Difference between revisions of "Function composition"

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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 to 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 (operator)|Atop]], [[Beside]], and [[Over]] can be visualised as follows:
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'''Function composition''' refers to the "gluing" together of two or more [[function]]s using a [[dyadic operator]] or a [[train]] such that the functions are applied to the [[argument]](s) as normal, but in a particular pattern specific to 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 patterns represented by the operators [[Atop (operator)|Atop]], [[Beside]], and [[Over]] can be visualised as follows:
  
:[[File:Compositions.png|frameless|497px]]
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{| class=wikitable
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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! Atop !! Beside !! Over
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|-
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|[[File:F⍤g.png|frameless|200px]]||[[File:F∘g.png|frameless|200px]]||[[File:F⍥g.png|frameless|200px]]
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|}
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The patterns represented by [[Tacit_programming#trains|trains]], the 2-train Atop and the 3-train Fork, can be visualised as follows:
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{| class=wikitable
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! 2-train !! 3-train
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|-
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|[[File:Fg.png|frameless|200px]]||[[File:Fgh.png|frameless|200px]]
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|}
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The <source lang=apl inline>∘</source> operator (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>):
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{| class=wikitable
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! Bind !! 3-train
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|-
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|[[File:A∘g;g∘A.png|frameless|200px]]||[[File:Agh.png|frameless|200px]]
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|}
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=== Summary of rules ===
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These are the rules applied in [[Dyalog APL]]:
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<source lang=apl inline>  (  g∘h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>
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<source lang=apl inline>⍺ (  g∘h) ⍵</source> {{←→}} <source lang=apl inline>   ⍺    g (  h ⍵)</source><br>
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<source lang=apl inline>  (A∘g  ) ⍵</source> {{←→}} <source lang=apl inline>   A    g      ⍵ </source><br>
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<source lang=apl inline>  (  g∘A) ⍵</source> {{←→}} <source lang=apl inline>   ⍵    g      A </source><br>
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<source lang=apl inline>  (  g⍤h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>
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<source lang=apl inline>⍺ (  g⍤h) ⍵</source> {{←→}} <source lang=apl inline>        g (⍺ h ⍵)</source><br>
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<source lang=apl inline>  (  g⍥h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>
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<source lang=apl inline>⍺ (  g⍥h) ⍵</source> {{←→}} <source lang=apl inline>(  h ⍺) g (  h ⍵)</source><br>
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<source lang=apl inline>  (f g h) ⍵</source> {{←→}} <source lang=apl inline>(  f ⍵) g (  h ⍵)</source><br>
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<source lang=apl inline>⍺ (f g h) ⍵</source> {{←→}} <source lang=apl inline>(f ⍵) g (⍺ h ⍵)</source><br>
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<source lang=apl inline>  (A g h) </source> {{←→}} <source lang=apl inline>   A    g (  h ⍵)</source><br>
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<source lang=apl inline>⍺ (A g h) ⍵</source> {{←→}} <source lang=apl inline>   A    g (⍺ h ⍵)</source><br>
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<source lang=apl inline>    (g h) ⍵</source> {{←→}} <source lang=apl inline>        g (  h ⍵)</source><br>
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<source lang=apl inline>⍺   (g h) ⍵</source> {{←→}} <source lang=apl inline>        g (⍺ h ⍵)</source>
  
 
== Additional compositions ==
 
== Additional compositions ==

Revision as of 18:31, 30 June 2022

Function composition refers to the "gluing" together of two or more functions using a dyadic operator or a train such that the functions are applied to the argument(s) as normal, but in a particular pattern specific to the used operator. The term function composition comes from traditional mathematics where it is used for a function when written as . APL generalises this idea to dyadic functions, allowing various patterns of application in addition to the simple application of one monadic function to the result of another monadic function. The patterns represented by the operators Atop, Beside, and Over can be visualised as follows:

Atop Beside Over
F⍤g.png F∘g.png F⍥g.png

The patterns represented by trains, the 2-train Atop and the 3-train Fork, can be visualised as follows:

2-train 3-train
Fg.png Fgh.png

The operator (in this context called Bind) and the 3-train can also be used with constant arrays, then treating the arrays (A) as constant functions, much as if they were used as operands to the Constant operator (A):

Bind 3-train
A∘g;g∘A.png Agh.png

Summary of rules

These are the rules applied in Dyalog APL:

  (  gh)         g (  h )
 (  gh)        g (  h )
  (Ag  )    A    g       
  (  gA)        g      A 
  (  gh)         g (  h )
 (  gh)         g ( h )
  (  gh)         g (  h )
 (  gh) (  h ) g (  h )
  (f g h) (  f ) g (  h )
(f g h) ( f ) g ( h )
  (A g h)    A    g (  h )
(A g h)    A    g ( h )
    (g h)         g (  h )
   (g h)         g ( h )

Additional compositions

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 {⍵⍵ ⍺⍺ }.

When Dyalog added Atop and Over, it was with the reasoning that these were the only compositions where the leftmost function acted as the "root" function in the evaluation tree, while the arguments were used each on their respective sides of the constituent functions:

AllCompositions.png

Of note here is f⍨∘g which is equivalent to — although with swapped operands — Reverse-compose (also called Before), and the mirrored version of Beside (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 swapped: {( ⍵⍵ ) ⍺⍺ ( ⍵⍵ )}. This operator can for example be used to implement three-way comparison:

Try it online!

      S  {( ⍵⍵ ) ⍺⍺ ( ⍵⍵ )}
      cmp  -S
      2 cmp 3
¯1
      3 cmp 3
0
      4 cmp 3
1

External links

APL features [edit]
Built-ins Primitives (functions, operators) ∙ Quad name
Array model ShapeRankDepthBoundIndex (Indexing) ∙ AxisRavelRavel orderElementScalarVectorMatrixSimple scalarSimple arrayNested arrayCellMajor cellSubarrayEmpty arrayPrototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ BoxNamespace
Concepts and paradigms Conformability (Scalar extension, Leading axis agreement) ∙ Scalar function (Pervasion) ∙ Identity elementComplex floorTotal array orderingTacit programming (Function composition, Close composition) ∙ Glyph
Errors LIMIT ERRORRANK ERRORSYNTAX ERRORDOMAIN ERRORLENGTH ERRORINDEX ERRORVALUE ERROR