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-This page contains examples that show APL's strengths. The examples require minimal background and have no special dependencies.
+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]].
-== Longer examples ==
-A list of more involved examples.
-[[Category: Lists]]
-* [[APL Wiki logo]]
-* [[Convex Hull]]
-* [[Conway's Game of Life]]
-* [[Ranking poker hands]]
 == Arithmetic mean ==
@@ Line 17: / Line 5: @@
 Here is an APL program to calculate the average (arithmetic mean) of a list of numbers, written as a [[dfn]]:
 <source lang=apl>
-       {(+⌿ω)÷≢ω}
+       {(+⌿⍵)÷≢⍵}
 </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.
-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>
        +⌿ 1 2 3 4 5 6
@@ Line 36: / Line 24: @@
 [[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>
-       {α,', ',ω}⌿
+       {⍺,', ',⍵}⌿
 </source>
 will transform a list of strings representing words into a comma-separated list:
 <source lang=apl>
-     {⍺,', ',⍵}⌿'cow' 'sheep' 'cat' 'dog'
+      {⍺,', ',⍵}⌿'cow' 'sheep' 'cat' 'dog'
 ┌────────────────────┐
 │cow, sheep, cat, dog│
 └────────────────────┘
 </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>ρω</source>, the number of its elements.
+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>
-       {(+/ω)÷ρω} 3 4.5 7 21
+       {(+⌿⍵)÷≢⍵} 3 4.5 7 21
 .875
 </source>
 === Tacit programming ===
+{{Main|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:
@@ Line 59: / Line 49: @@
 </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.
+This is a so called 3-train, also known as a ''fork''. It is evaluated like this:
-<source lang=apl>
+{|
-(+⌿÷≢)/ 3 4.5 7 21
+|<source lang=apl>(+⌿ ÷ ≢) 3 4.5 7 21</source>|| {{←→}} ||<source lang=apl>(+⌿ 3 4.5 7 21) ÷ (≢ 3 4.5 7 21)</source>
-.75 5.75 14
+|}
-</source>
-or, more verbosely
+Note that <source lang=apl inline>+⌿</source> is evaluated as a single derived function.
-<source lang=apl>
+The general scheme for monadic 3-trains is the following:
-      ave ← +⌿÷≢
+{|
-      ave 3 4.5 7 21
+|<source lang=apl>(f g h) ⍵</source>|| {{←→}} ||<source lang=apl>(f ⍵) g (h ⍵)</source>
-.875
+|}
-      mave ← ave/
-mave 3 4.5 7 21
+But other types of [[Tacit programming#Trains|trains]] are also possible.
-.75 5.75 14
-</source>
 ==Text processing==
@@ Line 104: / Line 92: @@
 </source>
 {{Works in|[[Dyalog APL]]}}
-Notice of you can read the [[tacit]] function <source lang=apl inline>≠⊆⊢</source> like an English sentence: ''The inequality partitions the right argument''.
+Notice that you can read the [[tacit]] function <source lang=apl inline>≠⊆⊢</source> like an English sentence: ''The inequality partitions the right argument''.
 === Indices of multiple elements ===
 <source lang=apl inline>∊</source> gives us a mask for elements (characters) in the left argument that are members of the right argument:
@@ Line 122: / Line 111: @@
 </source>
 {{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'
+1 0 0 1 0 0
+0 1 1 0 0 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 1 0
+</source>
 === 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>
        '()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
@@ Line 138: / Line 145: @@
 <source lang=apl>
        +\-⌿'()'∘.='plus(square(a),plus(square(b),times(2,plus(a,b)))'
+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'
+2 5
+      '()'⍳'plus(square(a),plus(square(b),times(2,plus(a,b)))'
+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 0['()'⍳'plus(square(a),plus(square(b),times(2,plus(a,b)))']
+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 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>
@@ Line 143: / Line 169: @@
 === 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>
 Represent both the grid of letters and the grille as character matrices.
@@ Line 168: / Line 194: @@
 ILIKEAPL
 </source>
+===References===
+<references/>
+{{APL development}}
-{{APL programming language}}
+[[Category:Examples]]

Simple examples: Difference between revisions

Simple examples (view source)

Revision as of 05:49, 26 August 2020

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