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Function composition: Difference between revisions
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'''Function composition''' refers to the "gluing" together of two [[function]]s using a [[dyadic operator]] such that the functions are applied to the [[argument]](s) as normal, but in a particular pattern specific the the used [[operator]]. The term [[wikipedia:function composition|function composition]] comes from [[traditional mathematics]] where it is used for a function <math>h(x)=f(g(x))</math> when written as <math> h(x) = (f \circ g)(x)</math>. APL generalises this idea to [[dyadic function]]s, allowing various patterns of application in addition to the simple application of one [[monadic function]] to the result of another monadic function. The three main patterns, represented by [[Atop]], [[Beside]], and [[Over]] can be visualised as follows: | '''Function composition''' refers to the "gluing" together of two [[function]]s using a [[dyadic operator]] such that the functions are applied to the [[argument]](s) as normal, but in a particular pattern specific the the used [[operator]]. The term [[wikipedia:function composition|function composition]] comes from [[traditional mathematics]] where it is used for a function <math>h(x)=f(g(x))</math> when written as <math> h(x) = (f \circ g)(x)</math>. APL generalises this idea to [[dyadic function]]s, allowing various patterns of application in addition to the simple application of one [[monadic function]] to the result of another monadic function. The three main patterns, represented by [[Atop]], [[Beside]], and [[Over]] can be visualised as follows: | ||
:[[File:Compositions.png|frameless]] | :[[File:Compositions.png|frameless|497px]] | ||
When any of these are applied monadically, the dotted branch falls away, and they are all equivalent to each other and to <math>(f \circ g)(x)</math> of traditional mathematics. | When any of these are applied monadically, the dotted branch falls away, and they are all equivalent to each other and to <math>(f \circ g)(x)</math> of traditional mathematics. | ||
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Of note is [[Reverse-compose]] <source lang=apl inline>⍛</source> (also called ''Before''), which is 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]]. | Of note is [[Reverse-compose]] <source lang=apl inline>⍛</source> (also called ''Before''), which is 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 | 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]]: | ||
[https://tio.run/##SyzI0U2pTMzJT///P1jhUdsEhWqNR71bFYAYQu3SBBFApKABIuDiWzVruZJzC8BadIMfdS7hetQ3FcQxUjBWMFEASRn//w8A Try it online!]<source lang=apl> | [https://tio.run/##SyzI0U2pTMzJT///P1jhUdsEhWqNR71bFYAYQu3SBBFApKABIuDiWzVruZJzC8BadIMfdS7hetQ3FcQxUjBWMFEASRn//w8A Try it online!]<source lang=apl> | ||
S ← {(⍵ ⍵⍵ ⍺) ⍺⍺ (⍺ ⍵⍵ ⍵)} | S ← {(⍵ ⍵⍵ ⍺) ⍺⍺ (⍺ ⍵⍵ ⍵)} | ||