Vocabulary/at

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[x] u@v y [x] u@n y Atop Conjunction

Rank -- depends on the rank of v -- WHY IS THIS IMPORTANT?



u@v is equivalent to (u@:v)"v. That is,

  u@v y  <-->      {{u (v y)}}"mv y
x u@v y  <-->  x {{u (x v y)}}"(lv,rv) y

where mv, lv, and rv are the monadic, left, and right ranks of v, respectively. You should understand Rank (") and At (@:) before trying to understand Atop (@) .

Forms the composition of verbs u and v; the resulting verb is applied independently to each v-cell of the argument(s).

This contrasts with u@:v which applies v to the entire argument(s) and then applies u on the entire filled and assembled result of v .

  • Operand u is always executed monadically on the result of each application of v
  • Operand v is executed either monadically or dyadically depending whether  u@v has been called monadically or dyadically.
   #@> 'Newton';'Einstein'
6 8

   2 3 <@, 4 5
+-------+
|2 3 4 5|
+-------+

See: More Information for a visual comparison of At (@:), Atop (@), Compose (&) and Appose (&:).


Noun right operand (u@n)

If the right operand of @ is a noun n, the resulting verb u@n is equivalent to u@(n"_) , where the constant verb (n"_) ignores its argument(s) and produces the result n , which in turn becomes the argument to u.

The resulting verb ignores its argument(s) and is customarily called with an empty array argument, e.g. ''.

   greet=: echo@'hello'
   greet''
hello

   100 i.@4 'apple'
0 1 2 3

".@'name' is a good way to take the value of the noun name at the point of execution rather than the point of parsing.


Common uses

Implement: f(g(x)) -- the mathematical composition of the two functions: f and g.

   mean=: +/ % #
   cat=: ,&1"1
   ]z=: i.2 3       NB. sample noun
0 1 2
3 4 5
   cat z            NB. appends 1 to each row of z
0 1 2 1
3 4 5 1

   mean@cat z       NB. mean of the ROWS of cat z
1 3.25

   mean@cat b.0     NB. rank of (mean@cat)
1 1 1

You can also implement: f(g(x)) with: Compose (&) or Cap ([:) or At (@:).

But see Rank in a hurry: an insidious rank problem for how and when these different methods give different results.


Should I use @ or @:?

Suppose your y is a list of boxes and you want to create a list of the number of items in the contents of each box. You want to apply # individually on the contents of each box, so you use @ to get

   #@> 4 5 6 7;1 2;3
4 2 1

If you used @:, the interpreter would open all the boxes and join them together, then applying # just once on the assembled result:

   #@:> 4 5 6 7;1 2;3
3

Suppose your y is a list of numbers and you want to total of the squares of all the numbers. You want to apply +/ to all the squares, so you use @: to get

   +/@:*: 3 4 5
50

If you used @, +/ would be applied on each result item of *: - and because the rank of *: is 0, you would be applying +/ on individual atoms:

   +/@* 3 4 5
9 16 25

Related Primitives

At (@:), Compose (&), Appose (&:), Hook ((u v)), Fork ((f g h))


More Information

1. Phrase  (u@:v)"v means "apply v followed by u on each cell of the operand(s) independently". The rank of a cell is given by the rank of the verb v . The results from the cells are collected to produce the overall result of  u@v .

An illustration of this explanation follows, extracted from the series of diagrams referenced below:

flow diagram of the atop conjunction

Atop (u@v) , At (u@:v) , Appose (&:) , and Compose (&) are also visualized along with verb application and rank here: a series of flow diagrams

2. So what's the difference between Atop (@) and Compose (&) ?

None at all, for the monads (u@v) and (u&v)

  u&v y ↔ u v y
  u@v y ↔ u v y

But the dyads are different:

  x u&v y ↔ (v x) u (v y)
  x u@v y ↔ u x v y

According to the J Dictionary -- &: is equivalent to & except that the ranks of the resulting function are infinite; the relation is similar to that between @: and @



Oddities

The J Dictionary's definition of [x] u@v y as equivalent to  u [x] v y has caused much confusion, because the statement must be read in the context of the rank of v .

The precise definition is as above.