Point-free style programming

• Programming paradigm in which function definitions do not identify the arguments
• Definitions just compose other functions by combining them
• Combinators
  • manipulate the arguments
• Introduced in J
• Now part of APL

Feel of APL

• Vectors
• Functions
  • monadic
  • dyadic
• Operators
  • reduce
  • scan
  • inner product
  • outer product
• Combinators (“trains”)
  • atop
  • fork

Operators in APL

      monadic               dyadic
+     identity              addition
-     negation              subtraction
×     sign                  multiplication
÷     1/x                   division
ι     counter
|     absolute value        magnitude

Monadic functions

      -10
¯10

      ÷10
0.1

      |10
10

      |¯10
10

Passing result to other monadic function

      |(÷(-3))
0.3333333333

• Can be shortened to

      |÷-3
0.3333333333

• Btw APL is an array language, so

      -÷⍳10
¯1 ¯0.5 ¯0.3333333333 ¯0.25 ¯0.2 ¯0.1666666667 ¯0.1428571429 ¯0.125 ¯0.1111111111 ¯0.1

From functions to trains

• Function defined using classic style

      bar ← {-÷|⍵}

      bar 3
¯0.3333333333

• Point-free style

      baz ← -(÷|)

      baz 3
¯0.3333333333

S-combinator

• Commonly used pattern in algorithms:

(⍺ f ⍵) g (⍺ h ⍵)

• or in Python/Go/C>

g(f(⍺, ⍵), h(⍺, ⍵))

• ⍺ - parameter passed to function f and h
• ⍵ - parameter passed to function f and h
• f is dyadic function
• h is dyadic function

• g is dyadic function

• It’s possible to shorten it to

(f g h)

• or even to

fgh

Example

• Initial expression:

(⍳10) ∘.× (⍳10)

 1  2  3  4  5  6  7  8  9  10
 2  4  6  8 10 12 14 16 18  20
 3  6  9 12 15 18 21 24 27  30
 4  8 12 16 20 24 28 32 36  40
 5 10 15 20 25 30 35 40 45  50
 6 12 18 24 30 36 42 48 54  60
 7 14 21 28 35 42 49 56 63  70
 8 16 24 32 40 48 56 64 72  80
 9 18 27 36 45 54 63 72 81  90
10 20 30 40 50 60 70 80 90 100

• Can be shortened to:

(⍳∘.×⍳)10

• Put it into the function:

multiplicationTable ← ⍳∘.×⍳

• that’s really it!

• Check it:

multiplicationTable 10