J Notation as a Tool of Thought

> Rank Is actually implicitly done by numpy using a mechanism called broadcasting.

Numpy's broadcasting is scalar conformability, to which the rank operator (and general conformability) provides a general case. Example in j:

     ] x =. i. 2 3 4                                                                                                                   
   0  1  2  3
   4  5  6  7
   8  9 10 11

  12 13 14 15
  16 17 18 19
  20 21 22 23
     ] y =. i. 3 4
  0 1  2  3
  4 5  6  7
  8 9 10 11
     x + y         NB. this will error because + expects that, if its arguments' shapes are not the same, one will be a prefix of the other
  |length error
  |   x    +y
                   NB. this is easy enough to fix, however
     x +"2 y       NB. +"2 is shorthand for +"2 2; meaning, choose rank-2 arrays from both the left and right arguments
   0  2  4  6
   8 10 12 14
  16 18 20 22

  12 14 16 18
  20 22 24 26
  28 30 32 34

Numpy will actually do this without the rank operator, because it uses suffix agreement rather than prefix agreement (which is absolutely bonkers—j used suffix agreement for about 5 minutes in 1990, before realising it was an awful idea). For for numpy, see if you can add:

  np.array([[[0, 1, 2, 3], [4, 5, 6, 7], [8, 9, 10, 11]], [[12, 13, 14, 15], [16, 17, 18, 19], [20, 21, 22, 23]]]) + np.array([[0, 1, 2], [3, 4, 5]])

Intelligently. (I'm sure it's not overly difficult to come up with a solution, but can you do it with a single higher-order function call which generalises to other argument shapes?)

(The j equivalent, (i. 2 3 4) + (i. 2 3) also works without trouble.)

  ------------------------------------------------------------------------

Another example, which may be more illustrative, is the ability to perform reductions along arbitrary axes. For example:

     ] x =. i. 4 3
  0  1  2
  3  4  5
  6  7  8
  9 10 11
     +/ x       NB. sum reduced along leading axis, the default, producing an array of shape 3
  18 22 26
     +/"1 x    NB. sum each rank-1 array (vector); or, reduce last axis, producing an array of shape 4
  3 12 21 30

  ------------------------------------------------------------------------

Another curiosity, which I have thus far neglected, is the extent to which numpy's being 'a little more verbose' is actually incredibly important in shaping the way you approach and think about problems. The great-uncle comment also addresses this, but Iverson probably says it better than either of us can: read https://www.jsoftware.com/papers/tot.htm

This seems very interesting for numerical computation, but at first glance few of these operations seem to apply directly to general computation (I'm sure that there are isomorphisms that can be used to apply them indirectly).

The example with sorting was the oddest from this point of view. The author praised the idea of the permutation vector as being a relatively direct mathematical specification of sorting (return the permutation of the original array in sorted order) while leaving out the 'sorted' part. Also, this focuses entirely on the result, but there is no word on the algorithm - 'how is the array sorted' is the question that computer science is designed to solve, as opposed to 'what are the fixed points of a sorted array'.

As such, as someone with no experience of J, I'm left wondering if this is another example of the infamous Haskell quicksort - well-picked examples of code which produces a desired result are extraordinarily terse and expressive, but actually implementing well-known algorithms is often just as verbose as C, give or take.

If you run the J GUI, written in J, it is going with blow your mind about how general purpose it is...

> there is no word on the algorithm - 'how is the array sorted' is the question that computer science is designed to solve, as opposed to 'what are the fixed points of a sorted array'

'[H]ow is the array sorted' is one question that computer science can be used to solve. However, assuming it has been solved (which, for many cases, it has), a much more interesting question is 'what can we do with a sorted array'; or, 'given that sorting has such-and-such time or memory complexity, how hard is this other algorithm?' The OP isn't showing off a particular sorting algorithm, he's showing off a sorting interface, because that's more novel in context.

Sorting according to a predicate could be done with outer product/table+reduction, but usually you don't need anything like that. Let's say you have an array of 3-vectors, and you want to sort the vectors according to the 2nd item in each vector. So your sort predicate is v1[1]

     ] x =. ?@] 9 3$10                                                                                                                 
  2 4 9
  3 8 2
  8 0 4
  3 9 6
  3 1 8
  2 6 8
  1 2 5
  9 4 0
  5 0 9
     (