Higher order functions

The Heart and Soul of Functional Programming

What is a higher order function?

• It's a function that takes another function as an argument.
• even 1 False even 2 True map even [1,2,3,4,5] [False,True,False,True,False] filter even [1,2,3,4,5] [2,4]
• even is a first-order function.
• map and filter are higher-order functions.

The types of higher order functions

• map    :: (a -> b)    -> [a] -> [b]
  filter :: (a -> Bool) -> [a] -> [a]
  

• The first argument is a function in both cases.
• They are also polymorphic. This is a powerful combination!

Is this a big deal?

• Yes!
• Higher-order functions are the "heart and soul" of functional programming!
• A higher-order function can be much more versatile than a first-order function, because a part of its behaviour can be controlled by the caller.
• We can replace many similar first-order functions by one higher-order function, factoring out the difference into a function argument.

Functions returning functions?

• If passing functions as arguments is powerful, how about returning functions as results?
• Compare these types:

  1. Int -> Int -> Int

  2. Int -> (Int -> Int)

  3. (Int -> Int) -> Int

• Quiz: how many arguments does the following functions have?

  • pick 1 = fst
    pick 2 = snd

  • id :: a -> a

Curried functions

• Compare these types:

  1. Int -> Int -> Int

  2. (Int,Int) -> Int

• Both are in effect functions that take two

  Int

  s and return an

  Int

  .
• We have seen that Haskell favours the first variant.
  • Such functions are called curried functions, after the logician Haskell Curry.
  • Incidentally (or not) the Haskell language is also named after Haskell Curry.
  • The predefined functions

    curry

    and

    uncurry

    convert between the two forms.

Curried functions

Uncurried vs curried functions as tables

• In math, functions are sometimes illustrated as tables (example)

• (Int,Int)->Int In Out (1,1) 2 (1,2) 3 (2,1) 3 (2,2) 4

  Int->Int->Int In Out 1 In  Out 1 2 2 3 2 In  Out 1 3 2 4

Live Demo

• Let's look at some illustrative examples
• Source code: HigherOrderFunctions.hs

Case study: sums and products of lists

• sum []     = 0
  sum (x:xs) = x + sum xs

• product []     = 1
  product (x:xs) = x * product xs

• Common pattern: combining the elements of a list with an operator
• Differences: the operator and the base case.

Factoring out the differences

• foldr op base [] = base
  foldr op base (x:xs) = x `op` foldr op base xs

• Note:

  x `f` y

  is the same as

  f x y

• What is the type of foldr?

• foldr :: (a -> b -> b) -> b -> [a] -> b

• foldr is also known as reduce. You already know map, so now you know the essence of MapReduce!

Code reuse

• We can now simplify the definitions of sum and product.

• sum     xs = foldr (+) 0 xs
  product xs = foldr (*) 1 xs

• And foldr is useful in many more cases!

• or     xs = foldr (||) False xs
  and    xs = foldr (&&) True xs
  concat xs = foldr (++) [] xs
  
  maximum (x:xs) = foldr max x xs

  • These are all examples of predefined functions in Haskell.

An intuition about foldr

• Think of

  foldr (&) z

  (where

  &

  is some operator) as function that
  • replaces

    :

    with

    &

    , and
  • replaces

    []

    with

    z

    .

•             xs == a : b : c : d : … : []

  
  

  foldr (&) z xs == a & b & c & d & … & z

  
  

  foldr (+) 0 xs == a + b + c + d + … + 0

  
  

  foldr (*) 1 xs == a * b * c * d * … * 1

Quiz

What does these function do?

• f1 xs = foldr (:) [] xs

• f2 xs ys = foldr (:) ys xs

• f3 xs = foldr snoc [] xs
    where snoc x ys = ys++[x]
  

• f4 f xs = foldr fc [] xs
    where fc x ys = f x:ys

Passing functions as arguments

• Sometimes the function we want to pass as an argument to a higher-order function hasn't been defined.
• We can solve it by naming and defining a helper function (as in f3 above).
• Let's also look at some other convenient way to create functions.

