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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]].
'''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 (<source inline lang=apl>⍺ ⍵</source>) and [[tradfn]]s (which have named arguments). Some APL dialects allow to combine functions into [[#trains|trains]] 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 ==
Line 11: Line 13:


== Derived functions ==
== Derived functions ==
Functions derived from an operator and operand are tacit.
Functions derived from a monadic operator and an operand, or from a dyadic operator and two operands are tacit functions:
<source lang=apl>
<source lang=apl>
       sum ← +/
       Sum ← +/
       sum ⍳10
       Sum ⍳10
55
55
      Dot ← +.×
      3 1 4 dot 2 7 1
17
</source>
== Derived operators ==
A dyadic operator with its right operand forms a tacit monadic operator:
<source lang=apl>
      1(+⍣2)10
12
      Twice ← ⍣2
      1 +Twice 10
12
</source>
</source>


== Trains ==
== Trains ==
A train is a series of functions in isolation. An isolated function is either surrounded by parentheses or named. 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. 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 2-train is an ''atop'':
=== 3-trains ===
A 3-train is a ''fork'', so denoted because its structure resembles a three-tines fork, or a three-pronged pitchfork. The two outer functions are applied first, and their results are used as arguments to the middle function:
{|
|<source lang=apl>  (f g h) ⍵</source>|| {{←→}} ||<source lang=apl>(  f ⍵) g (  h ⍵)</source>
|-
|<source lang=apl>⍺ (f g h) ⍵</source>|| {{←→}} ||<source lang=apl>(⍺ f ⍵) g (⍺ h ⍵)</source>
|}
The ''left tine'' of a fork can be an array:
{|
|<source lang=apl>  (A g h)</source>|| {{←→}} ||<source lang=apl>A g (  h ⍵)</source>
|-
|<source lang=apl>⍺ (A g h) ⍵</source>|| {{←→}} ||<source lang=apl>A g (⍺ h ⍵)</source>
|}
 
=== 2-trains ===
Most dialects define a 2-train is an ''atop'', equivalent to the function derived using the [[Atop (operator)|Atop]] operator. The left function is applied [[monadic function|monadically]] on the result of the right function:
{|
|<source lang=apl>  (g h) ⍵</source>|| {{←→}} ||<source lang=apl>g (  h ⍵)</source>
|-
|<source lang=apl>⍺ (g h) ⍵</source>|| {{←→}} ||<source lang=apl>g (⍺ h ⍵)</source>
|}
 
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> See [[Bind]].
 
[[J]] instead defines the 2-train as a [[hook]], equivalent to the function derived using the [[Withe]] operator. The left function is always applied [[dyadic function|dyadically]], taking as right argument, the result of applying the right function on the right argument. If there is no left argument, the sole argument is used also as left argument:
{|
|<source lang=apl>  (g h) ⍵</source>|| {{←→}} ||<source lang=apl>⍵ g (h ⍵)</source>
|-
|<source lang=apl>⍺ (g h) ⍵</source>|| {{←→}} ||<source lang=apl>⍺ g (h ⍵)</source>
|}
 
== Debugging ==
In [[Dyalog APL]], analysis of trains is assisted by a [[user command]] <source lang=apl inline>]Boxing on</source>. This is achieved by executing the command <source lang=apl inline>]Boxing on</source> 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:
<source lang=apl>
<source lang=apl>
  (g h) ⍵ ⬄ g (  h ⍵)
      ]Boxing on
⍺ (g h) ⍵ ⬄ g (⍺ h ⍵)
Was OFF
      ((+.×⍨⊢~∘.×⍨)1↓⍳)     ⍝ the train to be analysed
┌───────────────────────────────┬───────┐
│┌───────────┬─────────────────┐│┌─┬─┬─┐│
││┌───────┬─┐│┌─┬─┬───────────┐│││1│↓│⍳││
│││┌─┬─┬─┐│⍨│││⊢│~│┌───────┬─┐│││└─┴─┴─┘│
││││+│.│×││ │││ │ ││┌─┬─┬─┐│⍨││││      │
│││└─┴─┴─┘│ │││ │ │││∘│.│×││ ││││      │
││└───────┴─┘││ │ ││└─┴─┴─┘│ ││││      │
││          ││ │ │└───────┴─┘│││      │
││          │└─┴─┴───────────┘││      │
│└───────────┴─────────────────┘│      │
└───────────────────────────────┴───────┘
</source>
</source>


