Functions and patterns Haskell II Functions and patterns 22-Apr-19

Data Types Int + - * / ^ even odd Float + - * / ^ sin cos pi truncate Char ord chr isSpace isUpper … Bool && || not Lists : ++ head tail last init take Tuples fst snd Polymorphic: < <= == /= => > show

User-Defined Data Types data Color = Red | Blue toString Red = "red" toString Blue = "blue" data Tree a = Leaf a | Branch (Tree a) (Tree a) Can be tricky to use

Assorted Syntax Comments are -- to end of line or {- to -} (these may be nested) Types are capitalized, variables are not Indentation may be used in place of braces Infix operators: + - `mod` `not` Prefix operators: (+) (-) mod not Types: take :: Int -> [a] -> [a]

Layout The first nonblank character following where, let, or of determines the starting column let x = a + b y = a * b in y / x If you start too far to the left, that may end an enclosing clause You can use { } instead, but this is not usually done

Infinite Lists [1..5] == [1, 2, 3, 4, 5] [1..] == all positive integers [5, 10..32] == [5, 10, 15, 20, 25, 30] [5, 10..] == positive multiples of 5 [x*x | x <- [1..]] == squares of positive ints [x*x | x <- [1..], even x] == squares of positive even ints [(x, y) | x <- [1..10], y <- [1..10], x < y]

Functions are also data Functions are “first-class objects” Functions can be assigned Functions can be passed as parameters Functions can be stored in data structures There are operations on functions But functions can’t be tested for equality Theoretically very hard!

Anonymous Functions Form is \ parameters -> body Example: \x y -> (x + y) / 2 the \ is pronounced “lambda” the x and y are the formal parameters inc x = x + 1 this is shorthand for inc = \x -> x + 1 add x y = x + y this is shorthand for add = \x y -> x + y

Currying Technique named after Haskell Curry Functions only need one argument Currying absorbs an argument into a function f a b = (f a) b, where (f a) is a curried function (avg 6) 8 7.0

Slicing Functions may be “partially applied” inc x = x + 1 can be defined instead as inc = (+ 1) add x y = x + y can be defined instead as add = (+) negative = (< 0)

map map :: (a -> b) -> [a] -> [b] applies the function to all elements of the list Prelude> map odd [1..5] [True,False,True,False,True] Prelude> map (* 2) [1..5] [2,4,6,8,10]

filter filter :: (a -> Bool) -> [a] -> [a] Returns the elements that satisfy the test Prelude> filter even [1..10] [2,4,6,8,10] Prelude> filter (\x -> x>3 && x<10) [1..20] [4,5,6,7,8,9]

iterate iterate :: (a -> a) -> [a] -> [a] f x returns the list [x, f x, f f x, f f f x, …] Prelude> take 8 (iterate (2 *) 1) [1,2,4,8,16,32,64,128] Prelude> iterate tail [1..3] [[1,2,3],[2,3],[3],[], *** Exception: Prelude.tail: empty list

foldl foldl :: (a -> b -> a) -> a -> [b] -> a foldl f i x starts with i, repeatedly applies f to i and the next element in the list x Prelude> foldl (-) 100 [1..3] 94 94 = 100 - 1 - 2 - 3

foldl1 foldl1 :: (a -> a -> a) -> [a] -> a Same as: foldl f (head x) (tail x) Prelude> foldl1 (-) [100, 1, 2, 3] 94 Prelude> foldl1 (+) [1..100] 5050

flip flip :: (a -> b -> c) -> b ->a -> c Reverses first two arguments of a function Prelude> elem 'o' "aeiou" True Prelude> flip elem "aeiou" 'o' Prelude> (flip elem) "aeiou" 'o'

Function composition with (.) (.) :: (a -> b) -> (c -> a) -> (c -> b) (f . g) x is the same as f (g x) double x = x + x quadruple = double . double doubleFirst = (* 2) . head Main> quadruple 3 12 Main> doubleFirst [3..10] 6

span span :: (a -> Bool) -> [a] -> ([a], [a]) Break the lists into two lists those at the front that satisfy the condition the rest Main> span (<= 5) [1..10] ([1,2,3,4,5],[6,7,8,9,10]) Main> span (< 'm') "abracadabra" ("ab","racadabra")

break break :: (a -> Bool) -> [a] -> ([a], [a]) Break the lists into two lists those at the front that fail the condition the rest Main> break (== ' ') "Haskell is neat!" ("Haskell"," is neat!")

Function Definition I Functions are defined with = fact n = if n == 0 then 1 else n * fact (n - 1)

Function Definition II Functions are usually defined by cases fact n | n == 0 = 1 | otherwise = n * fact (n - 1) fact n = case n of 0 -> 1 n -> n * fact (n - 1) These are “the same”

Function Definition III You can separate the cases with “patterns” fact :: Int -> Int -- not essential fact 0 = 1 fact n = n * fact (n - 1) How does this work?

Pattern Matching Functions cannot in general be overloaded But they can be broken into cases Each case must have the same signature fact :: Int -> Int -- explicit signature fact 0 = 1 fact n = n * fact (n - 1) fact 5 won’t match the first, but will match the second

Pattern Types I A variable will match anything A wildcard, _, will match anything, but you can’t use the matched value A constant will match only that value Tuples will match tuples, if same length and constituents match Lists will match lists, if same length and constituents match However, the pattern may specify a list of arbitrary length

Pattern Types II (h:t) will match a nonempty list whose head is h and whose tail is t second (h:t) = head t Main> second [1..5] 2

Pattern Types III “As-patterns” have the form w@pattern When the pattern matches, the w matches the whole of the thing matched firstThree all@(h:t) = take 3 all Main> firstThree [1..10] [1,2,3]

Pattern Types IV (n+k) matches any value equal to or greater than k; n is k less than the value matched silly (n+5) = n Main> silly 20 15

Advantages of Haskell Extremely concise Easy to understand no, really! No core dumps Polymorphism improves chances of re-use Powerful abstractions Built-in memory management

Disadvantages of Haskell Unfamiliar Slow because compromises are less in favor of the machine

quicksort quicksort [] = [] quicksort (x:xs) = quicksort [y | y <- xs, y < x] ++ [x] ++ quicksort [y | y <- xs, y >= x]

The End