COSC 4 P 41 Functional Programming Some functions

COSC 4 P 41 – Functional Programming Some functions id id x : : a -> a = x const k _ : : a -> b -> a = k ($) f $ x : : (a -> b) -> a -> b = f x (. ) (f. g) x : : (b -> c) -> (a -> b) -> (a -> c) = f (g x) flip f x y : : (a -> b -> c) -> b -> a -> c = f y x © M. Winter 2. 1

COSC 4 P 41 – Functional Programming Recursion fac : : Int -> Int fac n | n == 0 | n > 0 = 1 = fac (n-1) * n fac : : Int -> Int fac n | n == 0 | n > 0 | otherwise = 1 = fac (n-1) * n = error ”fac only defined on natural numbers” © M. Winter 2. 2

COSC 4 P 41 – Functional Programming Primitive Recursion fun : : Int -> Int fun n | n == 0 = … | n > 0 = … fun (n-1) … data Nat = Zero | Succ Nat fun : : Nat -> Nat fun Zero = … fun (Succ n) = … (fun n) … © M. Winter 2. 3

COSC 4 P 41 – Functional Programming Tuples The type (t 1, t 2, …, tn) consists of tuples of values (v 1, v 2, …, vn) in which v 1: : t 1, …, vn: : tn. Examples: min. And. Max : : Int -> (Int, Int) min. And. Max x y | x >= y = (y, x) | otherwise = (x, y) type Shop. Item = (String, Int) A type definition is treated as a shorthand in Haskell – wherever a name like Shop. Item is used, it has exactly the same effect as if (String, Int) had been written. © M. Winter 2. 4

COSC 4 P 41 – Functional Programming Pattern Matching add. Pair : : (Int, Int) -> Int add. Pair (x, y) = x+y shift : : ((Int, Int) -> (Int, Int)) shift ((x, y), z) = (x, (y, z)) name : : Shop. Item -> String price : : Shop. Item -> Int name (n, p) = n price (n, p) = p fst : : (a, b) -> a fst (x, y) = x © M. Winter snd : : (a, b) -> b snd (x, y) = y 2. 5

COSC 4 P 41 – Functional Programming Currying and Uncurrying multiply : : Int -> Int multiply x y = x*y multiply. UC : : (Int, Int) -> Int multiply. UC (x, y) = x*y curry g x (x, y) g y uncurry f (x, y) x f y © M. Winter 2. 6

COSC 4 P 41 – Functional Programming Lists For the type t there is a Haskell type [t] of lists from t. [1, 2, 3, 4, 1, 4] : : [Int] [True] : : [Bool] [’a’, ’b’] : : [Char] ”aab” : : String but type String = [Char] [fac, (+1)] : : [Int -> Int] [] : : [a] © M. Winter 2. 7

COSC 4 P 41 – Functional Programming Lists for enumerated types Lists of numbers, characters and other enumerated types • [n. . m] is the list [n, n+1, …, m]; if n exceeds m, the list is empty. [2. . 7] = [2, 3, 4, 5, 6, 7] [3. 1. . 7. 0] = [3. 1, 4. 1, 5. 1, 6. 1] [’a’. . ’m’] = ”abcdefghijklm” • [n, p. . m] is the list of numbers whose first two elements are n and p with the numbers ascending in steps p-n up to m, [7, 6. . 3] = [7, 6, 5, 4, 3] [0. 0, 0. 3. . 1. 0] = [0. 0, 0. 3, 0. 6, 0. 9] [’a’, ’c’. . ’n’] = ”acegikm” © M. Winter 2. 8

COSC 4 P 41 – Functional Programming List comprehension Suppose that the list ex is [2, 4, 7], then the list comprehension [ 2*n | n <- ex ] will be [4, 8, 14]. Further examples: [ 2*n | n <- ex, even n, n > 3] = [8] [ m+n | (m, n) <- [(2, 3), (2, 1), (7, 8)] ] = [5, 3, 15] [ m+n | n <- ex, even n, m <- ex, odd m] = [9, 11] List comprehension is not a new feature of the language. Each expression using list comprehension can be translated into an expression of the core language, i. e. , without list comprehension. © M. Winter 2. 9

COSC 4 P 41 – Functional Programming Some list functions in Prelude. hs -- data [a] = [] | a : [a] head (x: _) : : [a] -> a = x last [x] last (_: xs) : : [a] -> a = x = last xs tail (_: xs) : : [a] -> [a] = xs init [x] init (x: xs) : : [a] -> [a] = [] = x : init xs null [] null (_: _) : : [a] -> Bool = True = False © M. Winter 2. 10

COSC 4 P 41 – Functional Programming (++) [] ++ ys (x: xs) ++ ys : : [a] -> [a] = ys = x : (xs ++ ys) map f xs : : (a -> b) -> [a] -> [b] = [ f x | x <- xs ] filter p xs : : (a -> Bool) -> [a] = [ x | x <- xs, p x ] concat : : [[a]] -> [a] = foldr (++) [] length : : [a] -> Int = foldl' (n _ -> n + 1) 0 (!!) (x: _) (_: xs) (_: _) [] © M. Winter : : !! 0 = !! n | n>0 = !! _ = [a] -> Int -> a x xs !! (n-1) error "Prelude. !!: negative index" error "Prelude. !!: index too large" 2. 11

COSC 4 P 41 – Functional Programming foldl : : (a -> b -> a) -> a -> [b] -> a foldl f z [] = z foldl f z (x: xs) = foldl f (f z x) xs foldl 1 f (x: xs) : : (a -> a) -> [a] -> a = foldl f x xs foldr : : (a -> b) -> b -> [a] -> b foldr f z [] = z foldr f z (x: xs) = f x (foldr f z xs) foldr 1 f [x] foldr 1 f (x: xs) : : (a -> a) -> [a] -> a = x = f x (foldr 1 f xs) iterate f x : : (a -> a) -> a -> [a] = x : iterate f (f x) reverse © M. Winter : : [a] -> [a] = foldl (flip (: )) [] 2. 12

COSC 4 P 41 – Functional Programming take n _ | n <= 0 take _ [] take n (x: xs) : : Int -> [a] = [] = x : take (n-1) xs drop n xs | n <= 0 drop _ [] drop n (_: xs) : : Int -> [a] = xs = [] = drop (n-1) xs and, or and or : : [Bool] -> Bool = foldr (&&) True = foldr (||) False any, all any p all p : : (a -> Bool) -> [a] -> Bool = or. map p = and. map p elem, not. Elem elem not. Elem © M. Winter : : Eq a => a -> [a] -> Bool = any. (==) = all. (/=) 2. 13

COSC 4 P 41 – Functional Programming sum, product sum product : : Num a => [a] -> a = foldl' (+) 0 = foldl' (*) 1 maximum, minimum : : Ord a => [a] -> a maximum = foldl 1 max minimum = foldl 1 min zip : : [a] -> [b] -> [(a, b)] = zip. With (a b -> (a, b)) zip. With z (a: as) (b: bs) zip. With _ _ _ unzip © M. Winter : : (a->b->c) -> [a]->[b]->[c] = z a b : zip. With z as bs = [] : : [(a, b)] -> ([a], [b]) = foldr ((a, b) ~(as, bs) -> (a: as, b: bs)) ([], []) 2. 14