1. Cse536 Functional Programming 1 7/14/2015 Lecture #2, Sept 29, 2004 Reading Assignments –Begin Chapter 2 of the Text Home work #1 can be found on the webpage, under the assignments link, and as an extra handout, given out in class today Today’s Topics –Definition by cases –Definition by patterns –Local definitions –Function types and prototyping –Overloaded functions –Ways to make functions –Tuples –Polymorphism –Operators as functions –Functions as Arguments –Functions returned as values

2. Cse536 Functional Programming 2 7/14/2015 Definition by Cases absolute x | x < 0 = -x | x >= 0 = x ? absolute 3 3 (6 reductions, 13 cells) ? absolute (- 5) 5 (9 reductions, 13 cells) ?

3. Cse536 Functional Programming 3 7/14/2015 Definition By Patterns Example on Booleans myand True False = False myand True True = True myand False False = False myand False True = False Order Matters –Variables in Patterns match anything myand2 True True = True myand2 x y = False –What happens if we reverse the order of the two equations above? Pattern may contain constructors constructors are always capitalized

4. Cse536 Functional Programming 4 7/14/2015 Patterns On lists File Contents hd (x:xs) = x tl (x:xs) = xs firstOf3 [x,y,z] = x Hugs Session ? hd [1,2,3] 1 ? firstOf3 [1,2,3] 1 ? firstOf3 [1,2,3,4] Program error: {firstOf3 [1, 2, 3, 4]}

5. Cse536 Functional Programming 5 7/14/2015 Rules for Patterns All the patterns (on the left) should have compatible types The cases should (but are not required to) be exhaustive There should be no ambiguity as to which case applies. Ordering fixes ambiguity if there is any. A Pattern is: –A variable x –A constructor applied to patterns x:xs or Branch(x,y,z) –A constant 3 or [] –A tuple of patterns (x,3,y:ys)

6. Cse536 Functional Programming 6 7/14/2015 Recursive Functions & Comments File Contents plus x y = if x == 0 then y else 1 + (plus (x-1) y) len [] = 0 len (x:xs) = 1 + (len xs) Hugs session ? plus 3 5 8 ? len [1,2,3,4] 4

7. Cse536 Functional Programming 7 7/14/2015 Comments -- A comment to the end of the line {- This is a comment to the next -} {- Nesting of {- -} pairs allowed -}

8. Cse536 Functional Programming 8 7/14/2015 Local Definitions Local Definitions: Where len2 [] = 0 len2 (x:xs) = one + z where z = len2 xs one = 1 Indentation matters ! Same sequence of tokens but different meaning where a = f x y -- y is a variable used as b = g z -- an arg to function f where a = f x y b = g z -- y is the name of fun -- being defined Location of makes a big difference RULE of thumb Definitions at the same scope should be indented equally far.

9. Cse536 Functional Programming 9 7/14/2015 Function Types & Prototyping Typing f a b c = a + b + c + 1 has type f :: Int -> Int -> Int -> Int Read as f:: Int -> (Int -> (Int -> Int)) Prototyping plus :: Int -> Int -> Int plus x y = if x == 0 then y else 1 + (plus (x-1) y) myand :: Bool -> Bool -> Bool myand True False = False myand True True = True myand False False = False myand False True = False

10. Cse536 Functional Programming 10 7/14/2015 Overloading and Classes ? :type difference difference:: Num a => a -> a -> a ? difference 3 4 ? difference 4.5 7.8 -3.3 ? The class Num is a predicate on types

11. Cse536 Functional Programming 11 7/14/2015 Ways to Create Functions By defining: plusone x = x+1 ? plusone 3 4 By operator section ? (3+) 5 8 ? map (3+) [2,3,4] [5, 6, 7] By lambda expression ? (\ x -> x+2) 5 7 ? map (\x -> x*2) [2,3,4] [4, 6, 8]

12. Cse536 Functional Programming 12 7/14/2015 Creating Functions (cont.) By currying (partial application) ? plus 3 plus 3 ? :type (plus 3) plus 3 :: Int -> Int ? map (plus 3) [3,4] [6, 7] By composition ? map (head. tail) [[2,3,4],[4,5,6]] [3, 5] By combinator: k x y = x – Functions which return functions ? map (k 3) [1,2,3] [3, 3, 3]

