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Here is an APL program to calculate the average (arithmetic mean) of a list of numbers, written as a [[dfn]]:
Here is an APL program to calculate the average (arithmetic mean) of a list of numbers, written as a [[dfn]]:
<source lang=apl>
<source lang=apl>
       {(+⌿ω)÷≢ω}  
       {(+⌿⍵)÷≢⍵}  
</source>
</source>
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.
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 <source lang=apl inline>ω</source> refers to the argument of the function, a list (or 1-dimensional array) of numbers. The <source lang=apl inline>≢</source> denotes the [[tally]] function, which returns here the length of (number of elements in) the argument <source lang=apl inline>ω</source>. The divide symbol <source lang=apl inline>÷</source> has its usual meaning.
The <source lang=apl inline></source> refers to the argument of the function, a list (or 1-dimensional array) of numbers. The <source lang=apl inline>≢</source> denotes the [[tally]] function, which returns here the length of (number of elements in) the argument <source lang=apl inline></source>. The divide symbol <source lang=apl inline>÷</source> has its usual meaning.


The parenthesised <source lang=apl inline>+⌿ω</source> denotes the sum of all the elements of <source lang=apl inline>ω</source>. The <source lang=apl inline>⌿</source> operator combines with the <source lang=apl inline>+</source> function: the <source lang=apl inline>⌿</source> fixes the <source lang=apl inline>+</source> function between each element of <source lang=apl inline>ω</source>, so that
The parenthesised <source lang=apl inline>+⌿⍵</source> denotes the sum of all the elements of <source lang=apl inline></source>. The <source lang=apl inline>⌿</source> operator combines with the <source lang=apl inline>+</source> function: the <source lang=apl inline>⌿</source> fixes the <source lang=apl inline>+</source> function between each element of <source lang=apl inline></source>, so that
<source lang=apl>
<source lang=apl>
       +⌿ 1 2 3 4 5 6
       +⌿ 1 2 3 4 5 6
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[[Operator]]s like <source lang=apl inline>⌿</source> can be used to derive new functions not only from [[primitive function]]s like <source lang=apl inline>+</source>, but also from defined functions. For example
[[Operator]]s like <source lang=apl inline>⌿</source> can be used to derive new functions not only from [[primitive function]]s like <source lang=apl inline>+</source>, but also from defined functions. For example
<source lang=apl>
<source lang=apl>
       {α,', ',ω}⌿
       {,', ',}⌿
</source>
</source>
will transform a list of strings representing words into a comma-separated list:
will transform a list of strings representing words into a comma-separated list:
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└────────────────────┘
└────────────────────┘
</source>
</source>
So back to our mean example. <source lang=apl inline>(+⌿ω)</source> gives the sum of the list, which is then divided by <source lang=apl inline>≢ω</source>, the number elements in it.
So back to our mean example. <source lang=apl inline>(+⌿⍵)</source> gives the sum of the list, which is then divided by <source lang=apl inline>≢⍵</source>, the number elements in it.
<source lang=apl>
<source lang=apl>
       {(+)÷≢ω} 3 4.5 7 21
       {(+⌿⍵)÷≢⍵} 3 4.5 7 21
8.875
8.875
</source>
</source>
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       (+⌿÷≢) 3 4.5 7 21
       (+⌿÷≢) 3 4.5 7 21
8.875
8.875
</source>
The operator <source lang=apl inline>⌿</source> can also be used to modify the <source lang=apl inline>(+⌿÷≢)</source> function to produce a moving average.
<source lang=apl>
      2 (+⌿÷≢)/ 3 4.5 7 21
3.75 5.75 14
</source>
or, more verbosely
<source lang=apl>
      ave ← +⌿÷≢
      ave 3 4.5 7 21
8.875
      mave ← ave⌿
      2 mave 3 4.5 7 21
3.75 5.75 14
</source>
</source>


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</source>
</source>
{{Works in|[[Dyalog APL]], [[NARS2000]], [[dzaima/APL]]}}
{{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:
<source lang=apl>
      '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
</source>
Then it is simply a matter of performing a sum-reduce <source lang=apl inline>+/</source> to calculate the total frequency of each character:<ref name="Marshall LambaConf 2019">[[Marshall Lochbaum]] used this example as part of his talk on [[Outer Product]] at LambdaConf 2019.</ref>
<source lang=apl>
      +/ 'abcd' ∘.= 'cabbage'
2 2 1 0
</source>


=== Parenthesis nesting level ===
=== Parenthesis nesting level ===
First we compare all characters to the opening and closing characters;
{{quote | "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]]. ''[https://www.jsoftware.com/papers/perlis78.htm 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 <source lang=apl inline>∘.=</source>. First we compare all characters to the opening and closing characters;
<source lang=apl>
<source lang=apl>
       '()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
       '()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
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<source lang=apl>
<source lang=apl>
       +\-⌿'()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
       +\-⌿'()'∘.='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
</source>
{{Works in|all APLs}}
==== Method B ====
Alternatively, we can utilise that if the [[Index Of]] function <source lang=apl inline>⍳</source> doesn't find what it is looking for, it returns the next index after the last element in the the lookup array:
<source lang=apl>
      '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
</source>
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:
<source lang=apl>
      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
</source>
The running sum is what we're looking for:
<source lang=apl>
      +\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
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
</source>
</source>
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=== Grille cypher ===
=== Grille cypher ===
A [https://en.wikipedia.org/wiki/Grille_(cryptography) grille] is a 500 year old method for encrypting messages.
A [[wikipedia:grille (cryptography)|grille]] is a 500 year old method for encrypting messages.
[[File:Grille.png|none|500px|frameless|left|The application of a grille cypher]]
[[File:Grille.png|none|500px|frameless|The application of a grille cypher]]
<p>
<p>
Represent both the grid of letters and the grille as character matrices.
Represent both the grid of letters and the grille as character matrices.
Line 158: Line 180:
ILIKEAPL
ILIKEAPL
</source>
</source>
===References===
<references/>
{{APL development}}


{{APL programming language}}
[[Category:Examples]]

Revision as of 22:44, 27 June 2020

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

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

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

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.

The application of a grille cypher

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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