Array Language Comparisons

In the table below, the + should be taken to mean any binary operand, like +, or -, or >. In the case where both arguments are collections, their shapes must agree in some way which is language-dependent. Hopefully I got all these characteristics right, but I'm less sure about MATLAB since I haven't used it for several years, nor Julia since I have not used it much at all.

These languages all support operations on (s)calars, (v)ectors, and higher-dimensional (m)atrices:

language	s+s	s+v	vk+vk	vk+ml×k	vk+mk×l	mk×l+mk×l	mk×l+ml×k
APL	y	y	e	n	y	e	leading
J	y	y	e	n	y	e	leading
K	y	y	e	n	y	e	leading
NumPy	y	y	e	y	n	e	trailing
Julia	y	y	e	y	n	e	trailing
MATLAB	y	y	e	n	y	e	leading

(NOTE: "e" in the table above means "element-wise")

They're also usually interpreted:

language	aot-compiled	jit-compiled	interpreted	static types	dynamic types
APL	co-dfns only	n	y	n	y
J	n	n	y	n	y
K	n	kx	y	n	y
NumPy	bytecode	pypy,numba	y	annotations	y
Julia	optional	y	y	annotations	y
MATLAB	optional	y	y	n	y

And differ slightly in their approaches to programming:

language	named functions	lambdas	scope	closures
APL	y (tradfns)	y (dnfs)	dynamic (dfns:lexical)	n
J	n	y	global, local	n
K	n	y	global, local	n
NumPy	y	y	dynamic (functions:lexical)	y (prefer classes)
Julia	y	y	lexical	y
MATLAB	y	y	dynamic	y (prefer classdef)

Isn't it nice to not have to write out a for-loop? The following is valid in APL, J, K, and some others:

   12 + 2 3 4
14 15 16

MATLAB and Octave do the same thing with a tiny bit more syntax in the form of square brackets:

octave> 12 + [2 3 4]
ans =
   14   15   16

BQN is similar, but has a special character for "stranding" a collection of values together (‿), as well as general list notation (⟨⟩):

12 + 2‿3‿4
⟨ 14 15 16 ⟩

NumPy is a little farther away due to being a library rather than built-in syntax, but if you squint you may still see it:

>>> a = numpy.array([2,3,4])
>>> 12 + a
array([14, 15, 16])

These language have different terminology for this behavior. Some K documentation says the + "is pervasive" or "penetrates" down to the elements of the arguments. J refers to this in terms of rank: "the + verb has rank 0". NumPy, Julia, and call this "broadcasting".

In the example (12 + 2 3 4) you could imagine it executing in one of two ways:

1. the operation 12+x is mapped over each "x" in 2 3 4
2. 12 is replicated and reshaped into 12 12 12, which adds elementwise with 2 3 4

Broadcasting is usually explained with the second formulation, but internally most array language implementations have some kinds of optimizations to avoid unnecessarily allocating memory for temporary values.

These languages also support adding collections to other collections when the shapes are identical:

   2 3 4 + 10 5 2
12 8 6

This is simply an elementwise operation. If the shapes differ, all of these languages consider it an error.

It gets a little more interesting when the shapes are not identical. Here's an example in J showing which shapes are compatible (i.y returns a range of numbers up to the product of y, with shape y):

   i.2 3  NB. iota with y argument (2 3)
0 1 2
3 4 5
   12 12 + i.2 3
12 13 14
15 16 17
   12 12 + i.3 2
|length error
|   12 12    +i.3 2
   12 12 12 + i.3 2
12 13
14 15
16 17

In the first addition, left argument 12 12 can add with the 2x3 right argument because the shape of the leading axes match. Similarly the last example adds 12 12 12 to a 3x2 argument. But when adding 12 12 to a 3x2 array, J makes a "length error" because the shapes are not compatible according to its rules.

Broadcasting in NumPy works a little differently:

>>> a23 = np.arange(6).reshape((2,3))  # 2x3 array
>>> a32 = np.arange(6).reshape((3,2))  # 3x2 array
>>> [12,12] + a32
array([[12, 13],
       [14, 15],
       [16, 17]])
>>> [12,12,12] + a23
array([[12, 13, 14],
       [15, 16, 17]])
>>> [12,12] + a23
ValueError: operands could not be broadcast together with shapes (2,) (2,3)

In NumPy style broadcasting, the trailing axes must match. Actually it's a little more complicated than this, and there are broadcasting rules which determine if the operation is valid. Broadcasting conceptually starts with trailing axis matching, but if there is a mismatch and the other dimension is a singleton dimension (i.e. the 1 dimension in a 2x1x3 tensor), then the singleton data is replicated to match the other dimension.

By contrast, J-style rank applies an operation to what they refer to as cells of the arguments. When the shapes are equal, the operation applies elementwise. Otherwise, the operation must match the leading axes of the cells of each argument. This means you can add a tensor with shape 2x3x4 to another with shape 2x3, but not shape 3x4 or even 2x3x3, but you can also override this rank and apply the + operation at a different level.

Finally, K emulates multi-dimensional data using vectors-of-vectors. It applies operations by default to the finest granularity, to leaf nodes of the structure, but if you want to override this then you must use different control structures to apply to "each", "eachleft", "eachright", or "eachprior". These do not cover all cases in the same general way as either broadcasting or rank, but in practice obtaining equivalent results is rare, and only slightly more awkward.