Functional Programming in

1
Functional Programming in
Peter Achten - Rinus Plasmeijer University of
Nijmegen www.cs.kun.nl/clean
2
Overview

• Predefined functions and operators on basic types
• Integer numbers
• Real numbers
• Booleans (truth values)
• Characters
• Predefined functions on
• a lists
• (a,b) tuples

3
Predefined functions on lists

• hd a ? a
• hd abort "Cannot take the head of empty
  list"
• hd x _ x
• tl a ? a
• tl abort "Cannot take the tail of
  empty list"
• tl _ xs xs

4
Take n-elements of a list

• Define a function that returns the first
  n-elements of a list.If the list is smaller than
  return just that list.
• Name and type
• take Int a ? a
• 1. The list is empty
• take n
• 2. The list is not empty
• trivial case zero elements to look for
• take 0 list
• at least one element to look for
• take n e es .

5
Take n-elements of a list

• Define a function that returns the first
  n-elements of a list.
• If the list is smaller than return just that
  list.
• Name and type
• take Int a ? a
• 1. The list is empty
• take n
• 2. The list is not empty
• trivial case zero elements to look for
• take 0 list
• at least one element to look for
• take n e es take (n-1) es

6
Take n-elements of a list

• Define a function that returns the first
  n-elements of a list.
• If the list is smaller than return just that
  list.
• Name and type
• take Int a ? a
• 1. The list is empty
• take n
• 2. The list is not empty
• trivial case zero elements to look for
• take 0 list
• at least one element to look for
• take n e es e take (n-1) es

7
Take n-elements of a list

• take Int a ? atake n take 0
  list take n e es e take (n-1)
  es
• Start take 2 1, 2, 3
• Start // definition of Start
• ? take 2 1 2 3 // definition
  of take, rule 3
• ? 1 take 1 2 3 // definition
  of take, rule 3
• ? 1 2 take 0 3 // definition
  of take, rule 2
• ? 1 2 // definition of take, rule
  2
• 1,2

8
Predefined functions on lists

• drop Int a ? a
• drop n drop 0 list list
• drop n e es drop (n-1) es
• Start ( take 2 1, 2, 3 , drop 2 1, 2,
  3 )
• ? (1, 2, 3)

9
List indexing

• (!!) infixl 9 a Int ? a // take n-th
  element of list
• (!!) x _ 0 x
• (!!) _ xs n
• ngt0 xs !! (n-1)
• (!!) _ n abort ("Cannot find element"
  toString n "in list")
• 1, 2, 3, 4, 5 !! 2 3

10
Sum and Length on Integers

• sum Int ? Int // take sum of list
• sum 0
• sum x xs x sum xs
• length a ? Int // take length of list
• length 0
• length _ xs 1 length xs

11
Sum, Length and Average on Integers

• sum Int ? Int // take sum of integer list
• sum 0
• sum x xs x sum xs
• length a ? Int // take length of any list
• length 0
• length _ xs 1 length xs
• -------
• average Int ? Int // take average of
  integer list
• average list sum list / length list
• Start average 1..10 5

12
Sum, Length and Average on Reals

• sum Real ? Real // take sum of real list
• sum 0.0
• sum x xs x sum xs
• length a ? Int // take length of any list
• length 0
• length _ xs 1 length xs
• -------
• average Real ? Real // take average of
  real list
• average list sum list / toReal (length list)
• Start average 1.0..10.0 5.5

13
Overloaded definitions of Sum and Average

• sum a ? a zero, a // take sum of list
• sum zero
• sum x xs x sum xs
• length a ? Int // take length of any list
• length 0
• length _ xs 1 length xs
• -------
• average a ? a , /, fromInt, zero a //
  take average of list
• average list sum list / fromInt (length list)
• Start ( average 1.0..10.0 5.5
• , average 1..10 5 )

14
Append and Flatten

• 1. . 5 6 ..10 1,2,3,4,5,6,7,8,9,10
• () infixr 5 a a ? a // appending
  lists
• () list list
• () x xs list x xs list
• flatten 1. . 5 , 6 ..10
  1,2,3,4,5,6,7,8,9,10
• flatten a ? a // flatten a list
• flatten
• flatten x xs x flatten xs

15
Naïve Reverse

• reverse 1. . 5 5,4,3,2,1
• reverse a ? a // reverse a list
• reverse
• reverse x xs reverse xs x

16
Naïve Reverse

• reverse reverse x xs
  reverse xs x
• () list list() x xs
  list x xs list
• reverse 1..5
• ? reverse 2..5 1 // definition of
  reverse, rule 2
• ? (reverse 3..5 2) 1 //
  definitions of reverse, rule 2
• ? (5 4) 3) 2) 1 //
  definition of , rule 2
• ? (((5 4) 3) 2) 1 //
  definition of , rule 1
• ? ((5,4 3) 2) 1 // definition
  of , rule 2
• ? (5 4 3 2) 1 //
  definition of , rule 2
• ? (5 4 3 2) 1 //
  definition of , rule 1
• ? (5,4,3 2) 1 // and so on
• ? 5,4,3,2,1 // normal form reached

