Bracket indexing

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Bracket indexing ([]), or simply Indexing, is a special primitive function which uses the postcircumfix notation X[Y] instead of a normal prefix function. The result of X[Y] is an array formed with items of X extracted by the index specification Y.

Indexing modes

Simple indexing

Most APL implementations support only this mode of indexing. In its simplest form, X[Y] on vector X and scalar Y extracts the item of X at index Y. In general, Y can be an array of any shape, with each item being a valid index in X; then X[Y] is a Y-shaped array which contains the indexed results.

'ABCDE'
B
'ABCDE'[2 31 2 3 4 5 1]
ABC
DEA

For higher-rank array X with rank n, the notation X[Y1;Y2;...;Yn] selects the indexes of X over each axis. If some Yk is omitted, it implies all indices of k-th axis is selected, which is equivalent to specifying (X)[k]. The resulting shape is the concatenation of shapes of Y1, Y2, ..., Yn.

A2 3 410×⍳24
10  20  30  40
50  60  70  80
90 100 110 120

130 140 150 160
170 180 190 200
210 220 230 240

A[1;1;1]
10

A[2;3 2;4 1]
240 210
200 170

A[;2;]
50  60  70  80
170 180 190 200

The major limitation of this indexing mode is that it only supports rectangular selection. For example, it is not possible to form X[1;1],X[2;2] from a matrix X by single indexing.

Choose indexing

In this mode, the index specification Y is a depth-2 nested array. Each item of Y is a vector whose length is the rank of X, and the result is a collection of items of X selected by each item of Y.

M
10 20 30 40
50 60 70 80

M[1 2]
20

M[2 2⍴⊂2 4]
80 80
80 80

M[(2 1)(1 2)]
50 20

'Z'[3⍴⊂]  ⍝ Scalar X can be indexed using enclosed empty vector
ZZZ

Reach indexing

In this mode, Y is a depth-3 nested array. Each item of Y is a vector of nested vectors which specify the index at each level of nesting (which is equivalent to the indexing by Pick). This allows to extract multiple items from a deeply nested array with a single indexing operation.

G('ABC' 1)('DEF' 2)('GHI' 3)('JKL' 4)
G2 3G,('MNO' 5)('PQR' 6)
G
┌───────┬───────┬───────┐
│┌───┬─┐│┌───┬─┐│┌───┬─┐│
││ABC1│││DEF2│││GHI3││
│└───┴─┘│└───┴─┘│└───┴─┘│
├───────┼───────┼───────┤
│┌───┬─┐│┌───┬─┐│┌───┬─┐│
││JKL4│││MNO5│││PQR6││
│└───┴─┘│└───┴─┘│└───┴─┘│
└───────┴───────┴───────┘

G[((1 2)1)((2 3)2)]
┌───┬─┐
DEF6
└───┴─┘

G[2 2⍴⊂(2 2)2]
5 5
5 5
G[⊂⊂1 1]
┌───────┐
│┌───┬─┐│
││ABC1││
│└───┴─┘│
└───────┘

Implementation support

Dyalog APL and NARS2000 support all three modes of indexing. NARS2000 even supports mixing Choose and Reach indexing modes. J does not have this notation at all.