APL

Iverson style array programming

https://aplwiki.com/wiki/Iverson_notation
Kenneth Iverson's notations
APL grew out of this idea

—

monadic: unary
dyadic: binary

Remarks:

Understanding the concept of ranks is very important in using APL.
Array is (more?) to APL as table is to lua.
APL evaluates expressions from right to left
* is exponentiation. Multiplication is ×

General

Comment: ⍝
Indexing starts from index origin (IO)

Operators

Reshape / shape (ρ )

https://aplwiki.com/wiki/Reshape

Two forms:

Dyadic: Reshape an array.
Monadic: Get shape of array

Dyadic examples:

> 2 4 ρ ⍳ 8
1 2 3 4
5 6 7 8

> 3 2 ρ ⍳ 6
1 2
3 4
5 6

> ⍝ Looks like truncation happens if shape doesn't fit
> 3 2 ρ ⍳ 8
1 2
3 4
5 6

Monadic example:

> ρ 2 4 ρ ⍳ 8
2 4

Index generator (⍳)

https://aplwiki.com/wiki/Index_Generator

> ⍳ 9
1 2 3 4 5 6 7 8 9

> ⍝ Set index origin so that ⍳ starts from 0
> ⎕IO ← 0
> ⍳ 9
0 1 2 3 4 5 6 7 8

Reverse along first axis (⊖)

> ⊖ ⍳ 5
5 4 3 2 1

> ⍝ Rows got reversed
> ⊖ 3 2 ρ ⍳ 6
5 6
3 4
1 2

Reverse along last axis (⌽)

> ⌽ 3 2 ρ ⍳ 6
2 1
4 3
6 5

> 3 2 ρ ⍳ 6
1 2
3 4
5 6

> ⍝ Columns got reversed
> ⌽ 3 3 ρ ⍳ 9
3 2 1
6 5 4
9 8 7

Transpose (⍉)

> ⍝ Columns got reversed
> ⍉ 2 3 ρ ⍳ 6
1 4
2 5
3 6

Drop (↓)

Drop first n elements of an array.

> 1 ↓ 1 2 3 4 5 6 7
2 3 4 5 6 7

> 2 ↓ 1 2 3 4 5 6 7
3 4 5 6 7

Take (↑)

Take first n elements of an array.

> 2 ↑ 1 2 3 4 5 6 7
1 2

> 4 ↑ 1 2 3 4 5 6 7
1 2 3 4

Replicate (/)

Duplication.
A dyadic operator.

> 2 / 3 4 5
3 3 4 4 5 5

> 1 / 3 4 5
3 4 5

> 3 / 3 4 5
3 3 3 4 4 4 5 5 5

> 2 / (3 4 5) (6 7)
3 4 5  3 4 5  6 7  6 7

Reduce along first axis (⌿)

> 2 4 ρ ι 8
1 2 3 4
5 6 7 8

> + ⌿ 2 4 ρ ι 8
6 8 10 12

First axis here is column-wise ??

Reduce along last axis (/)

> 2 4 ρ ι 8
1 2 3 4
5 6 7 8

> + / 2 4 ρ ι 8
10 26

Scan along first and last axis (⍀ and \)

>  2 4 ρ ι 8
1 2 3 4
5 6 7 8

>  + \ 2 4 ρ ι 8
1  3  6 10
5 11 18 26

>  + ⍀ 2 4 ρ ι 8
1 2  3  4
6 8 10 12

Factorial (!)

> !5
120

—

For an array, it's a map to all elements:

> 2 4 ρ ι 8
1 2 3 4
5 6 7 8

> ! 2 4 ρ ι 8
  1   2    6    24
120 720 5040 40320

Depth (≡)

Nesting depth.

> ≡ 2 4 ρ ι 8
1

> ≡ ι 8
1

Roll (?)

? n selects a number randomly from the first n numbers.

n must be greater than 0.

>  ? 0

DOMAIN ERROR
      ?0
      ^

Ceiling and floor (⌈ and ⌊)

Both are monadic.