17
Efficient Reverse with accumulator

• reverse a ? a // reverse a list
• reverse list reverse' list
• reverse' a a ? a // reverse with
  accumulator
• reverse' reversed_so_far
  reversed_so_far
• reverse' x xs reversed_so_far reverse'
  xs x reversed_so_far
• reverse 1..5 // definition of reverse
• ? reverse' 1..5 // definition of
  reverse', rule 2
• ? reverse' 2..5 1 // definition of
  reverse', rule 2
• ? reverse' 3..5 2 1 //
  definition of reverse', rule 2
• ? reverse' 4..5 3..1 // definition of
  reverse', rule 2
• ? reverse' 5 4..1 // definition of
  reverse', rule 2
• ? reverse' 5..1 // definition of
  reverse', rule 1
• ? 5..1 // normal form reached

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Local function definitions

• reverse a ? a // reverse a list
• reverse list reverse' list
• where
• reverse' a a ? a // reverse with
  accumulator
• reverse' reversed_so_far
  reversed_so_far
• reverse' x xs reversed_so_far reverse'
  xs x reversed_so_far
• After the 'where' one can define local function
  definitions
• They can only be used by the surrounding
  functions
• and the functions on the same level
• This is determined by the lay-out rule

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Local function definitions

• reverse a ? a // reverse a list
• reverse list reverse' list
• where
• reverse' a a ? a // reverse with
  accumulator
• reverse' reversed_so_far
  reversed_so_far
• reverse' x xs reversed_so_far reverse'
  xs x reversed_so_far
• The position of the first definition after a
  where defines a border line
• Each new definition on the same level must start
  at this line
• A previous definition continues if it starts to
  the right of the line
• A level is ended when a definition starts to the
  left of the line

20
Local function definitions

• reverse a ? a // reverse a list
• reverse list reverse' list
• where
• reverse' a a ? a // reverse with
  accumulator
• reverse' reversed_so_far
• reversed_so_far
• reverse' x xs reversed_so_far
• reverse' xs x reversed_so_far
• The position of the first definition after a
  where defines a border line.
• Each new definition on the same level must start
  at this line
• A previous definition continues if it starts to
  the right of the line
• A level is ended when a definition start to the
  left of the line

21
Local function definitions

• reverse a ? a // reverse a list
• reverse list reverse' list
• where
• reverse' a a ? a // reverse with
  accumulator
• reverse' reversed_so_far
  reversed_so_far
• reverse' x xs reversed_so_far reverse'
  xs x reversed_so_far

22
Higher-order functions

• Higher-order functions are functions which have
  a function as argument or as result.
• at_zero (Int ?Int) ? Int
• at_zero f f 0
• inc n n 1
• square n n n
• ident n n
• at_zero inc ? inc 0 ? 0 1 ? 1
• at_zero square ? square 0 ? 0 0 ? 0
• at_zero ident ? ident 0 ? 0

23
Higher-order functions

• Higher-order functions are functions which have
  a function as argument or as result.
• func_result Int ? (Int ? Int)
• func_result 0 inc
• func_result 1 square
• func_result f ident
• inc n n 1
• square n n n
• ident n n
• func_result 0 6 ? (func_result 0) 6 ? inc
  6 ? 6 1 ? 7
• func_result 1 6 ? (func_result 1) 6 ? square 6
  ? 6 6 ? 36
• func_result 2 6 ? (func_result 2) 6 ? ident
  6 ? 6

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Curried functions

• function application is left associative
• function x1 x2 xn
• ( ( ( function x1 ) x2 ) xn )
• function type is right associative
• function a1 a2 an?an1
• function (a1 ? (a2 ? ( ? (an?an1) )))
• plus Int Int ? Int plus (Int ? (Int ?
  Int))
• plus x y x y plus x y (() x)
  y

25
Map a function over a list

• Define a function map which applies a function
  to each argument of the list.
• Name and type
• map (a ? b) a ? b
• 1. The list is empty
• map f
• 2. The list is not empty
• map f x xs f x map f xs

26
Map a function over a list

• map (a ? b) a ? b
• map f
• map f x xs f x map f xs
• square Int ? Int
• square x x x
• Start map square 2, 3
• ? 4 , 9
• 2, 3 Int
• square (Int ? Int)
• map square 2, 3 Int

27
Map a function over a list

• map (a ? b) a ? b
• map f
• map f x xs f x map f xs
• Start map square 2, 3
• Start // definition of Start
• ? map square 2 3 // definition of
  map, rule 2
• ? square 2 map square 3 //
  definition of square
• ? 4 map square 3 // definition of
  map, rule 2
• ? 4 square 3 map square //
  definition of square
• ? 4 9 map square // definition of
  map, rule 1
• ? 4 9
• ? 4 , 9

28
Map a function over a list

• map (a ? b) a ? b
• map f
• map f x xs f x map f xs
• square Int ? Int
• square x x x
• Start map (()1) 2, 3
• ? 3 , 4
• 2, 3 Int
• () (Int Int ? Int)
• (() 1) (Int ? Int)

29
Map a function over a list

• map (a ? b) a ? b
• map f
• map f x xs f x map f xs
• Start map (()1) 2, 3
• Start // definition of Start
• ? map (()1) 2 3 // definition of
  map, rule 2
• ? (()1) 2 map (()1) 3 //
  definition of
• ? 3 map (()1) 3 // definition of
  map, rule 2
• ? 3 (()1) 3 map (()1) //
  definition of
• ? 3 4 map (()1) // definition of
  map, rule 1
• ? 3 4
• ? 3 , 4