1. Lee CSCE 314 TAMU 1 CSCE 314 Programming Languages Haskell: More on Functions and List Comprehensions Dr. Hyunyoung Lee

2. Lee CSCE 314 TAMU 2  Defining Functions  List Comprehensions  Recursion Outline

3. Lee CSCE 314 TAMU 3 Conditional Expressions As in most programming languages, functions can be defined using conditional expressions: if cond then e1 else e2 e1 and e2 must be of the same type else branch is always present abs :: Int -> Int abs n = if n >= 0 then n else –n max :: Int -> Int -> Int max x y = if x <= y then y else x take :: Int -> [a] -> [a] take n xs = if n <= 0 then [] else if xs == [] then [] else (head xs) : take (n-1) (tail xs)

4. Lee CSCE 314 TAMU 4 Guarded Equations As an alternative to conditionals, functions can also be defined using guarded equations. abs n | n >= 0 = n | otherwise = -n Prelude: otherwise = True Guarded equations can be used to make definitions involving multiple conditions easier to read: signum n | n < 0 = -1 | n == 0 = 0 | otherwise = 1 signum n = if n < 0 then -1 else if n == 0 then 0 else 1 compare with …

5. Lee CSCE 314 TAMU 5 Guarded Equations (Cont.) Guards and patterns can be freely mixed, the first equation whose pattern matches and guard is satisfied is chosen. take :: Int -> [a] -> [a] take n xs | n <= 0 = [] take n [] = [] take n (x:xs) = x : take (n-1) xs

6. Lee CSCE 314 TAMU 6 Pattern Matching Many functions are defined using pattern matching on their arguments. Pattern can be a constant value, or include one or more variables. not :: Bool -> Bool not False = True not True = False not maps False to True, and True to False.

7. Lee CSCE 314 TAMU 7 Functions can often be defined in many different ways using pattern matching. For example (&&) :: Bool  Bool  Bool True && True = True True && False = False False && True = False False && False = False True && True = True _ && _ = False can be defined more compactly by False && _ = False True && b = b However, the following definition is more efficient, because it avoids evaluating the second argument if the first argument is False: The underscore symbol _ is a wildcard pattern that matches any argument value.

8. Lee CSCE 314 TAMU 8  Patterns may not repeat variables. For example, the following definition gives an error: b && b = b _ && _ = False  Patterns are matched in order. For example, the following definition always returns False: _ && _ = False True && True = True

9. Lee CSCE 314 TAMU 9 List Patterns Internally, every non-empty list is constructed by repeated use of an operator (:) called “cons” that adds an element to the start of a list. [1,2,3,4] Means 1:(2:(3:(4:[]))). Functions on lists can be defined using x:xs patterns. head :: [a]  a head (x:_) = x tail :: [a]  [a] tail (_:xs) = xs head and tail map any non- empty list to its first and remaining elements. is this definition complete?

10. Lee CSCE 314 TAMU 10 Note:  x:xs patterns must be parenthesised, because application has priority over (:). For example, the following definition gives an error:  x:xs patterns only match non-empty lists: > head [] Error head x:_ = x  Patterns can contain arbitrarily deep structure: f (_: (_, x):_) = x g [[_]] = True

11. Lee CSCE 314 TAMU 11 Totality of Functions head (x:_) = x > head [] *** Exception: Prelude.head: empty list  (Total) function maps every element in the function’s domain to an element in its codomain.  Partial function maps zero or more elements in the function’s domain to an element in its codomain, and can leave some elements undefined.  Haskell functions can be partial. For example: > "10elements" !! 10 *** Exception: Prelude.(!!): index too large

12. Lee CSCE 314 TAMU 12 Lambda Expressions Functions can be constructed without naming the functions by using lambda expressions. x  x+x This nameless function takes a number x and returns the result x+x.  The symbol is the Greek letter lambda, and is typed at the keyboard as a backslash \.  In mathematics, nameless functions are usually denoted using the  symbol, as in x  x+x.  In Haskell, the use of the symbol for nameless functions comes from the lambda calculus, the theory of functions on which Haskell is based.

