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Tacit functions apply to implicit arguments following a small set of rules. This is in contrast to the explicit use of arguments in [[dfns]] (<source inline lang=apl>⍺ ⍵</source>) and [[tradfns]] (which have named arguments). Known dialects which implement trains are [[Dyalog APL]], [[dzaima/APL]], [[ngn/apl]] and [[NARS2000]].
[[File:Function compositions.png|thumb|right|Diagrams of [[function composition]]s, an important part of tacit programming.]]
'''Tacit programming''', also called '''[[wikipedia:Tacit_programming|point-free style]]''', refers to usage of tacit [[function]]s that are defined in terms of implicit [[argument]]s. This is in contrast to the explicit use of arguments in [[dfn]]s (<syntaxhighlight inline lang=apl>⍺ ⍵</syntaxhighlight>) and [[tradfn]]s (which have named arguments). Some APL dialects allow to combine functions into [[train]]s following a small set of rules. This allows creating complex [[derived function]]s without specifying any arguments explicitly.
 
Dialects which implement trains include [[Dyalog APL]], [[dzaima/APL]], [[ngn/apl]] and [[NARS2000]].


== Primitives ==
== Primitives ==
All [[primitive functions]] are tacit. Some APLs allow primitive functions to be named.
All [[primitive functions]] are tacit. Some APLs allow primitive functions to be named.
<source lang=apl>
<syntaxhighlight lang=apl>
       plus ← +
       plus ← +
       times ← ×
       times ← ×
       6 times 3 plus 5
       6 times 3 plus 5
48
48
</source>
</syntaxhighlight>


== Derived functions ==
== Derived functions ==
Functions derived from a monadic operator and an operand, or from a dyadic operator and two operands are tacit functions:
Functions derived from a monadic operator and an operand, or from a dyadic operator and two operands are tacit functions:
<source lang=apl>
<syntaxhighlight lang=apl>
       Sum ← +/
       Sum ← +/
       Sum ⍳10
       Sum ⍳10
Line 20: Line 23:
       3 1 4 dot 2 7 1
       3 1 4 dot 2 7 1
17
17
</source>
</syntaxhighlight>
== Derived operators ==
== Derived operators ==
A dyadic operator with its right operand forms a tacit monadic operator:
A dyadic operator with its right operand forms a tacit monadic operator:
<source lang=apl>
<syntaxhighlight lang=apl>
       1(+⍣2)10
       1(+⍣2)10
12
12
Line 29: Line 32:
       1 +Twice 10
       1 +Twice 10
12
12
</source>
</syntaxhighlight>


== Trains ==
== Trains ==
A train is a series of functions in isolation. An isolated function is either surrounded by parentheses or named. Below, <source lang=apl inline>⍺</source> and <source lang=apl inline>⍵</source> refer to the arguments of the train. <source lang=apl inline>f</source>, <source lang=apl inline>g</source>, and <source lang=apl inline>h</source> are functions (which themselves can be tacit or not), and <source lang=apl inline>A</source> is an array. The arguments are processed by the following rules:
A [[train]] is a series of functions in isolation. An isolated function is either surrounded by parentheses or named.


A 2-train is an ''atop'':
These rules are used for 3-trains, called [[fork]]s:
{|
|<syntaxhighlight lang=apl>  (f g h) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>(  f ⍵) g (  h ⍵)</syntaxhighlight>
|-
|<syntaxhighlight lang=apl>⍺ (f g h) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>(⍺ f ⍵) g (⍺ h ⍵)</syntaxhighlight>
|}
The ''left tine'' of a fork can be an array:
{|
{|
|<source lang=apl>  (g h) ⍵</source>|| {{←→}} ||<source lang=apl>g (  h ⍵)</source>
|<syntaxhighlight lang=apl>  (A g h) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>A g (  h ⍵)</syntaxhighlight>
|-
|-
|<source lang=apl>⍺ (g h) ⍵</source>|| {{←→}} ||<source lang=apl>g (⍺ h ⍵)</source>
|<syntaxhighlight lang=apl>⍺ (A g h) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>A g (⍺ h ⍵)</syntaxhighlight>
|}
|}
A 3-train is a ''fork'':
 
