1. CS 2104 – Prog. Lang. Concepts Functional Programming II Lecturer : Dr. Abhik Roychoudhury School of Computing From Dr. Khoo Siau Cheng’s lecture notes

2. reduce (op *) [2,4,6] 1 ==> 2 * (4 * (6 * 1)) ==> 48 reduce (fn (x,y)=>1+y) [2,4,6] 0 ==> 1 + (1 + (1 + 0)) ==> 3 [] : 2: 4: 6 + 1 + 1 + 10

3. Types: Classification of Values and Their Operators TypeValuesOperations booltrue,false=, <>, … int…,~1,0,1,2,…=,<>,<,+,div,… real..,0.0,.,3.14,..=,<>,<,+,/,… string“foo”,”\”q\””,…=,<>,… Basic Types Boolean Operations: e 1 andalso e 2 e 1 orelse e 2

4. Types in ML Every expression used in a program must be well- typed. –It is typable by the ML Type system. Declaring a type : 3 : int [1,2] : int list Usually, there is no need to declare the type in your program – ML infers it for you.

5. Structured Types Structured Types consist of structured values. Structured values are built up through expressions. Eg : (2+3, square 3) Structured types are denoted by type expressions. ::= | | * |  | list | …

6. Type of a Tuple A * B= set of ordered pairs (a,b) DataConstructor: (, )as in (a,b) Type Constructor :*as in A * B In general, (a 1,a 2,…,a n )belongs to A 1 *A 2 *…*A n. (1,2) : int * int (3.14159, x+3,true) : real * int * bool

7. Type of A List [1,2,3] : int list [3.14, 2.414] : real list [1, true, 3.14] : ?? Type Constructor :list A in A-list refers to any types: (int*int) list: [ ], [(1,3)], [(3,3),(2,1)], … int list list: [ ], [[1,2]], [[1],[0,1,2],[2,3],… A list= set of all lists of A -typed values. Not well-typed!!

8. fac : int -> int Function Types Declaring domain & co-domain Type Constructor : -> Data Construction via : 1. Function declaration : fun f x = x + 1 ; 2. Lambda abstraction : fn x => x + 1; Value Selection via function application: f 3  4 (fn x => x + 1) 3  4 A -> B = set of all functions from A to B.

9. datatype Days = Mo | Tu | We | Th | Fr | Sa | Su ; Selecting a summand via pattern matching: case d ofSa => “Go to cinema” |Su => “Extra Curriculum” |_ => “Life goes on” Sum of Types Enumerated Types New Type data / data constructors

10. Defining an integer binary tree: datatype IntTree = Leaf int | Node of (IntTree, int, IntTree) ; fun height (Leaf x) = 0 | height (Node(t1,n,t2))= 1 + max(height(t1),height(t2)) ; Combining Sum and Product of Types: Algebraic Data Types

11. Some remarks A functional program consists of an expression, not a sequence of statements. Higher-order functions are first-class citizen in the language. –It can be nameless List processing is convenient and expressive In ML, every expression must be well-typed. Algebraic data types empowers the language.

12. Outline More about Higher-order Function Type inference and Polymorphism Evaluation Strategies Exception Handling

13. Function with Multiple Arguments Curried functions accept multiple arguments fun twice f x = f (f x) ; Take 2 arguments Apply 1st argument Apply 2nd argument Curried function enables partial application. let val inc2 = twice (fn x => x + x) in (inc2 1) + (inc2 2) end; val it = 12 ;

14. Curried vs. Uncurried Curried functions fun twice f x = f (f x) ; twice (fn x => x+x) 3  12 Uncurried functions fun twice’ (f, x) = f (f x) ; twice’ (fn x => x+x, 3)  12

15. Curried Functions Curried functions provide extra flexibility to the language. compose f g = fn x => f (g x)  compose f g x = f (g x)  compose f = fn g => fn x => f (g x)  compose = fn f => fn g => fn x => f (g x)