13. Lee CSCE 314 TAMU 13 Why Are Lambda's Useful? 1.Lambda expressions can be used to give a formal meaning to functions defined using currying. For example: add x y = x + y square x = x * x add = \x -> (\y -> x + y) square = \x -> x * x means

14. Lee CSCE 314 TAMU 14 odds n = map f [0..n-1] where f x = x * 2 + 1 can be simplified to odds n = map (\x -> x * 2 + 1) [0..n-1] 2.Lambda expressions can be used to avoid naming functions that are only referenced once. For example: 3. Lambda expressions can be bound to a name (function argument) incrementer = \x -> x + 1 add (incrementer 5) 6

15. Lee CSCE 314 TAMU 15 Case Expressions take m ys = case (m, ys) of (n, _) | n [] (_, []) -> [] (n, x:xs) -> x : take (m-1) xs Pattern matching need not be tied to function definitions; they also work with case expressions. For example: length = \ls -> case ls of [] -> 0 (_:xs) -> 1 + length xs length [] = 0 length (_:xs) = 1 + length xs using a case expression and a lambda: (1) (2)

16. Lee CSCE 314 TAMU 16 Let and Where reserved s = -- using let let keywords = words “if then else for while” relops = words “== != =“ elemInAny w [] = False elemInAny w (l:ls) = w `elem` l || elemInAny w ls in elemInAny s [keywords, relops] The let and where clauses are used to create a local scope within a function. For example: unzip :: [(a, b)] -> ([a], [b]) unzip [] = ([], []) unzip ((a, b):rest) = let (as, bs) = unzip rest in (a:as, b:bs) (1) (2) reserved s = -- using where elemInAny s [keywords, relops] where keywords = words “if then else for while”... elemInAny w (l:ls) = w `elem` l || elemInAny w ls

17. Lee CSCE 314 TAMU 17 Let vs. Where f x -- using let-in expression = let a = w x in case () of _ | cond1 x -> a | cond2 x -> g a | otherwise -> f (h x a) The let … in … is an expression, whereas where blocks are declarations bound to the context. For example: f x -- using where block | cond1 x = a | cond2 x = g a | otherwise = f (h x a) where a = w x

18. Lee CSCE 314 TAMU 18 Sections An operator written between its two arguments can be converted into a curried function written before its two arguments by using parentheses. For example: > 1 + 2 3 This convention also allows one of the arguments of the operator to be included in the parentheses. For example: > (1+) 2 > (+2) 1 3 In general, if  is an operator then functions of the form (  ), (x  ) and (  y) are called sections. > (+) 1 2 3

19. Lee CSCE 314 TAMU 19 Why Are Sections Useful? Useful functions can sometimes be constructed in a simple way using sections. For example: - successor function ( \y->1+y ) - reciprocation function - doubling function - halving function (1+) (*2) (/2) (1/) Sometimes it is convenient or necessary to pass an operator as a parameter or when stating its type

20. Lee CSCE 314 TAMU 20 Exercises Consider a function safetail that behaves in the same way as tail, except that safetail maps the empty list to the empty list, whereas tail gives an error in this case. Define safetail using: (a)a conditional expression; (b)guarded equations; (c)pattern matching. Hint: the library function null :: [a]  Bool can be used to test if a list is empty. (1)

21. Lee CSCE 314 TAMU 21 Give three possible definitions for the logical or operator (||) using pattern matching. (2) Redefine the following version of (&&) using conditionals rather than patterns: (3) True && True = True _ && _ = False Do the same for the following version: (4) True && b = b False && _ = False

22. Lee CSCE 314 TAMU 22  Defining Functions  List Comprehensions  Recursion Outline

23. Lee CSCE 314 TAMU 23  A convenient syntax for defining lists  Set comprehension - In mathematics, the comprehension notation can be used to construct new sets from old sets. E.g., {(x 2,y 2 )|x ∈ {1,2,...,10}, y ∈ {1,2,...,10}, x 2 +y 2 ≤101}  Same in Haskell: new lists from old lists [(x^2, y^2) | x <- [1..10], y <- [1..10], x^2 + y^2 <= 101] generates: [(1,1),(1,4),(1,9),(1,16),(1,25),(1,36),(1,49),(1,64),(1,81),(1,100),(4,1),(4,4),(4,9),(4, 16),(4,25),(4,36),(4,49),(4,64),(4,81),(9,1),(9,4),(9,9),(9,16),(9,25),(9,36),(9,49), (9,64),(9,81),(16,1),(16,4),(16,9),(16,16),(16,25),(16,36),(16,49),(16,64),(16,81 ),(25,1),(25,4),(25,9),(25,16),(25,25),(25,36),(25,49),(25,64),(36,1),(36,4),(36,9 ),(36,16),(36,25),(36,36),(36,49),(36,64),(49,1),(49,4),(49,9),(49,16),(49,25),(4 9,36),(49,49),(64,1),(64,4),(64,9),(64,16),(64,25),(64,36),(81,1),(81,4),(81,9),(8 1,16),(100,1)] List Comprehensions

