# 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.

## Contents

## 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*.

### 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.

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