# Thinking functionally

Functions in Haskell are first class values, meaning they can be passed around like any other data (such are text or numbers), and also bound to names in the same way:

repl example
> notEven = (\x -> not (even x))
> notEven 3
True
> notEven 4
False
> test notEvenFunc = all (==False) [notEvenFunc (2*n) | n <- [1..10]] -- (1)!
> test notEven

-- a more concise definition
> equivalentNotEven = not . even -- (2)!
> equivalentNotEven 3
True
> equivalentNotEven 4
False
> test equivalentNotEven
True

-- even more direct!
> test (not . even) -- (3)!
True
1. Takes a function of type Int -> Bool and tests if it returns False for 10 even numbers.

2. See this section

3. It's not even necessary to assign a variable name to the input function at all: just pass in not . even as an argument.

Functions can also return functions:

repl example
> mkNotEven isEvenFunc = \n -> not (isEvenFunc n)
> (mkNotEven even) 3 -- (1)!
True

-- equivalently
> mkNotEven2 isEvenFunc = not . isEvenFunc
> (mkNotEven2 even) 3
True
1. The result of mkNotEven, when applied to even, is a function to tell if a number is odd.

Note

A common use case for functions returning functions is currying. Relatedly, there are functions curry and uncurry:

repl example
> conjunction x y = x && y
> :t conjunction
conjunction :: Bool -> Bool -> Bool
> conjunction True False
False

(uncurry conjunction) :: (Bool, Bool) -> Bool
> (uncurry conjunction) (True, False)
False

> conjunction2 (x, y) = x && y
> :t conjunction2
conjunction2 :: (Bool, Bool) -> Bool
> conjunction2 (True, False)
False
> :t (curry conjunction2)
(curry conjunction2) :: Bool -> Bool -> Bool
> (curry conjunction2) True False
False

## Composition¶

. chains together (or composes) functions:

repl example
import qualified Data.Text as T

> :t (=='a') -- (1)!
(=='a') :: Char -> Bool
> (=='a') 'b'
False

'h'

> :t ( (=='a') . T.head)
( (=='a') . T.head) :: Text -> Bool
> ( (=='a') . T.head) "hello"
False
1. (=='a') is the function which takes a Char and returns True if it is a otherwise False.

2. Here also, the repl gives a more general type. What is shown is more specific, but still true.

The composition operator . is not a special syntax; it is a function, typically written in infix position like + or *, with the following type:

(.) :: (Char -> Bool) -> (Text -> Char) -> (Text -> Bool)

Or in its general form:

(.) :: (b -> c) -> (a -> b) -> (a -> c)

### Pointfree code¶

Instead of writing func x = not (even x), one can write func = not . even, which avoids having to name a variable x at all.

import Graphics.Gloss.Data.Picture -- (1)!
picture :: Picture
picture =
rotate 90
$translate 20 20$ scale 30 30
circle 2
1. Requires the gloss package.
import Graphics.Gloss.Data.Picture -- (1)!
picture :: Picture
picture = transform (circle 2) where
transform =
rotate 90
. translate 20 20
. scale 30 30
1. Requires the gloss package.

Using flip:

> threeMinusN n = subtract n 3
> threeMinusN 6
-3

-- or (re the above example)
> translateXBy n = translate n 0
> threeMinusN = flip subtract 3
> threeMinusN 6
-3

-- or (re the above example)
> translateXBy = flip translate 0

## Map, fold, scan and zip¶

### Map¶

map :: (a -> b) -> ([a] -> [b])  -- (1)!
1. Your repl will display: (a -> b) -> [a] -> [b], leaving the brackets implicit.

map f ls gives the same result as the Python list comprehension [f(x) for x in ls]. That is, it applies a function f to each element of a list.

Info

In Haskell, one can also write a list comprehension, as: [f x | x <- list].

Note that map's type ranges over all types a and b. This means that it can change the type of the values of the list.

> map (> 5) [1..10]
[False,False,False,False,False,True,True,True,True,True]

> data Piece = Bishop | Knight deriving Show
> map show [Bishop, Knight, Knight]
["Bishop","Knight","Knight"]

### Folds¶

repl example
> foldr (+) 0 [1..10]
55
> :t foldr
foldr :: (a -> b -> b) -> b -> [a] -> b -- (1)!
1. Actually, the repl will give a more general type: foldr :: Foldable t => (a -> b -> b) -> b -> t a -> b. This generalizes foldr from lists to any kind of "container" type that is an instance of the Foldable typeclass.

The first argument (here (+)) is a function of type a -> (b -> b) (or here specifically: Int -> (Int -> Int)) to combine the list elements. The second argument (here 0) is an initial value, to be returned if the input list is the empty list []. The third argument is the input list to be folded, here [1,2,3,4,5,6,7,8,9,10].

This is by no means restricted to numerical code:

data Piece = Bishop | Knight | Rook deriving Show
findBestPiece = foldr best Bishop [Bishop, Knight, Rook, Bishop]

where

best piece1 piece2
| value piece1 >= value piece2 = piece1
| otherwise = piece2

value piece = case piece of
Bishop -> 3
Knight -> 3
Rook -> 5

Tip

In Haskell, it is often preferable to rely on functions like foldr instead of writing explicit recursion yourself.

sumList (x:xs) = x + sumList xs
sumList [x] = x
sumList = foldr 0 (+) -- (1)!
1. In fact, Haskell already provides this function, and calls it sum.

This is for two reasons:

1. It avoids buggy code. For example, sumList fails on the empty list [].
2. It is easier to understand. Explicit recursion can create "goto" like control flow.

Many programs can be expressed as folds (or unfolds!) over lists or other data structures, and Haskell has a range of intermediate and advanced libraries to write time/space efficient one-pass folds over complex data.

### Unfolds¶

repl example
import Data.List
> unfoldr (\x -> if x > 20 then Nothing else Just (even x, x + 3)) 0
[True,False,True,False,True,False,True]

Tip

Use laziness to unfold an infinite structure and then fold it back.

repl example
> let evens = unfoldr (\x -> Just (x + 2, x + 2)) 0 -- (1)!

> foldr (+) 0 (take 10 evens)
110

> any (>10) evens -- (2)!
True
1. An infinite list of even numbers.
2. any is really just a fold, and can be defined in terms of foldr.

### Scans¶

repl example
> scanl (+) 0 [1,1,1,1]
[0,1,2,3,4]

Tip

This corresponds to code that you would write with an accumulator in a non-functional language.

Under