16. fun f(x,y) = x + y f : int*int -> int Types of Multi-Argument Funs fun g x y = x + y g : int -> int -> int (g 3) : int -> int ((g 3) 4) : int Function application is left associative; -> is right associative

17. Outline More about Higher-order Function Type inference and Polymorphism Evaluation Strategies Exception Handling

18. Type Inference ML expressions seldom need type declaration. ML cleverly infers types without much help from the user. 2 + 2 ; val it = 4 : int fun succ n = n + 1 ; val succ = fn : int -> int

19. Explicit types are needed when type coercion is needed. fun add(x,y : real) = x + y ; fun add(x,y) = (x:real) + y; val add = fn : real*real -> real Helping the Type Inference

20. Conditional expression has the same type at both branch. fun abs(x) = if x>0 then x else 0-x ; val abs = fn : int -> int Every Expression has only One Type fun f x = if x > 0 then x else [1,2,3] val f = fn : Int -> ??? This is not type-able in ML.

21. Example of Type Inference fun f g = g (g 1) type(g) = t = int  t2 = t2  t3 int = t2, t2 = t3 type(g) = t = int  int type(f) = t  t1 = t  t3 = (int  int)  int t  t1 t int  t2 t2  t3 t3

22. Three Type Inference Rules (Application rule) If f x : t, then x : t’ and f : t’ -> t for some new type t’. (Equality rule) If both the types x : t and x : t’ can be deduced for a variable x, then t = t’. (Function rule) If t  u = t’  u’, then t = t’ and u = u’.

23. Example of Type Inference fun f g = g (g 1) Let g : t g (g (g 1)) : t rhs So, by function declaration, we have f : t g -> t rhs By application rule, let (g 1): t (g 1) g (g 1) : t rhs  g : t (g 1) -> t rhs. By application rule, (g 1) : t (g 1)  g : int -> t (g 1). By equality rule : t (g 1) = int = t rhs. By equality rule : t g = int -> int Hence, f : (int -> int) -> int

24. Parametric Polymorphism fun I x = x ; val I = fn : ’a -> ’a A Polymorphic function is one whose type contains type parameters. Type parameter (I 3)(I [1,2])(I square) Interpretation of val I = fn : ’a -> ’a for all type ‘a, function I takes an input of type ‘a and returns a result of the same type ‘a. A poymorphic function can be applied to arguments of more than one type.

25. Polymorphic Functions A polymorphic function is one whose type contains type parameter. fun map f [] = [] | map f (x::xs) = (f x) :: (map f xs) Type of map : (’a->’b) -> [’a] -> [’b] map (fn x => x+1) [1,2,3] => [2,3,4] map (fn x => [x]) [1,2,3] => [[1],[2],[3]] map (fn x => x) [“y”,“n”] => [“y”, “n”]

26. Examples of Polymorphic Functions fun compose f g = (fn x => f (g x)) t1  t2  t t4  t5 t4 t5  t6 t4  t6 type(f) = t1 = t5  t6 type(g) = t2 = t4  t5 range(compose) = t = t4  t6 type(compose) = t1  t2  t = (t5  t6)  (t4  t5)  (t4  t6)

27. Examples of Polymorphic Functions fun compose f g = (fn x => f (g x)) Let x:t x f:t f g:t g so compose: t f -> t g ->t rhs (fn x=>f (g x)):t rhs => t rhs = t x ->t (f(gx)) (g x):t (gx) ==> g: t x ->t (gx) and t g = t x ->t (g x) (f (g x)):t (f(gx)) ==> f:t (gx) ->t (f(gx)) and t f = t (gx) ->t (f(gx)) compose: (t (gx) ->t (f(gx)) )->(t x ->t (gx) )->(t x ->t (f(gx)) ) Rename the variables: compose: (’a->’b)->(’c->’a)->(’c->’b)