24. Lee CSCE 314 TAMU 24  The expression x <- [1..10] is called a generator, as it states how to generate values for x.  generators can be infinite, e.g.,  Comprehensions can have multiple generators, separated by commas. For example:  Multiple generators are like nested loops, with later generators as more deeply nested loops whose variables change value more frequently. > [(x,y) | x <- [1,2,3], y <- [4,5]] [(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)] List Comprehensions: Generators > take 3 [x | x <- [1..]] [1,2,3]

25. Lee CSCE 314 TAMU 25 > [(x,y) | y <- [4,5], x <- [1,2,3]] [(1,4),(2,4),(3,4),(1,5),(2,5),(3,5)]  For example: x  [1,2,3] is the last generator, so the value of the x component of each pair changes most frequently.

26. Lee CSCE 314 TAMU 26 Dependant Generators Later generators can depend on the variables that are introduced by earlier generators. [(x,y) | x <- [1..3], y <- [x..3]] The list [(1,1),(1,2),(1,3),(2,2),(2,3),(3,3)] of all pairs of numbers (x,y) such that x and y are elements of the list [1..3] and y  x. Using a dependant generator we can define the library function that concatenates a list of lists: concat :: [[a]] -> [a] concat xss = [x | xs <- xss, x <- xs] > concat [[1,2,3],[4,5],[6]] [1,2,3,4,5,6]

27. Lee CSCE 314 TAMU 27 Guards List comprehensions can use guards to restrict the values produced by earlier generators. [x | x <- [1..10], even x] list all numbers x s.t. x is an element of the list [1..10] and x is even Example: Using a guard we can define a function that maps a positive integer to its list of factors: factors :: Int -> [Int] factors n = [x | x <- [1..n], n `mod` x == 0] > factors 15 [1,3,5,15]

28. Lee CSCE 314 TAMU 28 A positive integer is prime if its only factors are 1 and itself. Hence, using factors we can define a function that decides if a number is prime: prime :: Int -> Bool prime n = factors n == [1,n] > prime 15 False > prime 7 True Using a guard we can now define a function that returns the list of all primes up to a given limit: primes :: Int -> [Int] primes n = [x | x <- [2..n], prime x] > primes 40 [2,3,5,7,11,13,17,19,23,29,31,37]

29. Lee CSCE 314 TAMU 29 The Zip Function A useful library function is zip, which maps two lists to a list of pairs of their corresponding elements. zip :: [a] -> [b] -> [(a,b)] > zip [ ’ a ’, ’ b ’, ’ c ’ ] [1,2,3,4] [( ’ a ’,1),( ’ b ’,2),( ’ c ’,3)] Using zip we can define a function that returns the list of all positions of a value in a list: positions :: Eq a => a -> [a] -> [Int] positions x xs = [i | (x ’,i) <- zip xs [0..n], x == x ’ ] where n = length xs - 1 > positions 0 [1,0,0,1,0,1,1,0] [1,2,4,7]

30. Lee CSCE 314 TAMU 30 Using pairs we can define a function that decides if the elements in a list are sorted: sorted :: Ord a => [a] -> Bool sorted xs = and [x  y | (x,y) <- pairs xs] > sorted [1,2,3,4] True > sorted [1,3,2,4] False Using zip we can define a function that returns the list of all pairs of adjacent elements from a list: pairs :: [a] -> [(a,a)] pairs xs = zip xs (tail xs) > pairs [1,2,3,4] [(1,2),(2,3),(3,4)]