> ⌈ 3.2
4

> ⌈ 3.5
4

> ⌈ 3
3

> ⌊ 3
3

> ⌊ 3.5
3

> ⌊ 2.9
2

Exponential (*)

> ⍳ 8
1 2 3 4 5 6 7 8

> 2 * ⍳ 8
2 4 8 16 32 64 128 256

Not yet sure how this came about though:

> * ⍳ 8
2.718281828 7.389056099 20.08553692 54.59815003 148.4131591 403.4287935
      1096.633158 2980.957987

Axis

https://aplwiki.com/wiki/Axis

Axes are dimensions along which the array is organized
Rank of an array = number of axes

≢ (monadic): tally. Number of major cells in an array. Like len() of python
≡ (monadic): depth. ie, maximum level of (array) nesting
- Depth and rank aren't same
⍤: rank
⊢: ??

Examples

> n ← 5 6 7
> n + 2
7 8 9

> +/n
18

> -/n
> ⍝ 5 - (6 - 7)
6

> ×/n
210

> */n
> ⍝ Because * is exponentiation
> ⍝ 5^(6^7) is quite a large number (a 195667-digit number)
∞

emacs

emacs: gnu-apl mode

https://github.com/lokedhs/gnu-apl-mode
Activate with M-x gnu-apl-mode
Shift to APL symbol keyboard by holding super key (ie, 'windows key')
- There are other ways like the APL-Z input method.

Keyboard

Default keyboard is like (from here):

╔════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦════╦═════════╗
║ ~  ║ !⌶ ║ @⍫ ║ #⍒ ║ $⍋ ║ %⌽ ║ ^⍉ ║ &⊖ ║ *⍟ ║ (⍱ ║ )⍲ ║ _! ║ +⌹ ║         ║
║ `◊ ║ 1¨ ║ 2¯ ║ 3< ║ 4≤ ║ 5= ║ 6≥ ║ 7> ║ 8≠ ║ 9∨ ║ 0∧ ║ -× ║ =÷ ║ BACKSP  ║
╠════╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦═╩══╦══════╣
║       ║ Q  ║ W⍹ ║ E⋸ ║ R  ║ T⍨ ║ Y¥ ║ U  ║ I⍸ ║ O⍥ ║ P⍣ ║ {⍞ ║ }⍬ ║  |⊣  ║
║  TAB  ║ q? ║ w⍵ ║ e∈ ║ r⍴ ║ t∼ ║ y↑ ║ u↓ ║ i⍳ ║ o○ ║ p⋆ ║ [← ║ ]→ ║  \⊢  ║
╠═══════╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩═╦══╩══════╣
║ (CAPS   ║ A⍶ ║ S  ║ D  ║ F  ║ G  ║ H⍙ ║ J⍤ ║ K  ║ L⌷ ║ :≡ ║ "≢ ║         ║
║  LOCK)  ║ a⍺ ║ s⌈ ║ d⌊ ║ f_ ║ g∇ ║ h∆ ║ j∘ ║ k' ║ l⎕ ║ ;⍎ ║ '⍕ ║ RETURN  ║
╠═════════╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═══╦╩═════════╣
║             ║ Z  ║ Xχ ║ C¢ ║ V  ║ B£ ║ N  ║ M  ║ <⍪ ║ >⍙ ║ ?⍠ ║          ║
║  SHIFT      ║ z⊂ ║ x⊃ ║ c∩ ║ v∪ ║ b⊥ ║ n⊤ ║ m| ║ ,⍝ ║ .⍀ ║ /⌿ ║  SHIFT   ║
╚═════════════╩════╩════╩════╩════╩════╩════╩════╩════╩════╩════╩══════════╝

(Depending on the font used, this may look different on different devices.)

vim

vim-apl: https://github.com/zoomlogo/vim-apl

Backtick is the prefix key by default:

`i gives ι
`$ gives ⍋
`r gives ρ

gnuapl

REPL:

Exit repl: )OFF
Run like: apl -s -f <file-name.apl>

Even more

Game of life with apl: https://www.youtube.com/watch?v=a9xAKttWgP4
ANN with apl: https://www.youtube.com/playlist?list=PLgTqamKi1MS3p-O0QAgjv5vt4NY5OgpiM