In APL (but not [[J]]), these rules are used for 2-trains, called [[atop]]s:
{|
{|
|<source lang=apl>  (f g h) ⍵</source>|| {{←→}} ||<source lang=apl>(  f ⍵) g (  h ⍵)</source>
|<syntaxhighlight lang=apl>  (g h) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>g (  h ⍵)</syntaxhighlight>
|-
|-
|<source lang=apl>⍺ (f g h) ⍵</source>|| {{←→}} ||<source lang=apl>(⍺ f ⍵) g (⍺ h ⍵)</source>
|<syntaxhighlight lang=apl>⍺ (g h) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>g (⍺ h ⍵)</syntaxhighlight>
|}
|}
The ''left tine'' of a fork can be an array:
 
Any train can be expressed in terms of [[function composition]] &mdash; except dyadic forks. Some common patterns are:
 
{|
{|
|<source lang=apl> (A g h)</source>|| {{←→}} ||<source lang=apl>A g (  h ⍵)</source>
|<syntaxhighlight lang=apl>(f g h) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>g⍨∘f⍨∘h⍨ ⍵</syntaxhighlight>
|-
|-
|<source lang=apl>⍺ (A g h) ⍵</source>|| {{←→}} ||<source lang=apl>A g (⍺ h ⍵)</source>
|<syntaxhighlight lang=apl>(f g f) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>g⍥f⍨ ⍵</syntaxhighlight>
|-
|<syntaxhighlight lang=apl>(⊢ g f) ⍵</syntaxhighlight>|| {{←→}} ||<syntaxhighlight lang=apl>g∘f⍨ ⍵</syntaxhighlight>
|}
|}


Only [[dzaima/APL]] allows <source lang=apl inline>(A h)</source>, which it treats as <source lang=apl inline>A∘h</source>.<ref>dzaima/APL: [https://github.com/dzaima/APL/blob/ceea05e25687988ed0980a4abf4b9249b736543f/docs/differences.txt#L19 Differences from Dyalog APL]. Retrieved 09 Jan 2020.</ref>
== Debugging ==
In [[Dyalog APL]], analysis of trains is assisted by a [[user command]] <syntaxhighlight lang=apl inline>]Boxing on</syntaxhighlight>. This is achieved by executing the command <syntaxhighlight lang=apl inline>]Boxing on</syntaxhighlight> and then entering a train without any parameters. A structure of the train will be displayed.


For example, the "accursed train" from the section below can be analysed like this:
<syntaxhighlight lang=apl>
      ]Boxing on
Was OFF
      ((+.×⍨⊢~∘.×⍨)1↓⍳)    ⍝ the train to be analysed
┌───────────────────────────────┬───────┐
│┌───────────┬─────────────────┐│┌─┬─┬─┐│
││┌───────┬─┐│┌─┬─┬───────────┐│││1│↓│⍳││
│││┌─┬─┬─┐│⍨│││⊢│~│┌───────┬─┐│││└─┴─┴─┘│
││││+│.│×││ │││ │ ││┌─┬─┬─┐│⍨││││      │
│││└─┴─┴─┘│ │││ │ │││∘│.│×││ ││││      │
││└───────┴─┘││ │ ││└─┴─┴─┘│ ││││      │
││          ││ │ │└───────┴─┘│││      │
││          │└─┴─┴───────────┘││      │
│└───────────┴─────────────────┘│      │
└───────────────────────────────┴───────┘
</syntaxhighlight>
Alternatively, a train can be represented in form of a tree:
<syntaxhighlight lang=apl>
      ]Boxing on -trains=tree
Was ON -trains=box
      ((+.×⍨⊢~∘.×⍨)1↓⍳)    ⍝ the train to be analysed
    ┌───┴───┐ 
  ┌─┴─┐  ┌─┼─┐
  ⍨ ┌─┼─┐ 1 ↓ ⍳
┌─┘ ⊢ ~ ⍨     
.    ┌─┘     
┌┴┐    .       
+ ×  ┌┴┐     
      ∘ ×     
</syntaxhighlight>
Or fully parenthesised:
<syntaxhighlight lang=apl>
      ]Boxing on -trains=parens
Was OFF -trains=box
      ((+.×⍨⊢~∘.×⍨)1↓⍳)    ⍝ the train to be analysed
(((+.×)⍨)(⊢~((∘.×)⍨)))(1↓⍳)
</syntaxhighlight>
=== Conversion to dfns ===
It can help understanding to convert a tacit function to a dfn. For many tacit functions, it is not immediately clear if the intention of the function is to be used monadically or dyadically, or even both. Such knowledge can be conveyed by comments, but sometimes it is possible to spot patterns that are exclusively monadic or dyadic: A function with a bound argument (for example <syntaxhighlight lang=apl inline>+∘1</syntaxhighlight>) can indicate a monadic function, and in some contexts, <syntaxhighlight lang=apl inline>=</syntaxhighlight>, which can only be used dyadically, would indicate a dyadic function. The website [https://tacit.help tacit.help] provides automated translation of most tacit functions, into both monadic and dyadic, fully parenthesised dfns.
== Examples ==
== Examples ==
One of the major benefits of tacit programming is the ability to convey a short, well-defined idea as an isolated expression. This aids both human readability ([[semantic density]]) and the computer's ability to interpret code, potentially executing special code for particular [[idiom]]s.
One of the major benefits of tacit programming is the ability to convey a short, well-defined idea as an isolated expression. This aids both human readability ([[semantic density]]) and the computer's ability to interpret code, potentially executing special code for particular [[idiom]]s.


