Tacit programming

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.

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:

A 2-train is an atop:

(g h) ⍵	$\Leftrightarrow$	g ( h ⍵)
⍺ (g h) ⍵	$\Leftrightarrow$	g (⍺ h ⍵)

A 3-train is a fork:

(f g h) ⍵	$\Leftrightarrow$	( f ⍵) g ( h ⍵)
⍺ (f g h) ⍵	$\Leftrightarrow$	(⍺ f ⍵) g (⍺ h ⍵)

The left tine of a fork can be an array:

(A g h)	$\Leftrightarrow$	A g ( h ⍵)
⍺ (A g h) ⍵	$\Leftrightarrow$	A g (⍺ h ⍵)

Only dzaima/APL allows (A h), which it treats as A∘h.

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 $2.625$ to the improper fraction $21 \over 8$ with

      (1∧⊢,÷)2.625
21 8

Alternatively, we can convert it to the mixed fraction $2{5 \over 8}$ 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 ${\textbf {a}}$ in the direction of another vector ${\textbf {b}}$:

${\textbf {a}}_{\textbf {b}}=({\textbf {a}}\cdot {\hat {\textbf {b}}}){\hat {\textbf {b}}}$

      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.

Tutorials

• Dyalog: version 14.0 release notes
• APL Cultivation: Transcribing to and reading trains
• APLtrainer: How to read trains in Dyalog APL code (video)
• APLtrainer: Function trains in APL (video)
• Dyalog Webinar: Train Spotting in Dyalog APL (video)
• Dyalog '13: Train Spotting in Version 14.0 (video)

Documentation

• Announcement
• Dyalog