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)

Constants

e: *1 (ie, e¹)
pi: ○1

Operators

Symbol	Monadic	Dyadic
¯	✓
×	?	✓
÷	✓	✓
ρ	?	✓
ι	✓	?
⊖	✓
⌽	✓
⍉	✓
↓		✓
↑		✓
`/`		✓
⌿
?	✓
⌈	✓
⌊	✓
○	✓	✓
`*`	✓	✓
⍟	✓	✓
≢	✓	✓
∪	✓	✓
∩	✗	✓
⊢	✓	✗

Arithmetic

¯: unary negation
×: multiplication
÷ (dyadic): division
÷ (monadic): reciprocal

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

—

> 5 ⍴ 0
⍝ No output

> 5 ⍴ 0
0 0 0 0 0

> 2 5 ⍴ 0
0 0 0 0 0
0 0 0 0 0

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.

Maximum number is ω. Count of numbers is α:

n must be greater than 0.

>  ? 0

DOMAIN ERROR
      ?0
      ^

Random values within a range:

Vector of random floats between 0 and 1:

> ⍝ Vector of length 5
> (5 ? 100) ÷ 100
0.75 0.92 0.14 0.32 0.34

> ⍝ Vector of length 5
> (5 ? 1000) ÷ 100
0.5 9.35 0.17 0.05 3.92

3x3 matrix of random floats between 0 and 1:

> 3 3 ⍴ (9 ? 1000) ÷ 1000
0.837 0.427 0.169
0.671 0.094 0.801
0.72  0.888 0.492

Ceiling and floor (⌈ and ⌊)

Both are monadic.

> ⌈ 3.2
4

> ⌈ 3.5
4

> ⌈ 3
3

> ⌊ 3
3

> ⌊ 3.5
3

> ⌊ 2.9
2

Circle function (○)

Monadic usage: multiply arg with pi
Dyadic usage: left argument selects a trigonometric function
- Positive left argument: sin, cos, tan, etc
- Negative left argument: sinh, cosh, tanh, etc
- Left arg range: -12 to 12
- -7 to 7: circular functions
- Others: spherical functions

x	Formula
0	π * y
1	sin(y)
2	cos(y)
3	tan(y)
4	arcsin(y)
5	arccos(y)
6	arctan(y)
7	π * y
-1	sinh(y)
-2	cosh(y)
-3	tanh(y)
-4	arsinh(y)
-5	arcosh(y)
-6	artanh(y)

Exponential (*)

> ⍳ 8
1 2 3 4 5 6 7 8

> 2 * ⍳ 8
2 4 8 16 32 64 128 256

Default base is e.

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

log (⍟)

Default base is e:

> ⍟4
1.386294361

> ⍟2.717
0.9995283304

> ⍟2.71828
0.9999993273

Other bases:

> 10 ⍟ 10000
4

> 2 ⍟ 2048
11

≢

Tally (monadic): Get first dimension of a matrix

> ⍝ 3D matrix
> ≢ 2 3 4 ρ ι 24
2

> ⍝ 2D matrix
> ≢ 3 8 ρ ι 24
3

> ⍝ Vector
> ≢ ι 24
24

> ⍝ Scalar
> ≢ 4
1

Inequality (Dyadic): not-equal-to

> 3 ≢ 3
0

> 3 ≢ 4
1

> (1 2 3) ≢ (1 2 3)
0

∪

Monadic: Take unique elements only
Dyadic: Union of args, with 'duplicates of α in ω removed'

> ⍝ Monadic 
> ∪ 1 2 3 3 3 4 5
1 2 3 4 5

> ⍝ Dyadic
> 1 2 3 3 4 ∪ 4 5 
1 2 3 3 4 5

Intersection (∩)

No monadic version

> 1 2 3 3 ∩ 4 5 2 1
1 2

Right/Identity (⊢)

Monadic: Returns its argument unchanged
Dyadic: Return right argument

This operator is same as ⎕←ω.

Left/hide (⊢)

Monadic: ??
Dyadic: Return left argument

> 1 5 6 ⊣ 2 3
1 5 6

> ⊣ 2 3
⊣ 2 3

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.

User defined functions

Delimited by ∇ and ∇ (del symbol)

Format:

∇ result ← fun_name arg_name
    ⍝ body
    result ← expression
∇

Examples

Monadic:

∇ double ← square x
    double ← x × x
∇

> square 3
9

Dyadic:

∇ sum ← x add  y
    sum ← x + y
∇

> 3 add 5
8

Niladic (no args):

Cannot have more than 2 args, but can use an array arg to get around that.