A 3-train is a ''fork'':
Alternatively, a train can be represented in form of a tree:
<source lang=apl>
<source lang=apl>
  (f g h) ⍵ ⬄ ( f ⍵) g ( h ⍵)
      ]Boxing on -trains=tree
⍺ (f g h) ⍵ ⬄ (⍺ f ⍵) g (⍺ h ⍵)
Was ON -trains=box
      ((+.×⍨⊢~∘.×⍨)1↓⍳)    ⍝ the train to be analysed
    ┌───┴───┐ 
  ┌─┴─┐  ┌─┼─┐
  ⍨ ┌─┼─┐ 1 ↓ ⍳
  ┌─┘ ⊢ ~ ⍨     
  .    ┌─┘     
┌┴┐    .       
+ ×  ┌┴┐     
      ∘ ×     
</source>
</source>
 
Or fully parenthesised:
The ''left tine'' of a fork (but not an atop) can be an array:
<source lang=apl>
<source lang=apl>
  (A g h) ⍵ ⬄ A g ( h ⍵)
      ]Boxing on -trains=parens
(A g h) ⍵ ⬄ A g (⍺ h ⍵)
Was OFF -trains=box
      ((+.×⍨⊢~∘.×⍨)1↓⍳)    ⍝ the train to be analysed
(((+.×)⍨)(⊢~((∘.×)⍨)))(1↓⍳)
</source>
</source>


== 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 [[idioms]].
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>
<source lang=apl>
       (+,-)2
       (+,-) 2     ⍝ ±2
2 ¯2
2 ¯2
       1 2 3 (+,-) 4
       5 (+,-) 2  ⍝ 5±2
5 6 7 ¯3 ¯2 ¯1
7 3
      (2 3⍴0) (+,-) 1
1 1 1 ¯1 ¯1 ¯1
1 1 1 ¯1 ¯1 ¯1     
</source>
</source>


Line 62: Line 130:


=== 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>
<source lang=apl>
       (1∧⊢,÷)2.625
       (1∧⊢,÷)2.625
21 8
21 8
</source>
</source>
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:
<source lang=apl>
<source lang=apl>
       (1∧0 1∘⊤,÷)2.625
       (1∧0 1∘⊤,÷)2.625
Line 105: Line 172:
0 0 2
0 0 2
</source>
</source>
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]].
{{APL syntax}}
 
===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.
<source lang=apl>
      ((+.×⍨⊢~∘.×⍨)1↓⍳)17 ⍝ Accursed train
666
</source>
First, <source lang=apl inline>((+.×⍨⊢~∘.×)1↓⍳)</source> is supplied with only one argument <source lang=apl inline>17</source> and is thus interpreted monadically.
 
Second, <source lang=apl inline>(+.×⍨⊢~∘.×⍨)1↓⍳</source> is a 4-train: reading right-to-left, the last 3 components are interpreted as the fork <source lang=apl inline>1↓⍳</source> and the 4-train is interpreted as the atop <source lang=apl inline>(+.×⍨⊢~∘.×⍨)(1↓⍳)</source>.
Similarly, <source lang=apl inline>(+.×⍨⊢~∘.×⍨)</source> is also a 4-train and interpreted as the atop <source lang=apl inline>+.×⍨(⊢~∘.×⍨)</source>.
 
Thus the accursed train is interpreted as <source lang=apl inline>((+.×⍨(⊢~∘.×⍨))(1↓⍳))17</source>. Having read the train, we now evaluate it monadically.
<source 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 15 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
</source>
Note that <source lang=apl inline>((⊢⍨∘.×⍨)1↓⍳)</source> is a train computing primes up to the given input.
 
A more satisfying variation of the accursed train is the following.
<source 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
</source>
 
== See also ==
 
* [[Function composition]]
 
== External links ==
=== Tutorials ===
* Dyalog: [https://help.dyalog.com/16.0/Content/RelNotes14.0/Function%20Trains.htm version 14.0 release notes]
* [[Dfns workspace]]: [https://dfns.dyalog.com/n_tacit.htm Translation of <nowiki>[dfns]</nowiki> into tacit form]
* [[APL Cultivation]]: [https://chat.stackexchange.com/rooms/52405/conversation/lesson-23-transcribing-to-and-reading-trains Transcribing to and reading trains]
* [[APLtrainer]]: [https://www.youtube.com/watch?v=kt4lMZbn-so How to read trains in Dyalog APL code] (video)
* [[APLtrainer]]: [https://www.youtube.com/watch?v=A2LqqBosvY0 Function trains in APL] (video)
* [[Dyalog webinar]]: [https://www.youtube.com/watch?v=Enlh5qwwDuY?t=440 Train Spotting in Dyalog APL] (video)
* [[Dyalog '13]]: [https://www.youtube.com/watch?v=7-93GzDqC08 Train Spotting in Version 14.0] (video)
 