Partial application

• We can apply functions to "too few" arguments
• take 2 "Haskell" "Ha" map (take 2) ["Hello","Haskell"] ["He","Ha"]
• The advantage with curried functions reveals itself!

Partial application of infix operators

• Operators can be used as functions by omitting one operand.

  • (*3) 5 == 15

  • map (*3) [1..5] == [3,6,9,12,15]

  • filter (>3) [1..5] == [4,5]

  • filter (3>) [1..5] == [1,2]

• This is called sections.

Function composition

• Example: remove white space from a string

  • removeSpaces "abc def \n ghi" = "abcdefghi"

• We can use isSpace from module Data.Char,

  • removeSpaces s = filter (not . isSpace) s

• Function composition is defined as

  • (f . g) x = f (g x)
    

• What is the type of (.)?

  (b->c)->(a->b)->(a->c)

Anonymous functions

• So far we have defined functions by equations where the function parameters appears on the left hand side.

  • snoc y ys = ys++[y]
    

• But there is another way

  • snoc = \ y ys -> ys++[y]
    

• If we are going to use a function only in one place, we can use it directly without giving it a name first

  • f3 xs = foldr snoc [] xs
      where snoc y ys = ys++[y]
    

  • f3 xs = foldr (\ y ys -> ys++[y]) [] xs

• We have the same choice for functions as for any other value.
• The \ is supposed to look a bit like a λ (the greek letter lambda) and comes from the λ-calculus, where functions are always written in this way.

Example

• Combine a list of lines into a string with newline characters:

  • unlines ["abc","def","ghi"] = "abc\ndef\nghi\n"

• One-line definition:

  • unlines ls = foldr (\xs ys->xs++"\n"++ys) [] ls

• or with a named helper function:

  • unlines ls = foldr join [] ls
      where join xs ys = xs ++ "\n" ++ ys

Eta conversion

• Consider definitions of the form

  • f x = g x

• We are saying that f is the same function as g. This can be said in a simpler way

  • f = g

• This is called Eta (η) conversion/reduction/expansion.
• Caveat: the monomorphism restriction in Haskell:
  • A definition without parameters and without a type signature is not allowed to be overloaded.

Eta conversion examples

or     = foldr (||) False
and    = foldr (&&) True
concat = foldr (++) []

removeSpaces = filter (not . isSpace)

unlines = foldr (\xs ys->xs++"\n"++ys) []

More standard higher-order functions

• Define

  • takeLine :: String -> String

• such that

  • takeLine "abc\ndef\nghi\n" = "abc"

• Generalize it to

  • takeWhile :: (a->Bool) -> [a] -> [a]
    

• Also define the complementary

  • dropWhile :: (a->Bool) -> [a] -> [a]
    

Lines

• Define

  • lines :: String -> [String]
    

• such that

  • lines "abc\ndef\nghi\n" = ["abc","def","ghi"]
    

• Generalize it to

  • segments :: (a->Bool) -> [a] -> [[a]]
    

Words

• Define

  • words :: String -> [String]
    

• such that

  • words "abc def  ghi" = ["abc","def","ghi"]
    

A bigger example: word counting

• Define a function that counts how many times words occur in a text and displays each word with its count.

• wordCounts :: String -> String

• putStr (wordCounts "hello clouds\nhello sky") clouds: 1 hello: 2 sky: 1
• Source code: HigherOrderFunctions.hs, towards the bottom of the file.

Lessons

• Higher-order functions take functions as arguments, making them flexible and useful in many situations.
• By writing higher-order functions to capture common patterns, we can reduce the amount of code we need to write dramatically.
• Anonymous functions, partial applications, function composition and sections help us create functions to pass as arguments, often eliminating the need for a separate function definition.
• Haskell provides many useful higher-order functions;
  • break problems into small parts, each of which can be solved by an existing function.

Higher order functions in other languages?

• Consider these generic sorting functions in Haskell:

  • sort   :: Ord a            => [a] -> [a]
    sortBy :: (a->a->Ordering) -> [a] -> [a]
    

• Compare with a generic sorting function in C:
  • void qsort(void *base, size_t nmemb, size_t size, int (*compar)(const void *, const void *));