Simple examples

This page contains examples that show APL's strengths. The examples require minimal background and have no special dependencies. If these examples are too simple for you, have a look at our advanced examples.

Arithmetic mean

Here is an APL program to calculate the average (arithmetic mean) of a list of numbers, written as a dfn:

      {(+⌿ω)÷≢ω}

It is unnamed: the enclosing braces mark it as a function definition. It can be assigned a name for use later, or used anonymously in a more complex expression.

The ω refers to the argument of the function, a list (or 1-dimensional array) of numbers. The ≢ denotes the tally function, which returns here the length of (number of elements in) the argument ω. The divide symbol ÷ has its usual meaning.

The parenthesised +⌿ω denotes the sum of all the elements of ω. The ⌿ operator combines with the + function: the ⌿ fixes the + function between each element of ω, so that

      +⌿ 1 2 3 4 5 6
21

is the same as

      1+2+3+4+5+6
21

Operators

Operators like ⌿ can be used to derive new functions not only from primitive functions like +, but also from defined functions. For example

      {α,', ',ω}⌿

will transform a list of strings representing words into a comma-separated list:

      {⍺,', ',⍵}⌿'cow' 'sheep' 'cat' 'dog'
┌────────────────────┐
│cow, sheep, cat, dog│
└────────────────────┘

So back to our mean example. (+⌿ω) gives the sum of the list, which is then divided by ≢ω, the number elements in it.

      {(+⌿)÷≢ω} 3 4.5 7 21
8.875

Tacit programming

Main article: Tacit

In APL’s tacit definition, no braces are needed to mark the definition of a function: primitive functions just combine in a way that enables us to omit any reference to the function arguments — hence tacit. Here is the same calculation written tacitly:

      (+⌿÷≢) 3 4.5 7 21
8.875

The operator ⌿ can also be used to modify the (+⌿÷≢) function to produce a moving average.

      2 (+⌿÷≢)/ 3 4.5 7 21
3.75 5.75 14

or, more verbosely

      ave ← +⌿÷≢
      ave 3 4.5 7 21
8.875
      mave ← ave⌿
      2 mave 3 4.5 7 21
3.75 5.75 14

Text processing

APL represents text as character lists (vectors), making many text operations trivial.

Split text by delimiter

≠ gives 1 for true and 0 for false. It pairs up a single element argument with all the elements of the other arguments:

      ','≠'comma,delimited,text'
1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 0 1 1 1 1

⊢ returns its right argument:

          ','⊢'comma,delimited,text'
comma,delimited,text

⊆ returns a list of runs as indicated by runs of 1s, leaving out elements indicated by 0s:

      1 1 0 1 1 1⊆'Hello!'
┌──┬───┐
│He│lo!│
└──┴───┘

We use the comparison vector to partition the right argument:

Try it now!

      ','(≠⊆⊢)'comma,delimited,text'
┌─────┬─────────┬────┐
│comma│delimited│text│
└─────┴─────────┴────┘

Works in: Dyalog APL

Notice of you can read the tacit function ≠⊆⊢ like an English sentence: The inequality partitions the right argument.

Indices of multiple elements

∊ gives us a mask for elements (characters) in the left argument that are members of the right argument:

      'mississippi'∊'sp'
0 0 1 1 0 1 1 0 1 1 0

⍸ gives us the indices where true (1):

      ⍸'mississippi'∊'sp'
3 4 6 7 9 10

We can combine this into an anonymous infix (dyadic) function:

      'mississippi' (⍸∊) 'sp'
3 4 6 7 9 10

Works in: Dyalog APL, NARS2000, dzaima/APL

Frequency of characters in a string

The Outer Product allows for an intuitive way to compute the occurrence of characters at a given location in a string:

      'abcd' ∘.= 'cabbage'
 0 1 0 0 1 0 0
 0 0 1 1 0 0 0
 1 0 0 0 0 0 0
 0 0 0 0 0 0 0

Then it is simply a matter of performing a sum-reduce +/ to calculate the total frequency of each character:

      +/ 'abcd' ∘.= 'cabbage'
 2 2 1 0

Parenthesis nesting level

We can expand on the use of ∘. in the above example to perform more complex calculations. First we compare all characters to the opening and closing characters;

      '()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1

An opening increases the current level, while a closing decreases, so we convert this to changes (or deltas) by subtracting the bottom row from the top row:

      -⌿'()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
0 0 0 0 1 0 0 0 0 0 0 1 0 ¯1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 ¯1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 ¯1 ¯1 ¯1

The running sum is what we're looking for:

      +\-⌿'()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
0 0 0 0 1 1 1 1 1 1 1 2 2 1 1 1 1 1 1 2 2 2 2 2 2 2 3 3 2 2 2 2 2 2 2 3 3 3 3 3 3 3 4 4 4 4 3 2 1

Works in: all APLs

Grille cypher

A grille is a 500 year old method for encrypting messages.

Represent both the grid of letters and the grille as character matrices.

      ⎕←(grid grille)←5 5∘⍴¨'VRYIALCLQIFKNEVPLARKMPLFF' '⌺⌺⌺ ⌺ ⌺⌺⌺ ⌺ ⌺ ⌺⌺⌺ ⌺⌺⌺  ⌺⌺'
┌─────┬─────┐
│VRYIA│⌺⌺⌺ ⌺│
│LCLQI│ ⌺⌺⌺ │
│FKNEV│⌺ ⌺ ⌺│
│PLARK│⌺⌺ ⌺⌺│
│MPLFF│⌺  ⌺⌺│
└─────┴─────┘

Retrieve elements of the grid where there are spaces in the grille.

      grid[⍸grille=' ']
ILIKEAPL

An alternative method using ravel.

      (' '=,grille)/,grid
ILIKEAPL