28. Outline More about Higher-order Function Type inference and Polymorphism Evaluation Strategies Exception Handling

29. Approaches to Expression Evaluation Different approaches to evaluating an expression may change the expressiveness of a programming language. Two basic approaches: –Innermost (Strict) Evaluation Strategy SML, Scheme –Outermost (Lazy) Evaluation Strategy Haskell, Miranda, Lazy ML

30. Innermost Evaluation Strategy To Evaluate the call : –(1) Evaluate ; –(2) Substitute the result of (1) for the formals in the body ; –(3) Evaluate the body of ; –(4) Return the result of (3) as the answer. let fun f x = x + 1 + x in f (2 + 3) end ; body formals actuals

31. fun f x = x + 2 + x ; f (2+3) ==> f (5) ==> 5 + 2 + 5 ==> 12 fun g x y = if (x < 3) then y else x; g 3 (4/0) ==> g 3  ==>  Also referred to as call-by-value evaluation. Occasionally, arguments are evaluated unnecessarily.

32. Outermost Evaluation Strategy To Evaluate : (1) Substitute actuals for the formals in the body ; (2) Evaluate the body ; (3) Return the result of (2) as the answer. fun f x = x + 2 + x ; f (2+3) ==> (2+3) + 2 + (2+3) ==> 12 fun g x y = if x < 3 then y else x ; g 3 (4/0) ==> if 3 < 3 then (4/0) else 3 ==> 3

33. It is possible to eliminate redundant computation in outermost evaluation strategy. fun f x = x + 2 + x ; f (2+3) ==> x + 2 + x ==> 5 + 2 + x ==> 7 + x ==> 7 + 5 ==> 12 5 x=(2+3) Note: Arguments are evaluated only when they are needed.

34. Why Use Outermost Strategy? Closer to the meaning of mathematical functions fun k x y = x ; val const1 = k 1 ; val const2 = k 2 ; Better modeling of real mathematical objects val naturalNos = let fun inf n = n :: inf (n+1) in inf 1 end ;

35. Hamming Number List, in ascending order with no repetition, all positive integers with no prime factors other than 2, 3, or 5. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15,...

36. n as a prime factor fun scale n [] = [] | scale n (x::xs) = (n*x) :: (scale n xs) scale 2 [1,2,3,4,5] = [2,4,6,8,10] scale 3 [1,2,3,4,5] = [3,6,9,12,15] scale 3 (scale 2 [1,2,3,4,5]) = [6,12,18,24,30]

37. Merging two Streams fun merge [] [] = [] | merge (x::xs) (y::ys) = if x < y then x :: merge xs (y::ys) else if x > y then y :: merge (x::xs) ys else x :: merge xs ys merge [2,4,6] [3,6,9] = [2,3,4,6,9]

38. Hamming numbers val hamming = 1 :: merge (scale 2 hamming) (merge (scale 3 hamming) (scale 5 hamming)) :: 1 merge scale 3 scale 5 scale 2

39. Outline More about Higher-order Function Type inference and Polymorphism Evaluation Strategies Exception Handling

40. Handle special cases or failure (the exceptions) occurred during program execution. hd []; uncaught exception hd Exception can be raised and handled in the program. exception Nomatch; exception Nomatch : exn fun member(a,x) = if null(x) then raise Nomatch else if a = hd(x) then x else member(a,tl(x))

41. fun member(a,x) = if null(x) then raise Nomatch else if a = hd(x) then x else member(a,tl(x)) member(3,[1,2,3,1,2,3]) ; val it = [3,1,2,3] : int list member(4,[]) ; uncaught exception Nomatch member(5,[1,2,3]) handle Nomatch=>[]; val it = [] : int list

42. Conclusion More about Higher-order Function –Curried vs Uncurried functions –Full vs Partial Application Type inference and Polymorphism –Basic Type inference rules –Polymorphic functions Evaluation Strategies –Innermost –Outermost Exception Handling is available in ML