31. Lee CSCE 314 TAMU 31 Exercises A triple (x,y,z) of positive integers is called pythagorean if x 2 + y 2 = z 2. Using a list comprehension, define a function (1) pyths :: Int -> [(Int,Int,Int)] that maps an integer n to all such triples with components in [1..n]. For example: > pyths 5 [(3,4,5),(4,3,5)] pyths n = [(x,y,z)| x<-[1..n], y<-[1..n], z<-[1..n], x^2+y^2==z^2]

32. Lee CSCE 314 TAMU 32 A positive integer is perfect if it equals the sum of all of its factors, excluding the number itself. Using a list comprehension, define a function (2) perfects :: Int  [Int] that returns the list of all perfect numbers up to a given limit. For example: > perfects 500 [6,28,496]

33. Lee CSCE 314 TAMU 33 (xs i * ys i )  i = 0 n-1 Using a list comprehension, define a function that returns the scalar product of two lists. The scalar product of two lists of integers xs and ys of length n is given by the sum of the products of the corresponding integers: (3) scalarProduct :: [Int] -> [Int] -> Int scalarProduct xs ys = sum[x*y|(x,y) <- zip xs ys]

34. Lee CSCE 314 TAMU 34  Defining Functions  List Comprehensions  Recursion Outline

35. Lee CSCE 314 TAMU 35 A Function without Recursion Many functions can naturally be defined in terms of other functions. factorial :: Int  Int factorial n = product [1..n] factorial maps any integer n to the product of the integers between 1 and n Expressions are evaluated by a stepwise process of applying functions to their arguments. For example: factorial 4 product [1..4] = product [1,2,3,4] = 1*2*3*4 = 24 =

36. Lee CSCE 314 TAMU 36 Recursive Functions Functions can also be defined in terms of themselves. Such functions are called recursive. factorial 0 = 1 factorial n = n * factorial (n-1) factorial maps 0 to 1, and any other positive integer to the product of itself and the factorial of its predecessor. factorial 3 3 * factorial 2 = 3 * (2 * factorial 1) = 3 * (2 * (1 * factorial 0)) = 3 * (2 * (1 * 1)) = 3 * (2 * 1) = = 6 3 * 2 =

37. Lee CSCE 314 TAMU 37 Note:  The base case factorial 0 = 1 is appropriate because 1 is the identity for multiplication: 1 * x = x = x * 1.  The recursive definition diverges on integers  0 because the base case is never reached: > factorial (-1) Error: Control stack overflow

38. Lee CSCE 314 TAMU 38 Why is Recursion Useful? Some functions, such as factorial, are simpler to define in terms of other functions. As we shall see, however, many functions can naturally be defined in terms of themselves. Properties of functions defined using recursion can be proved using the simple but powerful mathematical technique of induction.

39. Lee CSCE 314 TAMU 39 Recursion on Lists Lists have naturally a recursive structure. Consequently, recursion is used to define functions on lists. product :: [Int]  Int product [] = 1 product (n:ns) = n * product ns product maps the empty list to 1, and any non- empty list to its head multiplied by the product of its tail. product [2,3,4]2 * product [3,4] = 2 * (3 * product [4]) = 2 * (3 * (4 * product [])) = 2 * (3 * (4 * 1)) = 24 =

40. Lee CSCE 314 TAMU 40 Using the same pattern of recursion as in product we can define the length function on lists. length :: [a]  Int length [] = 0 length (_:xs) = 1 + length xs length maps the empty list to 0, and any non-empty list to the successor of the length of its tail. length [1,2,3] 1 + length [2,3] = 1 + (1 + length [3]) = 1 + (1 + (1 + length [])) = 1 + (1 + (1 + 0)) = 3 =

41. Lee CSCE 314 TAMU 41 Using a similar pattern of recursion we can define the reverse function on lists. reverse :: [a]  [a] reverse [] = [] reverse (x:xs) = reverse xs ++ [x] reverse maps the empty list to the empty list, and any non-empty list to the reverse of its tail appended to its head. reverse [1,2,3] reverse [2,3] ++ [1] = (reverse [3] ++ [2]) ++ [1] = ((reverse [] ++ [3]) ++ [2]) ++ [1] = (([] ++ [3]) ++ [2]) ++ [1] = [3,2,1] =