13. Cse536 Functional Programming 13 7/14/2015 Expressions, Values, and Types There are three types of distinguished entities in Haskell Expressions –5 + 3 –len [1,2,3] –rev [2,9] –len Values –8 –3 –[9,2] – > Types –Int –[ Int ] –[ a ] -> Int

14. Cse536 Functional Programming 14 7/14/2015 Tuples Heterogeneous Collection of a fixed width Tuple Expressions –(5+3, not True) –(tl [1,2,3], [2]++[3,4], “abc”) Evaluate to Tuple Values –(8, False) –([2,3], [2,3,4], “abc”) And have Tuple Types –(Int, Bool) –([Int], [Int], String)

15. Cse536 Functional Programming 15 7/14/2015 Typing Tuples ? :type (1,"x",True) (1,"x",True)::(Int,String,Bool) ? :type (1,2) (1,2) :: (Int,Int) ? :type (2,("x",3)) (2,("x",3))::(Int,(String,Int)) –Note: (Int,([Char],Int)) <> (Int,[Char],Int) Pattern matching on tuples ? (\ (a,b) -> a) (2,3) 2 ? (\ (a,b) -> b + 2) (2,3) 5

16. Cse536 Functional Programming 16 7/14/2015 Used when returning multiple values Function that splits a list into two pieces at some particular position split 2 [1,2,3,4] --> ([1,2],[3,4]) split 3 [1,2,3,4,5,6,7] ---> ([1, 2, 3],[4, 5, 6, 7]) split 0 [1,2,3] --> ([],[1,2,3]) split 0 x = ([],x) split n [] = ([],[]) split n (x:xs) = (x:ys,zs) where (ys,zs) = split (n-1) xs

17. Cse536 Functional Programming 17 7/14/2015 Polymorphism (FN pp 16, GITH pp3-5) 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 :: –(:) :: –split ::

18. Cse536 Functional Programming 18 7/14/2015 Operators as functions (FN pp 6-8 & 21 GITH pp 11-12) ? :type (:) (:) :: a -> [a] -> [a] ? :type (++) (++) :: [a] -> [a] -> [a] ? :type (+) (+) :: Num a => a -> a -> a Operator Precedence –level 9. !! –level 8 ^ –level 7 * / `div` `rem` `mod` –level 6 + - –level 5 : ++ \\ –level 4 == /= >= `elem` –level 3 && –level 2 || –level 1 (not used in the prelude)

19. Cse536 Functional Programming 19 7/14/2015 Associativity (FN pp 22-23) Right –2 : 3 : 4 : [] = 2 : (3 : (4 : [])) –Other Right Associative operators (:) (^) (++) (&&) (||) Left –7 - 6 - 2 = (7 - 6) - 2 –Other Left Associative operators (!!) (-) (*) (+) Non-associative –( )... (i.e. all the relational operators)

20. Cse536 Functional Programming 20 7/14/2015 Defining one’s own operators (FN pp 23) infix 4 `inlist` infix 3 -&- x `inlist` [] = False x `inlist` (y:ys) = if x==y then True else x `inlist` ys ? 3 `inlist` [1,2,4] False ? 3 `inlist` [1,2,3,5] True

21. Cse536 Functional Programming 21 7/14/2015 Functions as arguments (FN pp 25-28) Consider: mymap f [] = [] mymap f (x:xs) = (f x):(mymap f xs) The parameter f is a function! What is the type of mymap ? mymap :: (a -> b) -> [a] -> [b] What happens when it is applied? map add1 map ( \ x -> 3) [1,2,3]

22. Cse536 Functional Programming 22 7/14/2015 When do you define a higher order function? Abstraction 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

23. Cse536 Functional Programming 23 7/14/2015 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 ?

24. Cse536 Functional Programming 24 7/14/2015 Functions returned as values Consider: k x = (\ y -> x) ? (k 3) 5 3 Another Example: plusn n = (\ x -> x + n) ? (plusn 4) 5 9 Is plusn different from plus ? why? – plus x y = x + y

25. Cse536 Functional Programming 25 7/14/2015 Additional Examples The webpage includes a link (in the lecture notes section, under today’s date, Sept. 29, 2004) to some additional examaples 1.Numerical Functions 1.Differentiation and square root. pp 30-32 Fokker Notes 2.Primes. pp 29 Fokker Notes 3. Display Tool. pp 105 Reade Book 4.Numbers in Long Hand 5.Sorting. pp 106-109 Reade Book 6.Making Change. Bird & Wadler