1. Cse536 Functional Programming 1 6/10/2015 Lecture #6, Oct 13, 2004 Reading Assignments –Read chapter 5 of the Text Polymorphic and Higher-Order Functions –Read chapter 6 of the Text Shapes III: Perimeters of Shapes –Assignment #3 – Now in Assignments directory »Assigned: Oct. 13, 2004 »Due, in class: Wednesday, Oct. 20, 2004 Today’s Topics –Polymorphic Functions – Polymorphic datatypes »Type Constructors define polymorphic Constructor functions –Recursive datatypes – Higher Order functions –Perimeters of Shapes

2. Cse536 Functional Programming 2 6/10/2015 Polymorphic Length len :: [a] -> Int len [] = 0 len (x:xs) = 1 + len xs Polymorphic functions don’t “look at” their polymorphic arguments. They use the same code now matter what the type of their polymorphic arguments. “a” is a type variable. It is lowercase to distinguish it from types, which are uppercase.

3. Cse536 Functional Programming 3 6/10/2015 Polymorphism Consider: tag1 x = (1,x) ? :type tag1 tag1 :: a -> (Int,a) Other functions have types like this consider (++) ? :type (++) (++) :: [a] -> [a] -> [a] ? :type ([1,2]++) ([1,2] ++) :: [Int] -> [Int] What are some other polymorphic functions and their types? –id :: –reverse :: –head :: –tail :: –(:) ::

4. Cse536 Functional Programming 4 6/10/2015 Polymorphic data structures Polymorphism originates from data structures that don’t care what kind of data they store. id :: a -> a -- The ultimate -- polymorphic function reverse :: [a] -> [a] -- lists tail :: [a] -> [a] head :: [a] -> a (:) :: a -> [a] -> [a] tag1 :: Num a => b -> (a,b) -- tuples How do we define new data structures with “holes” that can be polymorphic?

5. Cse536 Functional Programming 5 6/10/2015 Parametric Type Constructors Polymorphic types data Option a = NONE | SOME a – Note NONE is a constant, SOME is a function We can have: type example with that type Option Int SOME 3 Option String SOME "x" Option (Int,Bool) SOME (3,True) even Option (Option Int) Give some examples with the type Option (Option Int) The type parameter to Option causes it’s constructors to be polymorphic

6. Cse536 Functional Programming 6 6/10/2015 Why use Option? pos :: Eq a => a -> [a] -> Option Int pos a l = let find a [] n = NONE find a (x:xs) n = if a==x then SOME n else find a xs (n+1) in find a l 0 ? pos 4 [1,2,3,4,5] SOME 3 ? pos 4 [1,5,6,7] NONE Option is a type constructor since it constructs new types from old ones.

7. Cse536 Functional Programming 7 6/10/2015 Types Constructors with multiple inputs data Pair a b = Pair (a,b) data Pair2 a b = Pair2 a b In Haskell sometimes the Type constructor and the value constructor have the same name. ? :t Pair(2,"x") Pair (2,"x") :: Pair Int [Char] ? :t Pair ((),Pair (2,3)) Pair ((),Pair (2,3)) :: Pair () (Pair Int Int) Domain of constructors - To use tuples or not? That is the question! ? :t Pair Pair :: (a,b) -> Pair a b ? :t Pair2 Pair2 :: a -> b -> Pair2 a b

8. Cse536 Functional Programming 8 6/10/2015 Recursive types Well known and loved lists data [a] = [] | a : [a] You can define your own ! data Mylist a = Nil | Cons (a,Mylist a) But be warned! The two are not the same type. ? :t Cons Cons :: (a,Mylist a) -> Mylist a ? Cons(3,2:[]) ERROR: Cannot derive instance in expression *** Expression : Cons (3,[ 2 ]) *** Required instance: Monad Mylist Lists are predefined, but could be defined this way

9. Cse536 Functional Programming 9 6/10/2015 Recursive type constructors with multiple inputs data Twolist a b = Twonil | Acons (a,Twolist a b) | Bcons (b,Twolist a b) append2 Twonil ys = ys append2 (Acons(a,xs)) ys = Acons(a,append2 xs ys) append2 (Bcons(b,xs)) ys = Bcons(b,append2 xs ys) rev2 Twonil = Twonil rev2 (Acons(a,t)) = append2 (rev2 t) (Acons(a,Twonil)) rev2 (Bcons(b,t)) = append2 (rev2 t) (Bcons(b,Twonil)) ? :t rev2 rev2 :: Twolist a b -> Twolist a b ? append2 (Acons(2,Twonil))(Bcons(4,Twonil)) Acons (2,Bcons (4,Twonil))

10. Cse536 Functional Programming 10 6/10/2015 Recursive type constructors with 0 inputs data Nat = Zero | Succ Nat; Is there any polymorphism here? Why not? Write the addition function for Nat? add :: Nat -> Nat -> Nat add = Write a fun from positive Int to Nat. toNat :: Int -> Nat ? toNat 3 Succ (Succ (Succ Zero)) toNat =

