You've reproduced the most trivial example, which many languages make easy. I would be very interested to see another language with a rank operator, for instance.

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

Challenge for you: rewrite a nontrivial program in one of those frameworks, with the following restrictions:

- No iteration (including implicit iterations—map, filter; reduce is ok)

- No loops whatsoever. Recursion is ok, but should be avoided wherever possible.

- No explicitly named arguments; everything in pointfree style.

(I know, map/filter can be implemented recursively. But compared with the equivalent constructs in apl, they're about 10× as verbose, and harder to reason about and understand. Even reduce is somewhat of a gimme.)

Rank Is actually implicitly done by numpy using a mechanism called broadcasting. For example:

  >>> np.array([10, 20, 30]) + np.array([[1,2,3], [4,5,6], [7,8,9]])
  array([[11, 22, 33],
       [14, 25, 36],
       [17, 28, 39]])

Sieves exist in numpy, called masks:

  >>>np.array([10, 20, 30]) > 15
  array([False, True, True])

Of course they can be operated on just like any other numpy array.

Grades exist in numpy:

  >>>np.array([5,4,3,2,1]).argsort()
  array([4, 3, 2, 1, 0])

Of course they are a little more verbose since all of those operations are from the library and not native to python.

To do sieves in J like the blog post mentioned, the equivalent numpy would be:

>>> arr = np.array([10, 20, 30])

>>> arr[arr > 15]

array([20, 30])

> 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

What is prefix/suffix agreement? Could you give a simple example of why prefix beats suffix?

I'm curious and have never heard these terms before, and googling didn't help.

To broadcast operations (such as addition) between arrays in NumPy, trailing dimensions have to be equal (or be of length 1).

In the example given above the 3D array and 2D array have shape (lengths of dimensions):

   (2, 3, 4)
      (2, 3)

That is - the suffixes do not agree (4 != 3 and 3 != 2) and NumPy raises an error.

However, for the same operation in J the prefixes agree:

   (2, 3, 4)
   (2, 3)

and the addition gives the expected result.

To add the arrays with these shapes in NumPy, one method is transpose each array (reverse order of the dimensions), add these arrays, and then transpose back:

  (a.T + b.T).T

Thinking about these as a "verbose imperative language" programmer, I'd say suffix agreement seems to make more sense to me at first glance, and I'd like to hear more about why it might be worse.

For why I think it makes sense: I think of multidimensional arrays as being arrays of arrays, and "normal" index lookup operating on the first dimension. If I have a

    float[100][3]

in some context I might think of it as 100 vec3s, and I might want to do some vec3 operation on each of them. I might want to dot them all with my some other vector, or add them all to some other vector. I almost never have 100 scalars and want to apply one scalar to all elements of the corresponding vec3.

But I guess maybe this is all widely agreed on, and maybe the contentious part is just index order? Like, maybe you'd say "100 vec3s" is actually

    float[3][100]

in which case prefix agreement would make more sense.

Sibling explains the difference. To explain why prefix is better—first let's look at the base case where argument shapes match. Let's assume that addition is defined on scalars; we don't have to explain that 11=5+6. Then let's consider vectors: there are two forms of scalar broadcasting for vectors. The first is addition of two equal-length vectors, for which each item of the left argument will be broadcast to the corresponding item of the right argument: 10 14=3 5+7 9. The second is the addition of a scalar to a vector (or vice versa), where the scalar is matched up with each item of the vector: 10 14=6 10+4.

For the first case, we can say that:

  R[i] = x[i] + y[i]  (when x and y are vectors)

(Where i is any valid array subscript; and R, x, and y are the names conventionally given to the result, left argument, and right argument of some function, respectively.)

Assume that we extend our scalar rule to arbitrary dimensions (that is, scalar+n-dimensional array will do what we expect)—there is actually a good reason for this, but we'll just assume it for now; it's a pretty intuitive rule.

Then the simple recursive rule I gave above gives you prefix agreement for any two argument shapes; just replace 'vector' with 'nonscalar'. Here are the rules for suffix agreement:

  R[i] = x[i] + y[i]   (when x and y have the same rank)
  R[i] = x    + y[i]   (when y has bigger rank)
  R[i] = x[i] + y      (when x has bigger rank)

The prefix agreement rule is simple: add each element of x to its corresponding element in y. The suffix agreement rule is much more complex (3 rules, as opposed to just 1), and with higher-ranked arrays it gets harder to reason about which elements go together.

