# Simple examples

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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 ${\displaystyle \Leftrightarrow }$ (+⌿ 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) ⍵ ${\displaystyle \Leftrightarrow }$ (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!'
┌──┬───┐
Helo!
└──┴───┘

We use the comparison vector to partition the right argument:

','(≠⊆⊢)'comma,delimited,text'
┌─────┬─────────┬────┐
commadelimitedtext
└─────┴─────────┴────┘
Works in: Dyalog APL

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'
Works in: APL2, APLX, GNU APL

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 English at 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
Works in: all APLs

#### 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
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

### References

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