=== Plus and minus ===
=== Plus and minus ===
<source lang=apl>
<syntaxhighlight lang=apl>
       (+,-) 2    ⍝ ±2
       (+,-) 2    ⍝ ±2
2 ¯2
2 ¯2
       5 (+,-) 2  ⍝ 5±2
       5 (+,-) 2  ⍝ 5±2
7 3
7 3
</source>
</syntaxhighlight>


=== Arithmetic mean ===
=== Arithmetic mean ===
<source lang=apl>
<syntaxhighlight lang=apl>
       (+⌿÷≢) ⍳10      ⍝ Mean of the first ten integers
       (+⌿÷≢) ⍳10      ⍝ Mean of the first ten integers
5.5
5.5
       (+⌿÷≢) 5 4⍴⍳4    ⍝ Mean of columns in a matrix
       (+⌿÷≢) 5 4⍴⍳4    ⍝ Mean of columns in a matrix
1 2 3 4
1 2 3 4
</source>
</syntaxhighlight>


=== Fractions ===
=== Fractions ===
We can convert decimal numbers to fractions. For example, we can convert <math>2.625</math> to the improper fraction <math>21\over 8</math> with
We can convert decimal numbers to fractions. For example, we can convert <math>2.625</math> to the improper fraction <math>\tfrac{21}{8}</math> with
<source lang=apl>
<syntaxhighlight lang=apl>
       (1∧⊢,÷)2.625
       (1∧⊢,÷)2.625
21 8
21 8
</source>
</syntaxhighlight>
Alternatively, we can convert it to the mixed fraction <math>2{5\over 8}</math> with
Alternatively, we can convert it to the mixed fraction <math>2\tfrac{5}{8}</math> with a mixed fraction:
A mixed fraction:
<syntaxhighlight lang=apl>
<source lang=apl>
       (1∧0 1∘⊤,÷)2.625
       (1∧0 1∘⊤,÷)2.625
2 5 8
2 5 8
</source>
</syntaxhighlight>


=== Is it a palindrome? ===
=== Is it a palindrome? ===
<source lang=apl>
<syntaxhighlight lang=apl>
       (⌽≡⊢)'racecar'
       (⌽≡⊢)'racecar'
1
1
       (⌽≡⊢)'racecat'
       (⌽≡⊢)'racecat'
0
0
</source>
</syntaxhighlight>


=== Split delimited text ===
=== Split delimited text ===
<source lang=apl>
<syntaxhighlight lang=apl>
       ','(≠⊆⊢)'comma,delimited,text'
       ','(≠⊆⊢)'comma,delimited,text'
┌─────┬─────────┬────┐
┌─────┬─────────┬────┐
Line 105: Line 161:
│space│delimited│text│
│space│delimited│text│
└─────┴─────────┴────┘
└─────┴─────────┴────┘
</source>
</syntaxhighlight>


