Hook

Hook operators in BQN

A hook is an asymmetrical form of function composition that first applies one of the composed functions to one argument, then applies the other function to one argument and the result. In J, a 2-train is a hook, while I adds the mirror image to give two functions hook (h) and backhook (H). BQN uses two modifiers Before (⊸) and After (⟜), which also serve the purpose of the Bind operator.

The meaning of 2-train as hook was first proposed in Iverson and McDonnell's paper Phrasal Forms introducing trains^[1], and soon included in J. This definition specifies that (F G) y is y F G y and x (F G) y is x F G y. However, Roger Hui later opined that this definition was better suited to a conjunction (dyadic operator) than an element of syntax,^[2] and defined to 2-train to represent Atop instead when he led the introduction of trains to Dyalog APL.

In I and BQN, there are two hooks in order to maintain symmetry: for example, BQN defines Before (⊸) to be the 2-modifier {(𝔽𝕨⊣𝕩)𝔾𝕩} ("𝔾's left argument comes from 𝔽") and After (⟜) to be {(𝕨⊣𝕩)𝔽𝔾𝕩} ("𝔽's right argument comes from 𝔾"). In the two-argument case these functions are identical to Reverse Compose and Beside respectively, but in the one-argument case they differ because the argument is used twice: the second function application takes it as an argument directly in addition to the result of the first function application.

Like Reverse Compose, the two hooks can be used together to form a split-compose construct.

    3‿¯1‿4 ×⊸×⟜| ¯2‿¯7‿1
⟨ 2 ¯7 1 ⟩

Works in: BQN

This definition behaves differently that the Compose-based one when only one argument is given: in that case, it becomes a monadic 3-train.

The name "hook" was chosen based on the hook shape of a function call diagram such as the one below, taken from Phrasal Forms.

⍺(fg)⍵ ←→ ⍺fg⍵

     f
    / \
   ⍺   g
        \
         ⍵

External links

J Dictionary, NuVoc
BQN tutorials

References

↑ Ken Iverson and Eugene McDonnell. Phrasal forms at APL89.
↑ Roger Hui. Hook Conjunction?. J Wiki essays. 2006. Accessed 2021-02-08.

APL built-ins [edit]
Primitives (Timeline)	Functions
Scalar
		Monadic	Conjugate ∙ Negate ∙ Signum ∙ Reciprocal ∙ Magnitude ∙ Exponential ∙ Natural Logarithm ∙ Floor ∙ Ceiling ∙ Factorial ∙ Not ∙ Pi Times ∙ Roll ∙ Type ∙ Imaginary ∙ Square Root ∙ Round
		Dyadic	Add ∙ Subtract ∙ Times ∙ Divide ∙ Residue ∙ Power ∙ Logarithm ∙ Minimum ∙ Maximum ∙ Binomial ∙ Comparison functions ∙ Boolean functions (And, Or, Nand, Nor) ∙ GCD ∙ LCM ∙ Circular ∙ Complex ∙ Root
Non-Scalar
		Structural	Shape ∙ Reshape ∙ Tally ∙ Depth ∙ Ravel ∙ Enlist ∙ Table ∙ Catenate ∙ Reverse ∙ Rotate ∙ Transpose ∙ Raze ∙ Mix ∙ Split ∙ Enclose ∙ Nest ∙ Cut (K) ∙ Pair ∙ Link ∙ Partitioned Enclose ∙ Partition
		Selection	First ∙ Pick ∙ Take ∙ Drop ∙ Unique ∙ Identity ∙ Stop ∙ Select ∙ Replicate ∙ Expand ∙ Set functions (Intersection ∙ Union ∙ Without) ∙ Bracket indexing ∙ Index ∙ Cartesian Product ∙ Sort
		Selector	Index generator ∙ Grade ∙ Index Of ∙ Interval Index ∙ Indices ∙ Deal ∙ Prefix and suffix vectors
		Computational	Match ∙ Not Match ∙ Membership ∙ Find ∙ Nub Sieve ∙ Encode ∙ Decode ∙ Matrix Inverse ∙ Matrix Divide ∙ Format ∙ Execute ∙ Materialise ∙ Range
Operators	Monadic	Each ∙ Commute ∙ Constant ∙ Replicate ∙ Expand ∙ Reduce ∙ Windowed Reduce ∙ Scan ∙ Outer Product ∙ Key ∙ I-Beam ∙ Spawn ∙ Function axis ∙ Identity (Null, Ident)
Operators	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)
Quad names	Index origin ∙ Comparison tolerance ∙ Migration level ∙ Atomic vector

[1] Ken Iverson and Eugene McDonnell. Phrasal forms at APL89.

[2] Roger Hui. Hook Conjunction?. J Wiki essays. 2006. Accessed 2021-02-08.

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[2]

Hook

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