∇ result ← sigmoid x
    result ← 1 ÷ 1 + *-x
∇

dfn

Pronunciation: deefns
Compact local functions, possibly anonymous
- Support in gnu-apl limited ??
Terms:
- ⍺: left arg
- ⍵: right arg
- ∇: self reference (useful for recursion)
'A dfn is a sequence of expressions inside { and }, separated by ⋄ or newlines.'
Return value: result of last evaluated expression

> cubed ← {⍵ × ⍵ × ⍵}
> cubed 3
27

> mult ← {α × ⍵}
> 4 mult 3
12

gnuapl

REPL:

Exit repl: )OFF
Run like: apl -s -f <file-name.apl>
Workspace:
- )CLEAR: clear workspace
- )LOAD myname: load saved workspace
- )DROP ws_name: delete a workspace
- )SAVE myname: save workspace
- )ERASE sym_name: delete a symbol from current workspace
- )COPY: copy some symbols
- )FNS: print names of functions defined in current workspace
- )OPS: print names of operators defined in current workspace
- )NMS: print all names defined in current workspace
- )VARS: print all user defined variables in current workspace
Run script: )IN myscript.apl
Load file: data ← ⎕FIO[49] 'test_y.csv'

Functions

⎕FIO['fopen'] instead of the less intuitive ⎕FIO[3], or even ⎕FIO[fopen]
- https://www.gnu.org/software/apl/apl-intro.html#CH_3.4.14.13
- The 3 is an axis argument

⎕UCS: Universal Character Set

Can be used to convert between encodings for characters ?? Like character to integer unicode code point?? and the other way around

From https://www.gnu.org/software/apl/apl-intro.html#CH_3.4.3:

⍝ Unicode → integer conversion
⎕UCS 'APL ⍴⍪ $¥£€⍧ ÄÖÜäöüß'
65 80 76 32 9076 9066 32 36 165 163 8364 9063 32 196 214 220 228 246 252 223

⍝ And back to Unicode
⎕UCS 65 80 76 32 9076 9066 32 36 165 163 8364 9063 32 196 214 220 22 8 246 252 223
APL ⍴⍪ $¥£€⍧ ÄÖÜäöüß

Editors

Emacs

Mode: gnu-apl-mode

https://github.com/lokedhs/gnu-apl-mode

—

Interactive session: M-x gnu-apl
Use APL-Z input method: C-x RET C-\
Show APL keyboard layout: C-c C-k or M-x gnu-apl-show-keyboard

Symbol entry:
- Use super-key if WM doesn't hijack it
  - Eg: Sup+i: ι
- Toggle input: C-\ APL-Z
  - Escape: .
  - Eg: . e becomes ∈

vim

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

Backtick is the prefix key by default:

`i gives ι
`$ gives ⍋
`r gives ρ

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.)

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
Other APL inspired language (mostly proprietary implementations): J, K, Q
- J: Like APL but uses only ASCII
  - https://github.com/jsoftware/jsource
- K3: A version of K
- Q: built on top of K4 but proprietary
Fun fact: APL inspired S, which in turn inspired R
Scripts:
- Read csv: https://www.gnu.org/software/apl/Bits_and_Pieces/CSV.apl.html
- https://www.gnu.org/software/apl/Bits_and_Pieces/Util_JSA.html

Symbol	Monadic	Dyadic
¯	✓
×	?	✓
÷	✓	✓
ρ	?	✓
ι	✓	?
⊖	✓
⌽	✓
⍉	✓
↓		✓
↑		✓
`/`		✓
⌿
?	✓
⌈	✓
⌊	✓
○	✓	✓
`*`	✓	✓
⍟	✓	✓
≢	✓	✓
∪	✓	✓
∩	✗	✓
⊢	✓	✗

Symbol	Monadic	Dyadic
¯	✓
×	?	✓
÷	✓	✓
ρ	?	✓
ι	✓	?
⊖	✓
⌽	✓
⍉	✓
↓		✓
↑		✓
`/`		✓
⌿
?	✓
⌈	✓
⌊	✓
○	✓	✓
`*`	✓	✓
⍟	✓	✓
≢	✓	✓
∪	✓	✓
∩	✗	✓
⊢	✓	✗

General

Constants

Operators

Arithmetic

Reshape / shape (ρ )

Index generator (⍳)

Reverse along first axis (⊖)

Reverse along last axis (⌽)

Transpose (⍉)

Drop (↓)

Take (↑)

Replicate (/)

Reduce along first axis (⌿)

Reduce along last axis (/)

Scan along first and last axis (⍀ and \)

Factorial (!)

Depth (≡)

Roll (?)

Ceiling and floor (⌈ and ⌊)

Circle function (○)

Exponential (*)

log (⍟)

≢

∪

Intersection (∩)

Right/Identity (⊢)

Left/hide (⊢)

Axis

More

Examples

emacs

User defined functions

Examples

dfn

gnuapl

Functions

Editors

Emacs

vim

Keyboard

Even more

Symbol	Monadic	Dyadic
¯	✓
×	?	✓
÷	✓	✓
ρ	?	✓
ι	✓	?
⊖	✓
⌽	✓
⍉	✓
↓		✓
↑		✓
`/`		✓
⌿
?	✓
⌈	✓
⌊	✓
○	✓	✓
`*`	✓	✓
⍟	✓	✓
≢	✓	✓
∪	✓	✓
∩	✗	✓
⊢	✓	✗