=== Documentation ===
* [https://help.dyalog.com/16.0/Content/RelNotes14.0/Function%20Trains.htm Announcement]
* [https://help.dyalog.com/latest/Content/Language/Introduction/Trains.htm Dyalog]
 
== References ==
<references/>
 
{{APL syntax}}[[Category:Tacit programming| ]]

Revision as of 19:06, 1 September 2021

Tacit programming, also called point-free style, refers to usage of tacit functions that are defined in terms of implicit arguments. This is in contrast to the explicit use of arguments in dfns (⍺ ⍵) and tradfns (which have named arguments). Some APL dialects allow to combine functions into trains following a small set of rules. This allows creating complex derived functions without specifying any arguments explicitly.

Dialects which implement trains include Dyalog APL, dzaima/APL, ngn/apl and NARS2000.

Primitives

All primitive functions are tacit. Some APLs allow primitive functions to be named.

      plus ← +
      times ← ×
      6 times 3 plus 5
48

Derived functions

Functions derived from a monadic operator and an operand, or from a dyadic operator and two operands are tacit functions:

      Sum ← +/
      Sum ⍳10
55

      Dot ← +.×
      3 1 4 dot 2 7 1
17

Derived operators

A dyadic operator with its right operand forms a tacit monadic operator:

      1(+⍣2)10
12
      Twice ← ⍣2
      1 +Twice 10
12

Trains

A train is a series of functions in isolation. An isolated function is either surrounded by parentheses or named. Below, and refer to the arguments of the train. f, g, and h are functions (which themselves can be tacit or not), and A is an array. The arguments are processed by the following rules:

3-trains

A 3-train is a fork, so denoted because its structure resembles a three-tines fork, or a three-pronged pitchfork. The two outer functions are applied first, and their results are used as arguments to the middle function:

  (f g h) ⍵
(  f ⍵) g (  h ⍵)
⍺ (f g h) ⍵
(⍺ f ⍵) g (⍺ h ⍵)

The left tine of a fork can be an array:

  (A g h)
A g (  h ⍵)
⍺ (A g h) ⍵
A g (⍺ h ⍵)

2-trains

Most dialects define a 2-train is an atop, equivalent to the function derived using the Atop operator. The left function is applied monadically on the result of the right function:

  (g h) ⍵
g (  h ⍵)
⍺ (g h) ⍵
g (⍺ h ⍵)

Only dzaima/APL allows (A h), which it treats as A∘h.[1] See Bind.

J instead defines the 2-train as a hook, equivalent to the function derived using the Withe operator. The left function is always applied dyadically, taking as right argument, the result of applying the right function on the right argument. If there is no left argument, the sole argument is used also as left argument:

  (g h) ⍵
⍵ g (h ⍵)
⍺ (g h) ⍵
⍺ g (h ⍵)

Debugging

In Dyalog APL, analysis of trains is assisted by a user command ]Boxing on. This is achieved by executing the command ]Boxing on 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:

      ]Boxing on
Was OFF
      ((+.×⍨⊢~∘.×⍨)1↓⍳)     ⍝ the train to be analysed
┌───────────────────────────────┬───────┐
│┌───────────┬─────────────────┐│┌─┬─┬─┐│
││┌───────┬─┐│┌─┬─┬───────────┐│││1│↓│⍳││
│││┌─┬─┬─┐│⍨│││⊢│~│┌───────┬─┐│││└─┴─┴─┘│
││││+│.│×││ │││ │ ││┌─┬─┬─┐│⍨││││       │
│││└─┴─┴─┘│ │││ │ │││∘│.│×││ ││││       │
││└───────┴─┘││ │ ││└─┴─┴─┘│ ││││       │
││           ││ │ │└───────┴─┘│││       │
││           │└─┴─┴───────────┘││       │
│└───────────┴─────────────────┘│       │
└───────────────────────────────┴───────┘

Alternatively, a train can be represented in form of a tree:

