Function composition: Difference between revisions

The operator represented by [[Jot]] (<source lang=apl inline>∘</source>, in this context called [[Bind]]) and the 3-train can also be used with constant arrays, then treating the arrays (<source lang=apl inline>A</source>) as constant functions, much as if they were used as operands to the [[Constant]] operator (<source lang=apl inline>A⍨</source>):

<source lang=apl inline>  (f⍛g  ) ⍵</source> {{←→}} <source lang=apl inline>(  f ⍵) g      ⍵ </source><br>

<source lang=apl inline>⍺ (f⍛g  ) ⍵</source> {{←→}} <source lang=apl inline>(  f ⍺) g      ⍵ </source><br>

<source lang=apl inline>  (  g∘h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>

<source lang=apl inline>⍺ (  g∘h) ⍵</source> {{←→}} <source lang=apl inline>   ⍺    g (  h ⍵)</source><br>

<source lang=apl inline>  (  g⍤h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>

<source lang=apl inline>⍺ (  g⍤h) ⍵</source> {{←→}} <source lang=apl inline>        g (⍺ h ⍵)</source><br>

<source lang=apl inline>  (  g⍥h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>

<source lang=apl inline>⍺ (  g⍥h) ⍵</source> {{←→}} <source lang=apl inline>(  h ⍺) g (  h ⍵)</source><br>

<source lang=apl inline>  (  g h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>

<source lang=apl inline>⍺ (  g h) ⍵</source> {{←→}} <source lang=apl inline>        g (⍺ h ⍵)</source><br>

<source lang=apl inline>  (f g h) ⍵</source> {{←→}} <source lang=apl inline>(  f ⍵) g (  h ⍵)</source><br>

<source lang=apl inline>⍺ (f g h) ⍵</source> {{←→}} <source lang=apl inline>(⍺ f ⍵) g (⍺ h ⍵)</source><br>

<source lang=apl inline>  (A∘g  ) ⍵</source> {{←→}} <source lang=apl inline>   A    g      ⍵ </source><br>

<source lang=apl inline>  (  g∘A) ⍵</source> {{←→}} <source lang=apl inline>   ⍵    g      A </source><br>

<source lang=apl inline>  (A g h) ⍵</source> {{←→}} <source lang=apl inline>   A    g (  h ⍵)</source><br>

<source lang=apl inline>⍺ (A g h) ⍵</source> {{←→}} <source lang=apl inline>   A    g (⍺ h ⍵)</source><br>

Additional compositions are possible, even without using an argument more than once or applying a function to its own result. However, most of these are rather trivial shuffled-around versions of the above three. For example, one could define an operator identical to Atop, only that it applies the right operand to the result of the left operand, that is <source lang=apl inline>{⍵⍵ ⍺ ⍺⍺ ⍵}</source>.

Of note here is <source lang=apl inline>f⍨∘g⍨</source> which is equivalent to — although with swapped operands — [[Reverse-compose]] <source lang=apl inline>⍛</source> (also called ''Before''), and the mirrored version of Beside <source lang=apl inline>∘</source> (also known as ''Compose'' and ''After''), because it is the only such variation that has been implemented, namely in [[dzaima/APL]] and [[Extended Dyalog APL]].  

A compositional operator that isn't just a shuffled around version of the basic three, is one that applies one operand between the other operand's dyadic result and the result of that other operand's result when [[swap]]ped: <source lang=apl inline>{(⍵ ⍵⍵ ⍺) ⍺⍺ (⍺ ⍵⍵ ⍵)}</source>. This operator can for example be used to implement [[wikipedia:three-way comparison|three-way comparison]]:

</source>{{Works in|[[Dyalog APL]], [[NARS2000]], [[ngn/apl]]}}