=== Component of a vector in the direction of another vector ===
=== Component of a vector in the direction of another vector ===
Sometimes a train can make an expression nicely resemble its equivalent definition in traditional mathematical notation. As an example, here is a program to compute the component of a vector <math>\textbf{a}</math> in the direction of another vector <math>\textbf{b}</math>:
Sometimes a train can make an expression nicely resemble its equivalent definition in traditional mathematical notation. As an example, here is a program to compute the component of a vector <math>\textbf{a}</math> in the direction of another vector <math>\textbf{b}</math>:
:::<math>\textbf{a}_\textbf{b} = (\textbf{a}\cdot\hat{\textbf{b}})\hat{\textbf{b}}</math>
:::<math>\textbf{a}_\textbf{b} = (\textbf{a}\cdot\hat{\textbf{b}})\hat{\textbf{b}}</math>
<source lang=apl>
<syntaxhighlight lang=apl>
       Root ← *∘÷⍨              ⍝ Nth root
       Root ← *∘÷⍨              ⍝ Nth root
       Norm ← 2 Root +.×⍨      ⍝ Magnitude (norm) of numeric vector in Euclidean space
       Norm ← 2 Root +.×⍨      ⍝ Magnitude (norm) of numeric vector in Euclidean space
Line 117: Line 173:
       3 5 2 InDirOf 0 0 1      ⍝ Trivial example
       3 5 2 InDirOf 0 0 1      ⍝ Trivial example
0 0 2
0 0 2
</source>
</syntaxhighlight>
For a more parallel comparison of the notations, see the [[Comparison_with_traditional_mathematics#Practical_example|comparison with traditional mathematics]].  
For a more parallel comparison of the notations, see the [[Comparison_with_traditional_mathematics#Practical_example|comparison with traditional mathematics]].
 
===The Number of the Beast===
The following expression for computing the [[wikipedia:666 (number)|number of the Beast]] (and of [[I.P. Sharp]]'s APL-based email system, [[666 BOX]]) nicely illustrates how to read a train.
<syntaxhighlight lang=apl>
      ((+.×⍨⊢~∘.×⍨)1↓⍳)17 ⍝ Accursed train
666
</syntaxhighlight>
First, <syntaxhighlight lang=apl inline>((+.×⍨⊢~∘.×)1↓⍳)</syntaxhighlight> is supplied with only one argument <syntaxhighlight lang=apl inline>17</syntaxhighlight> and is thus interpreted monadically.
 
Second, <syntaxhighlight lang=apl inline>(+.×⍨⊢~∘.×⍨)1↓⍳</syntaxhighlight> is a 4-train: reading right-to-left, the last 3 components are interpreted as the fork <syntaxhighlight lang=apl inline>1↓⍳</syntaxhighlight> and the 4-train is interpreted as the atop <syntaxhighlight lang=apl inline>(+.×⍨⊢~∘.×⍨)(1↓⍳)</syntaxhighlight>.
Similarly, <syntaxhighlight lang=apl inline>(+.×⍨⊢~∘.×⍨)</syntaxhighlight> is also a 4-train and interpreted as the atop <syntaxhighlight lang=apl inline>+.×⍨(⊢~∘.×⍨)</syntaxhighlight>.
 
Thus the accursed train is interpreted as <syntaxhighlight lang=apl inline>((+.×⍨(⊢~∘.×⍨))(1↓⍳))17</syntaxhighlight>. Having read the train, we now evaluate it monadically.
<syntaxhighlight lang=apl>
      ((+.×⍨(⊢~∘.×⍨))(1↓⍳))17 ⍝ Accursed train as an atop over a fork atop a fork
      +.×⍨(⊢~∘.×⍨)1↓⍳17      ⍝ Atop evalution
      +.×⍨(⊢1↓⍳17)~∘.×⍨1↓⍳17  ⍝ Fork evalution
      +.×⍨(1↓⍳17)~∘.×⍨1↓⍳17  ⍝ ⊢ evaluation
      +.×⍨2 3 5 7 11 13 17    ⍝ numbers 2 through 17 without those appearing in their multiplication table are primes
666                          ⍝ the sum of the squares of the primes up to 17
</syntaxhighlight>
Note that <syntaxhighlight lang=apl inline>((⊢⍨∘.×⍨)1↓⍳)</syntaxhighlight> is a train computing primes up to the given input.
 
