Vocabulary/hook

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[x] (u v) y Hook Invisible Conjunction

Rank Infinity -- operates on [x and] y as a whole -- WHY IS THIS IMPORTANT?

Makes a single new verb out of the two verbs: u v which equates to the phrase: (x or y) u v y

 u=: * NB. Dyad: multiply left and right args
 v=: +&1 NB. Monad: add 1 to y
 y=: 7
 v y NB. action of v alone on y [7+1]
8
 y u y NB. action of u alone on y and y [7*7]
49
 y u (v y) NB. action of first applying v to right-arg of u [7*8]
56
 y u v y NB. J works right to left, hence (...) redundant
56
 (u v) y NB. make Hook out of u and v
56
 nn1=: u v NB. common math phrase: n(n+1) defined as a Hook
 nn1 y
56
 x=: 10
 x u v y NB. without hook
80
 x (u v) y NB. with hook; same result, but (u v) can be used within longer tacit expression
80

Details

Two adjacent verbs when not followed by a noun (as in [x] (u v) y) create a hook that executes v followed by u:

We are talking here about after modifiers have been applied, so that there are only verbs and nouns. ({.~ 2&{) is a hook made from the two verbs {.~ and 2&{ .

x (u v) y ⇔ x u v y result
(u v) y ⇔ y u v y |
 u
 / \
 x or y v
 |
 y

There is a huge difference between

 i. # 3 1 4 1 5 9
0 1 2 3 4 5

and

 (i. #) 3 1 4 1 5 9 NB. same as 3 1 4 1 5 9 i. # 3 1 4 1 5 9
6

The first is simply right-to-left execution of # 3 1 4 1 5 9 followed by i. 6 . The second creates a hook out of the verbs (i. #) and then executes that according to the definition above.

A Beginner's Error

When you find two verbs that perform a function you like, like i. and # in the first example above which creates a list of ascending numbers the same length as y, you might try to make a single verb out of them, with

 numberforeach =: i. # NB. create a list of numbers with length #y

but you will be disappointed when you use it:

 numberforeach 3 1 4 1 5 9
6

The assignment to the name numberforeach created a hook, because the two adjacent verbs were not followed by a noun.

One way to think of this is that when substituting for a name, like numberforeach here, you must put parentheses around the substitution to avoid strange effects. This is akin to ordinary mathematics, where if you know that y=x+2 and you want to substitute that into z=y^2^+2y+5, you use parentheses to get z=(x+2)^2^+2(x+2)+5. In the sentence

 numberforeach 3 1 4 1 5 9

when you substitute for the name, you end up with

 (i. #) 3 1 4 1 5 9

which is a hook and not at all what you wanted.

To do what you wanted you should have written numberforeach =: i.@:#

The substitution model described above is not at all how J actually executes sentences, but it will get you the right answers.

Common uses

1. To fetch the letters of y before the first '$'

 ({.~ i.&'$') 'abc$defg'
abc
 'abc$defg' {.~ i.&'$' 'abc$defg' NB. Same thing without the hook
abc

2. To reshape y to exactly the shape of x

 2 2 3 ($ ,) i. 2 4
0 1 2
3 4 5
6 7 0
1 2 3
 2 2 3 $ , i. 2 4
0 1 2
3 4 5
6 7 0
1 2 3

3. Combine a noun n with a modified form of itself. Example: n*(n+1)

 nn1=: * +&1 NB. a Hook formed from (*) and (+&1)
 nn1 7
56

More Information

1. When a hook appears in a tacit definition, where it is creating a verb (and therefore not being executed on a noun), it doesn't have to be enclosed in parentheses:

 fetchdollar =: {.~ i.&'$'
 fetchdollar 'first$string'
first

2. In a normal sentence that is executing on noun arguments to produce a noun result, a hook must be enclosed in parentheses

 (#: i.@(*/)) 3 2 NB. hook: odometer function
0 0
0 1
1 0
1 1
2 0
2 1
 #: i.@(*/) 3 2 NB. no hook, just right-to-left execution
0 0 0
0 0 1
0 1 0
0 1 1
1 0 0
1 0 1

3. When there are an even number of verbs in a row, the leftmost one will be the u side of a hook, with the v side coming from the other verbs, interpreted as forks. See the discussion of forks for details.

 ({.~ 3 <. #) 1 2 3 4 5 NB. Take up to 3 items of y
1 2 3
 ({.~ 3 <. #) 1 2
1 2

Related Primitives

Fork (f g h), Atop (u@v), At (u@:v), Compose (u&v), Appose (u&:v), Under (u&.v), Under (u&.:v)