# 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:

{(+⌿⍵)÷≢⍵}

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 `⍵`

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 `⍵`

. The divide symbol `÷`

has its usual meaning.

The parenthesised `+⌿⍵`

denotes the sum of all the elements of `⍵`

. The `⌿`

operator combines with the `+`

function: the `⌿`

fixes the `+`

function between each element of `⍵`

, 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

{⍺,', ',⍵}⌿

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. `(+⌿⍵)`

gives the sum of the list, which is then divided by `≢⍵`

, the number elements in it.

{(+⌿⍵)÷≢⍵} 3 4.5 7 21 8.875

### Tacit programming

*Main article: Tacit programming*

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

This is a so called 3-train, also known as a *fork*. It is evaluated like this:

(+⌿ ÷ ≢) 3 4.5 7 21 |
(+⌿ 3 4.5 7 21) ÷ (≢ 3 4.5 7 21) |

Note that `+⌿`

is evaluated as a single derived function.
The general scheme for monadic 3-trains is the following:

(f g h) ⍵ |
(f ⍵) g (h ⍵) |

But other types of trains are also possible.

## 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:

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

Notice that you can read the tacit function `≠⊆⊢`

like an English sentence: *The inequality partitions the right argument*.

Many dialects do not support the above tacit syntax, and use the glyph `⊂`

for partition primitive function. In such dialects, the following formulation can be used:

(','≠s)⊂s←'comma,delimited,text'

This assigns the text to the variable `s`

, then separately computes the partitioning vector and applies it.

### 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

### 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:^{[1]}

+/ 'abcd' ∘.= 'cabbage' 2 2 1 0

### Parenthesis nesting level

"Ken was showing some slides — and one of his slides had something on it that I was later to learn was an APL one-liner. And he tossed this off as an example of the expressiveness of the APL notation. I believe the one-liner was one of the standard ones for indicating the nesting level of the parentheses in an algebraic expression. But the one-liner was very short — ten characters, something like that — and having been involved with programming things like that for a long time and realizing that it took a reasonable amount of code to do, I looked at it and said, “My God, there must be something in this language.”"

Alan Perlis.

Almost Perfect Artifacts Improve only in Small Ways: APL is more French than Englishat APL78.

What was the one-liner for the nesting level of parentheses? It would take a bit of work to figure out, because at the time of the meeting Perlis described, no APL implementation existed. Two possibilities are explained here.

#### Method A

For this more complex computation, we can expand on the previous example's use of `∘.=`

. 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

#### Method B

Alternatively, we can utilise that if the Index Of function `⍳`

doesn't find what it is looking for, it returns the next index after the last element in the the lookup array:

'ABBA'⍳'ABC' 1 2 5 '()'⍳'plus(square(a),plus(square(b),times(2,plus(a,b)))' 3 3 3 3 1 3 3 3 3 3 3 1 3 2 3 3 3 3 3 1 3 3 3 3 3 3 1 3 2 3 3 3 3 3 3 1 3 3 3 3 3 3 1 3 3 3 2 2 2

Whenever we have a 1 the parenthesis level increases, and when we have a 2 it decreases. If we have a 3, it remains as-is. We can do this mapping by indexing into these values:

1 ¯1 0['()'⍳'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:

+\1 ¯1 0['()'⍳'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

### 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

### References

- ↑ Marshall Lochbaum used this example as part of his talk on Outer Product at LambdaConf 2019.

APL development [edit]
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