The monadic operator ⍨ is defined and modeled as follows:

     f⍨ ⍵  ←→  ⍵ f ⍵
   ⍺ f⍨ ⍵  ←→  ⍵ f ⍺

   {⍺←⍵ ⋄ ⍵ ⍺⍺ ⍺}

Reflex

Some common well-known functions can be written as f⍨ where f is itself a well-known function:

   +⍨    double
   ×⍨    square
   ?⍨    random permutation
   ⍳⍨    self-index, APL Amuse-Bouche 3

See http://www.jsoftware.com/jwiki/Essays/Reflexive for further examples.

Awareness of the importance of the reflexive case might have led us to avoid the mistake in the definition of the dyadic case of ⍋. That is, if

   ⍺⍋⍵ ←→ ⍺⌷⍨⊂⍋⍵

then ⍋⍨⍵ would be sort. Ken Iverson seemed to have had this awareness because that’s how he defined the dyadic case of ⍋ in J. (I say “seemed” because he expressed surprise when first shown this use of ⍋⍨.)

The monadic case f⍨ came relatively late. It was not in Operators and Functions (1978) nor Rationalized APL (1983), and only introduced in A Dictionary of APL (1987). It came to Ken Iverson when he explicitly looked to natural languages for inspiration, whence it became “obvious”: f⍨⍵ ←→ ⍵ f ⍵ is the reflexive voice (je m’appelle Roger) and ⍺ f⍨ ⍵ ←→ ⍵ f ⍺ is the passive voice (the programming competition was won by a 17-year-old student vs. a 17-year-old student won the programming competition), both having evolved in natural languages for effective communication and elegant expression.

Commute

The alternative definition of ⍺⍋⍵ above, while illustrating the importance of the reflexive case (f⍨ ⍵), also illustrates the passive case (⍺ f⍨ ⍵). ⍺⌷⍨⊂⍋⍵ can be read and understood as “⍺ indexed by the enclosed grade of ⍵“, a different (and in my mind a better) emphasis than (⊂⍋⍵)⌷⍺, “the enclosed grade of ⍵, indexed into ⍺“.

I note that Arianna Locatelli, winner of the 2015 APL Problem Solving Competition but an APL beginner, used ⍨ twenty-one times in her presentation at Dyalog ’15. For example,⍨ was used in the computation of the standard deviation (slide 13):

   sol5←{a←,⍵ ⋄ ⊃0.5*⍨(⍴a)÷⍨+/2*⍨a-(⍴a)÷⍨+/a}

I believe this formulation comes naturally because ⍨ can be used to write functions in the order that they are applied. Another way to put it is that ⍨ reduces the need for long-scope parentheses. For example:

   (2×a) ÷⍨ (-b) (+,-) 0.5 *⍨ (b*2)-4×a×c
   ((-b) (+,-) ((b*2)-4×a×c)*0.5) ÷ 2×a

That ⍨ is easy to implement, {⍺←⍵ ⋄ ⍵ ⍺⍺ ⍺}, is neither here nor there; its value as a tool of thought is easily demonstrated.

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