Under

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Under (⍢), or Dual (¨), is a primitive dyadic operator which takes two function operands. It pre-processes its argument(s) with the monadic right operand, before applying the left operand on/between the result(s), then undoing the pre-processing.[1]

Model

In dialects that support user-defined operators and invertible functions, it can be defined as (or similar): 

       _U_ ← {⍺←{⍵ ⋄ ⍺⍺} ⋄ ⍵⍵⍣¯1⊢(⍵⍵ ⍺)⍺⍺(⍵⍵ ⍵)}

Examples

      ⍝ https://leetcode.com/problems/add-two-numbers/
      2 4 3 +_U_(10⊥⌽) 5 6 4
7 0 8

History

The Dual operator, written ¨, appeared in 1983 both in Rationalized APL and SHARP APL[2] just after. In both cases it was defined to be a close composition, so that the overall function has the rank of the right operand; this allows ¨> to serve the purpose of Each from a nested APL. Iverson changed the name to Under for his 1987 A Dictionary of APL, and J uses this name as well. It includes both a close form &. like SHARP and a non-close form &.:.

However, Ken Iverson had defined the Dual operator with glyph ⍢ in his 1978 paper "Operators and Functions",[3] using a non-close form because function rank had not yet been invented. NARS featured this operator in 1981. Roger Hui proposed Under with the same definition as a potential addition to Dyalog APL at Dyalog '15, and it was added to both Extended Dyalog APL and dzaima/APL in 2018. Structural Under was developed by Marshall Lochbaum around 2017,[4] and was included in Extended Dyalog initially and added to dzaima/APL in 2020. Later dzaima would implement ⍢× and ⍢| in the same spirit, now featured in dzaima/APL and dzaima/BQN. The language BQN has included structural and computational Under, using the glyph ⌾, since its initial design in 2020.

External links

References

  1. ↑ BQN: Circle Jot (⌾)
  2. ↑ "Language Extensions of May 1983". SATN-45, 1983-05-02.
  3. ↑ Ken Iverson. Operators and Functions, §8 Composition and Duality. IBM Research Report #RC7091. 1978-04-26.
  4. ↑ Marshall Lochbaum. BQN's development history: Structural Under.
APL built-ins [edit]
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Dyadic Bind ∙ Compositions (Compose, Reverse Compose, Beside, Withe, Atop, Over) ∙ Inner Product ∙ Determinant ∙ Power ∙ At ∙ Under ∙ Rank ∙ Depth ∙ Variant ∙ Stencil ∙ Cut ∙ Direct definition (operator) ∙ Identity (Lev, Dex)
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