11. Cse536 Functional Programming 11 6/10/2015 Polymorphism from functions as arguments Another source of polymorphism comes from functions which take functions as arguments. applyTwice f x = f(f x) Main> :t applyTwice applyTwice :: (a -> a) -> a -> a What’s the type of z below? z f x y = (f x, f y)

12. Cse536 Functional Programming 12 6/10/2015 Polymorphism: Functions returned as values Consider: k x = (\ y -> x) ? (k 3) 5 3 –What’s the type of K ? Another Example: compose f g = \ x -> f (g x) –What’s the type of compose ?

13. Cse536 Functional Programming 13 6/10/2015 Higher order functions abstract control Higher order functions capture control patterns transList :: [Vertex] -> [Point] transList [] = [] transList (p:ps) = trans p : transList ps putCharList :: String -> [IO ()] putCharList [] = [] putCharList (c:cs) = putChar c : putChaList cs map f [] = [] map f (x:xs) = (f x):(map f xs)

14. Cse536 Functional Programming 14 6/10/2015 When do you define a higher order function? Recognizing patterns is the key mysum [] = 0 mysum (x:xs) = (+) x (mysum xs) myprod [] = 1 myprod (x:xs) = (*) x (myprod xs) myand [] = True myand (x:xs) = (&&) x (myand xs) Note the similarities in definition and in use ? mysum [1,2,3] 6 ? myprod [2,3,4] 24 ? myand [True, False] False

15. Cse536 Functional Programming 15 6/10/2015 Abstracting myfoldr op e [] = e myfoldr op e (x:xs) = op x (myfoldr op e xs) ? :t myfoldr myfoldr :: (a -> b -> b) -> b -> [a] -> b ? myfoldr (+) 0 [1,2,3] 6 ?

16. Cse536 Functional Programming 16 6/10/2015 Using foldr Once we’ve captured a pattern we should use it! mysum = foldr (+) 0 myprod = foldr (*) 1 myand = foldr (&&) True Study the definitions below. Define equivalent functions in terms of foldr. map f [] = [] map f (x:xs) = (f x):(map f xs) map f = foldr... append [] ys = ys append (x:xs) ys = x : (append xs ys) append xs ys = foldr...

17. Cse536 Functional Programming 17 6/10/2015 foldl Let: x = [1,2,3,4] foldr :: (a -> b -> b) -> b -> [a] -> b foldr (+) e x = 1 + (2 + (3 + e)) foldl :: (b -> a -> b) -> b -> [a] -> b foldl (+) e x = (((e + 1) + 2) + 3) + 4 –Can you define foldl ?

18. Cse536 Functional Programming 18 6/10/2015 Why foldl and foldr ? Grouping from the left or the right matters in some computations. Consider the two defintions concat1 :: [[a]] -> [a] concat1 = foldr (++) [] concat2 :: [[a]] -> [a] concat2 = foldl (++) [] Which is more efficient? concat1 [x,y,z] = x ++ (y ++ z) concat2 [x,y,z] = (x ++ y) ++ z

19. Cse536 Functional Programming 19 6/10/2015 Reverse rev [] = [] rev (x:xs) = (rev xs) ++ [x] What’s the cost (in number of (:)’s) for rev [1,2,3,4]? Can we do better? –Use an accumulating parameter rev2 [] ys = ys rev2 (x:xs) ys =... rev x = rev2 x [] Can we write rev2 as a fold? –Which one do we use foldr or foldl?

20. Cse536 Functional Programming 20 6/10/2015 Artihmetic Sequences Special syntax for computing lists with regular properties. [1.. 6] = [1,2,3,4,5,6] [1,3.. 9] = [1,3,5,7,9] Infinite lists too! take 9 [1,3..] = [1,3,5,7,9,11,13,15,17] take 5 [5..] = [5,6,7,8,9]

21. Cse536 Functional Programming 21 6/10/2015 The perimeter of a Shape To compute the perimeter we need a function with four equations (1 for each constructor of Shape) The first three are easy … perimeter :: Shape -> Float perimeter (Rectangle s1 s2) = 2*(s1+s2) perimeter (RtTriangle s1 s2) = s1 + s2 + sqrt(s1^2+s2^2) perimeter (Polygon pts) = foldl (+) 0 (sides pts) Provided we can compute a list of the length of the sides of a polygon. This shouldn’t be too difficult since we can compute the distance between two points with distBetween s1 s2 s1 s2

22. Cse536 Functional Programming 22 6/10/2015 Sides is easy recursively sides :: [Vertex] -> [Side] sides [] = [] sides (p:pts) = aux (p:pts) where aux (p1:p2:pts) = (distBetween p1 p2) : (aux (p2:pts)) aux (pn:[]) = distBetween pn p : [] -- aux [pn] = [distBetween pn p] But can we do it as a seasoned functional programmer might do it?

