⊣𝕩)𝔾𝕩}</code> ("<code>𝔾</code>'s left argument comes from <code>𝔽</code>") and After (<code>⟜</code>) to be <code>{(𝕨⊣𝕩)𝔽𝔾𝕩}</code> ("<code>𝔽</code>'s right argument comes from <code>𝔾</code>"). In the dyadic case these functions are identical to [[Reverse Compose]] and [[Beside]] respectively, but in the monadic case they differ because the argument is used twice: the second function application takes it as an argument directly in addition to the result of the first function application.
 
Like [[Reverse Compose]], the two hooks can be used together to form a [[split-compose]] construct.
<pre>
<pre>
     3‿¯1‿4 ×⊸×⟜| ¯2‿¯7‿1
     3 ¯1 4 ×⍛×∘| ¯2 ¯7 1
⟨ 2 ¯7 1 ⟩
2 ¯7 1
</pre>{{Works in|[[BQN]]}}
</pre>{{Works in|[[Kap]]}}
This definition behaves differently that the Compose-based one when only one argument is given: in that case, it becomes a monadic 3-[[train]].
This definition behaves differently that the Compose-based one when only one argument is given; in that case, it becomes a monadic 3-[[train]].


The name "hook" was chosen based on the hook shape of a function call diagram such as the one below, taken from ''Phrasal Forms''.
The name "hook" was chosen based on the hook shape of a function call diagram such as the one below, taken from ''Phrasal Forms''.
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* J [https://www.jsoftware.com/help/dictionary/dictf.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/hook NuVoc]
* J [https://www.jsoftware.com/help/dictionary/dictf.htm Dictionary], [https://code.jsoftware.com/wiki/Vocabulary/hook NuVoc]
* [https://mlochbaum.github.io/BQN/tutorial/combinator.html#before-and-after BQN tutorials]
* [https://mlochbaum.github.io/BQN/doc/hook.html BQN]


== References ==
== References ==
<references />
<references />
{{APL built-ins}}[[Category:Primitive operators]]
{{APL syntax}}[[Category:Primitive operators]]
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