Functional Programming Lecture 12 - more higher order functions

1
Functional ProgrammingLecture 12 - more higher
order functions
2
Recursion over two arguments

• The function zip zips together two lists, to
  produce a new list of pairs.
• zip a -gt b -gt (a,b)
• zip (xxs) (yys) (x,y) zip xs ys
• zip _ _
• or
• zip (xxs) (yys) (x,y) zip xs ys
• zip (yys)
• zip xs
• (Why not zip (xxs) ?)
• E.g. zip 1,2,3 a,b,c gt (1,a),
  (2,b), (3,c)
• zip 1,2,3 a,b gt (1,a),
  (2,b)

3

• The function unzip unzips a list of pairs to
  produce a pair of lists.
• unzip (a,b) -gt (a,b)
• tricky definition
• unzip (,)
• unzip (x,y)zs (xxs,yys)
• where (xs,ys) unzip zs
• What is unzip (1,a), (2,b), (3,c)?
• gt (1,2,3,a,b,c)
• What is unzip (zip xs ys) ?
• gt (xs,ys) if xs and ys have same
  length
• What is zip (unzip zs)?
• gt type error, unzip has produced a
  tuple!

4
Fold and Foldr

• Consider computing the sum of the elements of a
  list
• e1, e2, en i.e. e1 (e2 ( en )).
• sumlist Int -gt Int
• sumlist x x
• sumlist (xxs) x sumlist xs
• Now consider computing the maximum of the
  elements of a list e1, e2, en
• i.e. e1 max (e2 max ( max en)).
• maxlist Int -gt Int
• maxlist x x
• maxlist (xxs) max x (maxlist xs)
• In both cases, we are folding the binary
  function and max through a non-empty list.

5
Fold

• Define fold as the function
• fold (a -gt a -gt a) -gt a -gt a
• a binary function a list the result
• fold f x x
• fold f (xxs) f x (fold f xs)
• Note although x matches the second equation,
  we will always apply the first equation.
• Examples
• fold () False, True, False
• gt True
• fold () Hello , world, !
• gt Hello world!
• fold () 1..5
• gt 120 n.b infix op. Becomes
  prefix in ( )

6

• But fold has not been defined for an empty list.
• What would we like to define
• fold () and
• fold () as ?
• There is no general answer, we must provide one
  for each function we give to fold.
• The function which defines fold for an empty list
  is foldr.

7
Foldr

• Define foldr as the function
• foldr (a -gt b -gt b) -gt b -gt a -gt b
• a binary function empty value a list
  the result
• Note foldr has a more general type.
• foldr f st st
• foldr f st (xxs) f x (foldr f st xs)
• Note foldr means fold right. Usually the extra
  argument is the identity element, i.e. the
  stopping value st is the identity for f.
• foldr is defined in the standard prelude.

8

• Examples
• foldr () False False, True, False
• gt True
• foldr () True True, True
• gt True
• foldr () False True, True
• gt False
• foldr () 1 1..5
• gt 120 1 is an identity
  for
• foldr () 0 1..5
• gt 0 because 0 is a
  zero for
• fac n foldr () 1 1..n