23. Cse536 Functional Programming 23 6/10/2015 Visualize what’s happening The list of vertex, vs = [A,B,C,D,E] distance between (A,B) (B,C) (C,D) (D,E) (E,A) Can we compute these pairs as a list? [ (A,B),(B,C),(C,D),(D,E),(E,A)] Yes by zipping the following two lists: [A,B,C,D,E] [B,C,D,E,A] zip vs ((tail vs)++[head vs]) A B C DE

24. Cse536 Functional Programming 24 6/10/2015 Second version of sides sides2 :: [Vertex] -> [Side] sides2 pts = zipWith distBetween pts (tail pts ++ [head pts]) In class exercise, define: zipWith zipWith f...

25. Cse536 Functional Programming 25 6/10/2015 The perimeter of an ellipse is given by the summation of an infinite series. For an ellipse with radii r1 and r2 p = 2  r 1 (1 -  s i ) where s 1 = 1/4 e 2 s i+1 = s i (2i-1)(2i-3) e 2 for i>= 1 4i 2 e = sqrt(r 1 * r 1 - r 2 * r 2 )/r1 Given s i its is easy to compute the next value in the series s i+1 Perimeter of an Ellipse

26. Cse536 Functional Programming 26 6/10/2015 Computing the series nextEl:: Float -> Float -> Float -> Float nextEl e s i = s*(2*i-1)*(2*i-3)*(e^2) / (4*i^2) We want to compute [s 1,s 2,s 3, …] for fixed e aux s i = nextEl e s i [s 1, aux s 1 1, aux (aux s 1 1) 2, aux (aux (aux s 1 1) 2) 3, …] s i+1 = s i (2i-1)(2i-3) e 2 4i 2 Can we capture this pattern?

27. Cse536 Functional Programming 27 6/10/2015 Scanl (scan from the left) [s 1, s 2 = f s 1 1, s 3 = f s 2 2 = f(f s 1 1) 2, s 4 = f s 3 3 = f(f (f s 1 1) 2) 3, …] scanl :: (a -> b -> b) -> b -> [a] -> [b] scanl f seed [] = seed : [] scanl f seed (x:xs) = seed : scanl f newseed xs where newseed = f x seed scanl aux s1 [1,2..]

28. Cse536 Functional Programming 28 6/10/2015 We get the following pattern s 1 = 1/4 e 2 s1 = 0.25 * ((r1^2 - r2^2) / r1) where r1 and r2 are the radii of the ellipse [s 1, s 2 = aux s 1 1, s 3 = aux s 2 2, s 4 = aux s 3 3, …]

29. Cse536 Functional Programming 29 6/10/2015 r2 = 1.5 r1 = 2.1 e = sqrt (r1^2 - r2^2) / r1 = 0.699854 [s1 = 0.122449, s2 = 0.0112453, s3 = 0.00229496, s4 = 0.000614721, s5 = 0.000189685, …] Note how quickly the series gets real small...

30. Cse536 Functional Programming 30 6/10/2015 Putting it all together perimeter (Ellipse r1 r2) | r1 > r2 = ellipsePerim r1 r2 | otherwise = ellipsePerim r2 r1 where ellipsePerim r1 r2 = let e = sqrt (r1^2 - r2^2) / r1 s = scanl aux (0.25*e^2) (map intToFloat [2..]) aux s i = nextEl e s i test x = x > epsilon sSum = foldl (+) 0 (takeWhile test s) in 2*r1*pi*(1 - sSum) This is s 1

31. Cse536 Functional Programming 31 6/10/2015 Assignment #3, Also in the assignments web page Be sure and include a listing (1-2 pages) which shows how you tested your programs in the Hugs system. 1) Write a function "wordsL" such that (wordsL "This is a sentence.") returns ["This", "is","a","sentence."]. It breaks a string into a list of strings. It breaks the strings (and throws away the separators) at blanks ( ' ' ), tabs ( '\t' ), and newlines ( '\n' ). Hint: takeWhile and dropWhile are very useful. e.g. (takeWhile even [2,4,5,6,7]) => [2,4]. There is a function called words in the prelude that does exactly this. Write your own version. 2) Exercise 3.1 page 49 of the text 3) Use the IO monad to write a function when given two strings as input, opens a file with the first strings name, and copys it to another file, with the name of the second string. copy :: String -> String -> IO (). Be sure and test your function in the input you hand in. 4) Write a program "pal" which reads a string from the user, and then prints on the screen if that string is a palindrome. (A palindrome is a string which reads the same both forwards and backwards. For example "hah" and "abCba".) pal :: IO () 5) Using the primitive drawing functions of chapter 4 of the text, open a graphics window and draw a picture of a "house".