Tacit programming

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Diagrams of function compositions, an important part of tacit programming.

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.


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

      plus  +
      times  ×
      6 times 3 plus 5

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

      Dot  +.×
      3 1 4 dot 2 7 1

Derived operators

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

      Twice  2
      1 +Twice 10


A train is a series of functions in isolation. An isolated function is either surrounded by parentheses or named.

These rules are used for 3-trains, called forks:

  (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 )

In APL (but not J), these rules are used for 2-trains, called atops:

  (g h) 
g (  h )
 (g h) 
g ( h )

Any train can be expressed in terms of function composition — except dyadic forks. Some common patterns are:

(f g h) 
(f g f) 
( g f) 


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

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

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 +1) can indicate a monadic function, and in some contexts, =, which can only be used dyadically, would indicate a dyadic function. The website tacit.help provides automated translation of most tacit functions, into both monadic and dyadic, fully parenthesised dfns.


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 4⍴⍳4    ⍝ Mean of columns in a matrix
1 2 3 4


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

21 8

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

      (10 1⊤,÷)2.625
2 5 8

Is it a palindrome?


Split 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

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 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

External links


In text form



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 structure