1. © M. Winter COSC 4P41 – Functional Programming 4.14.1 Patterns of computation over lists Applying to all – mapping map :: (a -> b) -> [a] -> [b] map f xs = [f x | x <- xs] Selecting elements – filtering filter :: (a -> Bool) -> [a] -> [a] filter p xs = [x | x<- xs, p x] Combing the items – folding foldr :: (a -> b -> b) -> b -> [a] -> b foldr f s [] = s foldr f s (x:xs) = f x (foldr f s xs)

2. © M. Winter COSC 4P41 – Functional Programming 4.24.2 Folding Binary (associative) operation op on a type a with neutral element e (Examples: (+,0), (*,1), (++,[]), (&&,True), (||,False), (.,id) ) foldr op e [a 0,a 1,…,a n-1,a n ] ( or foldl ) = a 0 `op` a 1 `op` … `op` a n-1 `op` a n foldr op e [] = e Binary (associative) operation op on a type a (without a neutral element) (Examples: max, min, lcm, gcd ) foldr1 op [a 0,a 1,…,a n-1,a n ] ( or foldl ) = a 0 `op` a 1 `op` … `op` a n-1 `op` a n foldr1 op [] is not defined (causes an error) If op is not associative the foldl and foldr versions may result in different functions.

3. © M. Winter COSC 4P41 – Functional Programming 4.34.3 Folding Binary function with arguments from different types Examples: reverse :: [a] -> [a] reverse = foldl (flip (:)) [] sort :: Ord a => [a] -> [a] sort = foldr insert [] Notice that foldr (flip (:)) [] and foldl insert [] don’t work.

4. © M. Winter COSC 4P41 – Functional Programming 4.44.4 Generalization Consider the following function: getWord :: String -> String getWord []= [] getWord (x:xs) | isSpace x= [] | otherwise= x : getWord xs We can generalize this function to have a test – or property – as a parameter. getUntil :: (a -> Bool) -> [a] -> [a] getUntil … getWord = getUntil isSpace

5. © M. Winter COSC 4P41 – Functional Programming 4.54.5 Functions as values Function composition (.) :: (b -> c) -> (a -> b) -> (a -> c) (f. g) x = f (g x) infixl 9 >.> (>.>) :: (a -> b) -> (b -> c) -> (a -> c) g >.> f = f. g

6. © M. Winter COSC 4P41 – Functional Programming 4.64.6 Lambda notation Instead of naming and defining a function, we can write it down directly. \n -> 2*n The following definitions of f are equivalent: f x y = result f = \x y -> result Example: multiples :: Int -> [Int] -> [Int] multiples n = filter (\x -> x `mod` n == 0)

7. © M. Winter COSC 4P41 – Functional Programming 4.74.7 Example: creating an index Example: Input: ”cathedral doggerel cathedral\nbattery doggerel cathedral\n cathedral” Output: battery2 cathedral1, 2, 3 doggerel1, 2 makeIndex :: Doc -> [([Int],Word)] type Doc= String type Line= String type Word= String

8. © M. Winter COSC 4P41 – Functional Programming 4.84.8 Example (cont’d) makeIndex = lines >.>-- Doc-> [Line] numLines >.>-- [Line]-> [(Int,Line)] allNumWords >.>-- [(Int,Line)]-> [(Int,Word)] sortLs >.>-- [(Int,Word)] -> [(Int,Word)] makeLists >.>-- [(Int,Word)]-> [([Int],Word)] amalgamate >.>-- [([Int],Word)]-> [([Int],Word)] shorten-- [([Int],Word)]-> [([Int],Word)]