42. Lee CSCE 314 TAMU 42 Multiple Arguments Functions with more than one argument can also be defined using recursion. For example:  Zipping the elements of two lists: zip :: [a]  [b]  [(a,b)] zip [] _ = [] zip _ [] = [] zip (x:xs) (y:ys) = (x,y) : zip xs ys drop :: Int  [a]  [a] drop n xs | n <= 0 = xs drop _ [] = [] drop n (_:xs) = drop (n-1) xs  Remove the first n elements from a list: (++) :: [a]  [a]  [a] [] ++ ys = ys (x:xs) ++ ys = x : (xs ++ ys)  Appending two lists:

43. Lee CSCE 314 TAMU 43 Laziness Revisited Laziness interacts with recursion in interesting ways. For example, what does the following function do? numberList xs = zip [0..] xs > numberList “abcd” [(0,’a’),(1,’b’),(2,’c’),(3,’d’)]

44. Lee CSCE 314 TAMU 44 Laziness with Recursion expensiveLen [] = 0 expensiveLen (_:xs) = 1 + expensiveLen xs Recursion combined with lazy evaluation can be tricky; stack overflows may result in the following example: stillExpensiveLen ls = len 0 ls where len z [] = z len z (_:xs) = len (z+1) xs cheapLen ls = len 0 ls where len z [] = z len z (_:xs) = let z’ = z+1 in z’ `seq` len z’ xs > expensiveLen [1..10000000] -- takes quite long > stillExpensiveLen [1..10000000] -- also takes long > cheapLen [1..10000000] -- less memory and time

45. Lee CSCE 314 TAMU 45 Quicksort The quicksort algorithm for sorting a list of integers can be specified by the following two rules:  The empty list is already sorted;  Non-empty lists can be sorted by sorting the tail values  the head, sorting the tail values  the head, and then appending the resulting lists on either side of the head value.

46. Lee CSCE 314 TAMU 46 Using recursion, this specification can be translated directly into an implementation: qsort :: [Int] -> [Int] qsort [] = [] qsort (x:xs) = qsort smaller ++ [x] ++ qsort larger where smaller = [a | a <- xs, a <= x] larger = [b | b x]  This is probably the simplest implementation of quicksort in any programming language! Note:

47. Lee CSCE 314 TAMU 47 For example (abbreviating qsort as q): q [3,2,4,1,5] q [2,1]++ [3] ++q [4,5] q [1]q []++ [2] ++q []q [5]++ [4] ++ [1] [] [5]

48. Lee CSCE 314 TAMU 48 Exercises (1) Without looking at the standard prelude, define the following library functions using recursion: and :: [Bool]  Bool  Decide if all logical values in a list are true: concat :: [[a]]  [a]  Concatenate a list of lists: and [] = True and (b:bs) = b && and bs concat [] = [] concat (xs:xss) = xs ++ concat xss

49. Lee CSCE 314 TAMU 49 (!!) :: [a]  Int  a  Select the nth element of a list: elem :: Eq a  a  [a]  Bool  Decide if a value is an element of a list: replicate :: Int  a  [a]  Produce a list with n identical elements: replicate 0 _ = [] replicate n x = x : replicate (n-1) x (!!) (x:_) 0 = x (!!) (_:xs) n = (!!) xs (n-1) elem x [] = False elem x (y:ys) | x==y = True | otherwise = elem x ys

50. Lee CSCE 314 TAMU 50 (2)Define a recursive function merge :: [Int]  [Int]  [Int] that merges two sorted lists of integers to give a single sorted list. For example: > merge [2,5,6] [1,3,4] [1,2,3,4,5,6] merge [] ys = ys merge xs [] = xs merge (x:xs) (y:ys) = if x <= y then x: merge xs (y:ys) else y: merge (x:xs) ys

51. Lee CSCE 314 TAMU 51 (3) Define a recursive function 1. Lists of length  1 are already sorted; 2. Other lists can be sorted by sorting the two halves and merging the resulting lists. msort :: [Int]  [Int] that implements merge sort, which can be specified by the following two rules: halves xs = splitAt (length xs `div` 2) xs msort [] = [] msort [x] = [x] msort xs = merge (msort ys) (msort zs) where (ys,zs) = halves xs