Tacit programming

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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 ( ) and tradfns (which have named arguments). Known dialects which implement trains are 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 an operator and operand are tacit.

      sum  +/
      sum 10


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 2-train is an atop:

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

A 3-train is a fork:

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

The left tine of a fork (but not an atop) can be an array:

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


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
      1 2 3 (+,-) 4
5 6 7 ¯3 ¯2 ¯1
      (2 30) (+,-) 1
1 1 1 ¯1 ¯1 ¯1
1 1 1 ¯1 ¯1 ¯1

Arithmetic mean

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

Top-heavy fraction as decimal

      (1∧⊢,÷) 1.125
9 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 a in the direction of another vector b.

      Sqrt  *.5              ⍝ Square root 
      Norm  Sqrt+.×          ⍝ 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

In particular, the definition of InDirOf resembles the definition in traditional mathematical notation:

Traditional notation APL
(Sqrt+.×) b
(÷Norm) b
a +.× b
a (⊢×+.×)Unit b

APL features [edit]
Built-ins Primitive functionPrimitive operatorQuad name
Array model ShapeRankDepthBoundIndex (Indexing) ∙ AxisRavelRavel orderElementScalarVectorMatrixSimple scalarSimple arrayNested arrayCellMajor cellSubarrayEmpty arrayPrototype
Data types Number (Boolean, Complex number) ∙ Character (String) ∙ BoxNamespace
Concepts and paradigms Leading axis theoryScalar extensionConformabilityScalar functionPervasionGlyphIdentity elementComplex floorTotal array ordering
APL syntax [edit]
General Comparison with traditional mathematicsPrecedenceTacit programming
Array Numeric literalStringStrand notationObject literalArray notation
Function ArgumentFunction valenceDerived functionDerived operatorNiladic functionMonadic functionDyadic functionAmbivalent functionTradfnDfnFunction train
Operator OperandOperator valenceTradopDopDerived operator
Assignment MultipleIndexedSelectiveModified
Other Function axisBranchQuad nameSystem commandUser commandKeywordDot notationFunction-operator overloading