      ]Boxing on -trains=tree
Was ON -trains=box
      ((+.×⍨⊢~∘.×⍨)1↓⍳)     ⍝ the train to be analysed
     ┌───┴───┐  
   ┌─┴─┐   ┌─┼─┐
   ⍨ ┌─┼─┐ 1 ↓ ⍳
 ┌─┘ ⊢ ~ ⍨      
 .     ┌─┘      
┌┴┐    .        
+ ×   ┌┴┐       
      ∘ ×

Or fully parenthesised:

      ]Boxing on -trains=parens
Was OFF -trains=box
      ((+.×⍨⊢~∘.×⍨)1↓⍳)     ⍝ the train to be analysed
(((+.×)⍨)(⊢~((∘.×)⍨)))(1↓⍳)

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 idioms.

Plus and minus

      (+,-) 2     ⍝ ±2
2 ¯2
      5 (+,-) 2   ⍝ 5±2
7 3

Arithmetic mean

      (+⌿÷≢) ⍳10       ⍝ Mean of the first ten integers
5.5
      (+⌿÷≢) 5 4⍴⍳4    ⍝ Mean of columns in a matrix
1 2 3 4

Fractions

We can convert decimal numbers to fractions. For example, we can convert to the improper fraction with

      (1∧⊢,÷)2.625
21 8

Alternatively, we can convert it to the mixed fraction with a mixed fraction:

      (1∧0 1∘⊤,÷)2.625
2 5 8

Is it a palindrome?

      (⌽≡⊢)'racecar'
1
      (⌽≡⊢)'racecat'
0

Split delimited text

      ','(≠⊆⊢)'comma,delimited,text'
┌─────┬─────────┬────┐
│comma│delimited│text│
└─────┴─────────┴────┘
      ' '(≠⊆⊢)'space delimited text'
┌─────┬─────────┬────┐
│space│delimited│text│
└─────┴─────────┴────┘

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 in the direction of another vector :

      Root ← *∘÷⍨              ⍝ Nth root
      Norm ← 2 Root +.×⍨       ⍝ Magnitude (norm) of numeric vector in Euclidean space
      Unit ← ⊢÷Norm            ⍝ Unit vector in direction of vector ⍵
      InDirOf ← (⊢×+.×)∘Unit   ⍝ Component of vector ⍺ in direction of vector ⍵
      3 5 2 InDirOf 0 0 1      ⍝ Trivial example
0 0 2

For a more parallel comparison of the notations, see the comparison with traditional mathematics.

The Number of the Beast

The following expression for computing the number of the Beast (and of I.P. Sharp's APL-based email system, 666 BOX) nicely illustrates how to read a train.

      ((+.×⍨⊢~∘.×⍨)1↓⍳)17 ⍝ Accursed train
666

First, ((+.×⍨⊢~∘.×)1↓⍳) is supplied with only one argument 17 and is thus interpreted monadically.

Second, (+.×⍨⊢~∘.×⍨)1↓⍳ is a 4-train: reading right-to-left, the last 3 components are interpreted as the fork 1↓⍳ and the 4-train is interpreted as the atop (+.×⍨⊢~∘.×⍨)(1↓⍳). Similarly, (+.×⍨⊢~∘.×⍨) is also a 4-train and interpreted as the atop +.×⍨(⊢~∘.×⍨).

Thus the accursed train is interpreted as ((+.×⍨(⊢~∘.×⍨))(1↓⍳))17. Having read the train, we now evaluate it monadically.

      ((+.×⍨(⊢~∘.×⍨))(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 15 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

Note that ((⊢⍨∘.×⍨)1↓⍳) is a train computing primes up to the given input.

A more satisfying variation of the accursed train is the following.

      (⍎⊢,⍕∘≢)'((+.×⍨⊢~∘.×⍨)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

See also

External links

Tutorials

Documentation

References

  1. dzaima/APL: Differences from Dyalog APL. Retrieved 09 Jan 2020.


APL syntax [edit]
General Comparison with traditional mathematicsPrecedenceTacit programming (Train, Hook, Split composition)
Array Numeric literalStringStrand notationObject literalArray notation (design considerations)
Function ArgumentFunction valenceDerived functionDerived operatorNiladic functionMonadic functionDyadic functionAmbivalent functionDefined function (traditional)DfnFunction train
Operator OperandOperator valenceTradopDopDerived operator
Assignment MultipleIndexedSelectiveModified
Other Function axisBracket indexingBranchStatement separatorQuad nameSystem commandUser commandKeywordDot notationFunction-operator overloadingControl structureComment