There's an even deeper synergy, though, which goes between prefix agreement and forks. Forks are a generalisation of a convention from calculus. There's a convention that, given functions f and g, (f+g) is a legal function such that (f+g)(x) <=> f(x)+g(x). (And ditto for other arithmetic operators.) Apl does not distinguish between user-defined functions (like f) and infix builtins (like +); x f y denotes the calling of function f with arguments x and y, same as x + y denotes the calling of function + with arguments x and y. So the f+g rule is generalised: we say that, given any functions f, g, and h, (f g h) x is equivalent to (f x) g (h x).

What does this have to do with prefix agreement? Well, all we have to do is remember that an array, as a relation of indices to elements, is very close to a function. In fact, I think that it's appropriate to say that an array is semantically a function, though its syntactic role is different. So saying x[i] is like saying 'apply function x to argument i'. Which leads very nicely—right into prefix agreement:

  (f g h) x  <=> (f x)  g (h x)
  . . . 
  (x + y)[i] <=> (x[i]) + (y[i])

> are a little more verbose

That's putting it mildly.

NOTABUG CLOSED WONTFIX

I have want to chime in. Agree, it is great that ideas from APL have made it into the language... But there is, fortunately, a long and fruitful path still to explore... I love that J is like an alternative path to functional approaches like Haskell and how it plays in terms of providing expressive power. Still, I will not create code in J that is maintained jointly with other users from other domains (which is the case usually when you are using numpy, for example)

I like how APL made it into Common Lisp wholesale. There's a library that compiles APL to Common Lisp, allowing you to mix APL and Lisp code, working on the same data structures with both languages.

https://github.com/phantomics/april

Example session I just did, copying APL code from https://aplwiki.com/wiki/John_Scholes%27_Conway%27s_Game_of_...:

  CL-USER> (april:with-april-context ((:space *lifespace*))
             (april:april "Show←{'·⌺'[⎕IO+⍵]}")
             (april:april "glider←3 3⍴1 1 1 1 0 0 0 1 0")
             (april:april-f "Show glider")
             (print (april:april "glider")) ; to prove it's a Lisp array
             (terpri)
             (april:april "grid←¯10 ¯10↑glider")
             (april:april-f "Show grid")
             (terpri)
             (april:april "life ← {⊃1 ⍵ ∨.∧ 3 4 = +/ +⌿ ¯1 0 1 ∘.⊖ ¯1 0 1 ⌽¨ ⊂⍵}")
             (terpri)
             (april:april-f "Show¨{life⍣⍵⊢grid}¨0,⍳3")
             (terpri))
  ⌺⌺⌺
  ⌺··
  ·⌺·

  #2A((1 1 1) (1 0 0) (0 1 0))
  ··········
  ··········
  ··········
  ··········
  ··········
  ··········
  ··········
  ·······⌺⌺⌺
  ·······⌺··
  ········⌺·


   ··········  ··········  ··········  ··········
   ··········  ··········  ··········  ··········
   ··········  ··········  ··········  ··········
   ··········  ··········  ··········  ··········
   ··········  ··········  ··········  ··········
   ··········  ··········  ··········  ··········
   ··········  ········⌺·  ·······⌺⌺·  ·······⌺⌺·
   ·······⌺⌺⌺  ·······⌺⌺·  ·······⌺·⌺  ······⌺⌺··
   ·······⌺··  ·······⌺·⌺  ·······⌺··  ········⌺·
   ········⌺·  ··········  ··········  ··········

(APL being compiled to Lisp is important, not just because it gives you proper interoperability out of the box, but also because on implementations like SBCL this means APL code will get compiled, through Lisp, to native code. And you can do that at runtime too.)

Now I wish I could find something that does the same for J, because I'm sure as hell not going to buy an APL-friendly keyboard.

> APL being compiled to Lisp is important, not just because it gives you proper interoperability out of the box, but also because on implementations like SBCL this means APL code will get compiled, through Lisp, to native code

Except that simd in lisp isn't great and needs to be done mostly by hand[1]; and simd is super important for making apl performant. Also because gc attributes that are good for lisp aren't great for apl. Also because compiling to native code doesn't actually really affect performance of apl, and might actually harm it for larger programs.

To be clear, april is a super-cool project, but not because of any of its performance characteristics.

> Now I wish I could find something that does the same for J, because I'm sure as hell not going to buy an APL-friendly keyboard.

You don't need a special keyboard, just a layout. I don't know how it works on windows (there's an 'IME', I think?) or macos, but x comes with a keyboard layout. I use this:

  setxkbmap -layout us,apl -variant dyalog -option grp:rwin_switch

Which makes it so that holding down the right super key and pressing any key produces an appropriate apl symbol. You may want to use a different key to switch; check 'man xkeyboard-config' and then search for 'Switching to another layout' for other options. Try: modifier key (whatever it is)+r, for ⍴ (rho).

1. https://www.reddit.com/r/lisp/comments/fkfgjn/sbcl_with_simd...