A more satisfying variation of the accursed train is the following.
<syntaxhighlight lang=apl>
      (⍎⊢,⍕∘≢)'((+.×⍨⊢~∘.×⍨)1↓⍳)'                    ⍝ Accursed train 2.0
      ⍎(⊢,⍕∘≢)'((+.×⍨⊢~∘.×⍨)1↓⍳)'                    ⍝ 4-train intepreted as an atop
      ⍎(⊢'((+.×⍨⊢~∘.×⍨)1↓⍳)'),⍕∘≢'((+.×⍨⊢~∘.×⍨)1↓⍳)' ⍝ fork evaluation
      ⍎'((+.×⍨⊢~∘.×⍨)1↓⍳)','17'                      ⍝ ⊢ evaluation and ⍕∘≢ evaluation
      ⍎'((+.×⍨⊢~∘.×⍨)1↓⍳)17'                        ⍝ , evaluation
666                                                  ⍝ ⍎ executes original Accursed train
</syntaxhighlight>
 
== External links ==
== External links ==
== Tutorials ==
=== Tutorials ===
* Dyalog: [http://help.dyalog.com/16.0/Content/RelNotes14.0/Function%20Trains.htm version 14.0 release notes]
<div style="column-width:30em">
* APL Cultivation: [https://chat.stackexchange.com/rooms/52405/conversation/lesson-23-transcribing-to-and-reading-trains Transcribing to and reading trains]
==== In text form ====
* APLtrainer: [https://www.youtube.com/watch?v=kt4lMZbn-so How to read trains in Dyalog APL code] (video)
* [[Learning APL]]: [https://xpqz.github.io/learnapl/tacit.html Trainspotting]
* APLtrainer: [https://www.youtube.com/watch?v=A2LqqBosvY0 Function trains in APL] (video)
* [[Documentation_suites#Dyalog_APL|Dyalog documentation]]: [https://help.dyalog.com/16.0/Content/RelNotes14.0/Function%20Trains.htm version 14.0 release notes]
* Dyalog Webinar: [https://www.youtube.com/watch?v=Enlh5qwwDuY?t=440 Train Spotting in Dyalog APL] (video)
* [[Dfns workspace]]: [https://dfns.dyalog.com/n_tacit.htm Translation of <nowiki>[dfns]</nowiki> into tacit form]
* Dyalog '13: [https://www.youtube.com/watch?v=7-93GzDqC08 Train Spotting in Version 14.0] (video)
* [[Dfns workspace]]: [https://dfns.dyalog.com/n_dft.htm Display of function tree]
* [[Dfns workspace]]: [https://dfns.dyalog.com/n_fork.htm Simulation of fork syntax]
* [[APL Cultivation]]: [https://chat.stackexchange.com/rooms/52405/conversation/lesson-23-transcribing-to-and-reading-trains Transcribing to and reading trains]
* gitonthescene: [https://gist.github.com/gitonthescene/666c77ee3ed0ae0a79cf8e057584b7fd Forks: Spoon fed]
* gitonthescene: [https://gist.github.com/gitonthescene/5e9c25ab9edd2f2ce0d5ad38d8a8b2b4 Training day]
 


== Documentation ==
==== Videos ====
* [http://help.dyalog.com/16.0/Content/RelNotes14.0/Function%20Trains.htm Announcement]
* [[APLtrainer]]: [https://www.youtube.com/watch?v=kt4lMZbn-so How to read trains in Dyalog APL code]
* [http://help.dyalog.com/latest/Content/Language/Introduction/Trains.htm Dyalog]
* [[APLtrainer]]: [https://www.youtube.com/watch?v=A2LqqBosvY0 Function trains in APL]
* [[Dyalog webinar]]: [https://www.youtube.com/watch?v=Enlh5qwwDuY?t=440 Train Spotting in Dyalog APL]
* [[Dyalog '13]]: [https://www.youtube.com/watch?v=7-93GzDqC08 Train Spotting in Version 14.0]
</div>


==References==
== References ==
<references/>
<references/>
{{APL syntax}}[[Category:Paradigms]]
 
{{APL syntax}}[[Category:Tacit programming| ]]
Retrieved from ‘https://aplwiki.com/wiki/Special:MobileDiff/3849...10049’

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