1. 6-Jul-16 Haskell II Functions and patterns

2. 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

3. User-Defined Data Types 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

4. It’s all about types! Getting the types right is critical to programming in Haskell In GHCi, :type x or :t x will tell you the type of x GHCi> :t "abc" "abc" :: [Char] The :: can be read as “has the type” [Char] is the representation for “list of char” Is "abc" really a list of characters? GHCi> :t ['a', 'b', 'c'] ['a', 'b', 'c'] :: [Char] GHCi> ['a', 'b', 'c'] == "abc" True

5. More type definitions GHCi> :t ["ab", "cd"] ["ab", "cd"] :: [[Char]] [[Char]] is a list of lists of characters = a list of strings GHCi> :t [] [] :: [a] Here, a is a “type variable”— [] could be a list of any type GHCi> :t [[]] [[]] :: [[a]] [[a]] is a list of lists, all of which have the same type GHCi> :t head head :: [a] -> a head takes a list of any type a and returns a single value of type a

6. Type restrictions GHCi> :t even even :: Integral a => a -> Bool Integral a is a type restriction—it says a must be some Integral type The -> indicates a function; in this case, a function from some type a (restricted to be integral) to a Boolean. GHCi> :t (5, 'a', "abc") (5, 'a', "abc") :: Num t => (t, Char, [Char]) (5, 'a', "abc") is a tuple consisting of a Num eric type t, a Char acter, and a string (list of Char ) Unlike lists, tuples can (and often do) contain values of different types

7. Types of functions GHCi> :t mod mod :: Integral a => a -> a -> a GHCi> mod 20 6 2 GHCi> 20 `mod` 6 2 Functions in Haskell take a single argument a -> a -> a associates as (a -> a) -> a GHCi> (mod 20) 6 2 GHCi> :t (mod 20) (mod 20) :: Integral a => a -> a

8. Types of higher-order functions GHCi> :t map map :: (a -> b) -> [a] -> [b] GHCi> :t flip flip :: (a -> b -> c) -> b -> a -> c GHCi> flip map [1, 2, 3, 4] even [False,True,False,True] GHCi> :t flip map flip map :: [a] -> (a -> b) -> [b] compare this with map :: (a -> b) -> [a] -> [b]

9. 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]

10. Layout Grouping is done by indentation, not by braces or parentheses Actually, braces can be used, but that’s unusual The first nonblank character following where, let, or of determines the starting column let x = a + b y = a * b in y / x Every expression in the same group must begin in the same column Make sure your text editor replaces tabs with spaces

11. 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 integers [x*x | x <- [1..], even x] == squares of positive even integers [(x, y) | x <- [1..10], y <- [1..10], x < y]

13. 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!

14. 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

15. 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

16. 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)

17. Point free style Functions that take parameters (“points”) can be written as functions with implicit parameters GHCi> let square_all xs = map (\x -> x^2) xs GHCi> square_all [1, 2, 3, 4, 5] [1,4,9,16,25] GHCi> let square_all = map (\x -> x^2) GHCi> square_all [1, 2, 3, 4, 5] [1,4,9,16,25] GHCi> let square_all = map (^2) GHCi> square_all [1, 2, 3, 4, 5] [1,4,9,16,25] Point free style can result in easier to read, less cluttered code It can also result in obscure code, so use with care 17

18. 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]

19. 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]

20. 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

21. 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

22. 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

23. 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' True Prelude> (flip elem) "aeiou" 'o' True

24. 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

25. 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")

26. 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!")

27. Function Definition I Functions are defined with = fact :: Int -> Int -- explicit type fact n = if n == 0 then 1 else n * fact (n - 1) There is no requirement that you explicitly state the type of each function you define Haskell is superb at inferring the types of functions Using Hadley-Milner type inference However: All experienced Haskell programmers do explicitly state the type of each function Doing so catches almost all programming errors! Haskell programming is all about getting the types right

28. Function Definition II Functions are usually defined by cases fact :: Int -> Int fact n | n == 0 = 1 | otherwise = n * fact (n - 1) fact :: Int -> Int fact n = case n of 0 -> 1 n -> n * fact (n - 1) These are equivalent definitions

29. Function Definition III You can separate the cases with “patterns” fact :: Int -> Int fact 0 = 1 fact n = n * fact (n - 1) How does this work?

30. 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

31. 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

32. 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

33. 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]

34. 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 This is the only arithmetical pattern; it does not generalize to any other pattern

35. 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

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

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

38. The End