Beautiful that it can get actually compiled... I dreamt of that, but in particular, with J, it'd be very hard as the actual implementation is data driven, i.e., lexical analysis alone would require building a kind of interpreter for chained operations to work efficiently under all circumstances or compile all possible variants of execution.

Emacs has an excellent APL keyboard implementation (via gnu-apl-mode) which uses . as a prefix for the apl keys.

Regarding the keyboard, I use something like this https://unix.stackexchange.com/questions/547705/finding-apl-... and well, a picture of the mappings, but somehow they kind of end sticking in your head...

Comparing numpy to J is like comparing a dirty rag to a designer suit. The examples you've listed are trivial; the power of J and other array languages cannot be appreciated from afar.

The philosophy behind array languages runs much deeper than adding two arrays or transposing matrices.

Can you give an example of something that can be done in J that can't be done easily in numpy?

I have only passing experience with numpy, but I suspect the following "shift a matrix in all 8 directions", used in the famous APL/J game of life solution [0][1], would require multiple statements as well as being more verbose in numpy (I'm guessing you'd need to use "roll" multiple times?):

    (>,{;~i:1) |. i.4 4

Try it online: https://tio.run/##y/r/PzU5I19Bw06n2rou08pQU6FGTyFTz0TB5P9/AA

[0] https://www.youtube.com/watch?v=a9xAKttWgP4&t=5s

[1] https://tio.run/##y/qvpKeepmBrpaCuoKNgoGAFxLp6Cs5BPm7/NWJtjb...

This 100 character version is probably the cleanest.

    M = np.reshape(np.arange(16),(4,4))
    np.array([np.roll(M,(i,j),(0,1)) 
              for i in [1,0,-1]
              for j in [1,0,-1]])

That requires you to break out of the array paradigm and drop back down to procedural looping. Which is what you'd expect since numpy is array operations bolted onto python as a library and J is array operations baked into the language itself.

It's more about expressiveness and the ability to stay at a "high level" than actually the ability to just do some computation that is really hard elsewhere.

After all, most programming languages can do "the same stuff" - but no one would compare Haskell to PHP.

In most of the examples in the article, the author is using just one or two verbs put together. Imagine if all of the verbs in your language worked seamlessly at the "concept" level, rather than the "write the loop" level.

Even functional constructs like map/filter/reduce take up a lot of words. One or two characters can do the same thing, and be read faster and more idiomatically by an astute practitioner.

That's my take on it (as an on and off Q/Kdb user, not a J dude -- yet..)

It is not about "cannot be done". Most high-level languages are Turing complete, hence every program in one can be written as some program in another.

The real clincher is expressiveness: shorter code is better, modular code is better, and languages which avoid unnecessary repetitiveness of boiler plate code is better. Different languages provide advantages along one or more of these parameters.

So can you give an example of something that can be expressed simply in APL/J/K that can't be expressed simply with numpy?

And to be clear I don't think "less characters" makes the expression more simple. Maybe "less statements" or "less operators"

Here's Conway's Game of Life by Arthur Whitney (creator of K):

  life:{3=a-x*4=a:2{+(0+':x)+1_x,0}/x}

I'm guessing a numpy implementation would be between one and two orders of magnitude more verbose, even if you're just comparing the number of operations.

It doesn't quite seem fair to compare numpy to one of the most famous K code golf's ever created, but here is my attempt, 140 characters compared to Arthur's 30, so 5 times longer. Not worry about padding the edges would cut it to 100. I don't think the gap is as wide as you think it is.

  def life(M):
    MP = np.pad(M,[(1,1),(1,1)])
    N = sum(np.roll(MP,(i,j),(0,1))
             for i in [1,0,-1]
             for j in [1,0,-1])
    return (3==N-MP*(4==N))[1:-1,1:-1]

Speedtest (presumably memory bound):

  100 x 100 matrix, numpy  0.2ms, K  0.2ms
  200 x 200 matrix, numpy  0.5ms, K  0.8ms
  500 x 500 matrix, numpy  5.2ms, K  8.0ms
  1000x1000 matrix, numpy 20.0ms, K 36.0ms
  5000x5000 matrix, numpy  0.5s , K  1.2s

Interestingly the K code is designed to return a boolean matrix, but operates much more slowly for me on a boolean matrix compared to an integer matrix, with the result that:

  q)\t klife M

Takes 1.2 seconds whilst

  q)\t klife klife M

Takes 24 seconds. So whilst the idea seems to be that klife is used with scan/over to run a number of iterations, it's actually a bad idea to run it more than once.

Exactly, one of the things I like about numpy is how it drew from APL and J. It is a more verbose language, and truth be told, J still has more expressive power (non regular processing of data, even state-machine-like processing) which can still influence numpy, or Julia.