Understanding Partial Functions and Recursion in OCaml Programming

Munawar Hafiz 2219 SC, UIUC http://www.cs.illinois.edu/class/cs421/ Programming Languages and Compilers (CS 421)‏ Based in part on slides by Mattox Beckman, as updated by Vikram Adve, Gul Agha, and Elsa Gunter

Partial application of functions let add_three x y z = x + y + z;; # let h = add_three 5 4;; val h : int -> int = <fun> # h 3;; - : int = 12 # h 7;; - : int = 16

Functions as arguments # let thrice f x = f (f (f x));; val thrice : ('a -> 'a) -> 'a -> 'a = <fun> # let g = thrice plus_two;; val g : int -> int = <fun> # g 4;; - : int = 10 # thrice (fun s -> "Hi! " ^ s) "Good-bye!";; - : string = "Hi! Hi! Hi! Good-bye!"

Save the Environment! A closure is a pair of an environment and an association of a sequence of variables (the input variables) with an expression (the function body), written: f  < (v1,…,vn)  exp, f > Where f is the environment in effect when f is defined (if f is a simple function)‏

Curried vs Uncurried • Recall val add_three : int -> int -> int -> int = <fun> • How does it differ from # let add_triple (u,v,w) = u + v + w;; val add_triple : int * int * int -> int = <fun> • add_three is curried; • add_triple is uncurried

Curried vs Uncurried # add_triple (6,3,2);; - : int = 11 # add_triple 5 4;; Characters 0-10: add_triple 5 4;; ^^^^^^^^^^ This function is applied to too many arguments, maybe you forgot a `;' # fun x -> add_triple (5,4,x);; : int -> int = <fun>

Functions Over Lists # let rec map f list = match list with [] -> [] | (h::t) -> (f h) :: (map f t);; val map : ('a -> 'b) -> 'a list -> 'b list = <fun> # map plus_two fib5;; - : int list = [10; 7; 5; 4; 3; 3] # map (fun x -> x - 1) fib6;; : int list = [12; 7; 4; 2; 1; 0; 0]

Iterating over lists # let rec fold_left f a list = match list with [] -> a | (x :: xs) -> fold_left f (f a x) xs;; val fold_left : ('a -> 'b -> 'a) -> 'a -> 'b list -> 'a = <fun> # fold_left (fun () -> print_string) () ["hi"; "there"];; hithere- : unit = ()

Iterating over lists # let rec fold_right f list b = match list with [] -> b | (x :: xs) -> f x (fold_right f xs b);; val fold_right : ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b = <fun> # fold_right (fun s -> fun () -> print_string s) ["hi"; "there"] ();; therehi- : unit = ()

Encoding Recursion with Fold # let rec append list1 list2 = match list1 with [ ] -> list2 | x::xs -> x :: append xs list2;; val append : 'a list -> 'a list -> 'a list = <fun> Base Case Operation Recursive Call # let append list1 list2 = fold_right (fun x y -> x :: y) list1 list2;; val append : 'a list -> 'a list -> 'a list = <fun> # append [1;2;3] [4;5;6];; - : int list = [1; 2; 3; 4; 5; 6]

Example of Tail Recursion # let prod list = let rec prod_aux l acc = match l with [] -> acc | (y :: rest) -> prod_aux rest (acc * y) in prod_aux list 1;; val prod : int list -> int = <fun> # let prod list = List.fold_left (fun acc -> fun y -> acc * y) 1 list;; val prod : int list -> int = <fun>

Lambda Lifting • You must remember the rules for evaluation when you use partial application # let add_two = (+) (print_string "test\n"; 2);; test val add_two : int -> int = <fun> # let add2 =(* lambda lifted *) fun x ->(+) (print_string "test\n"; 2)x;; val add2 : int -> int = <fun>

CPS Transformation • Step 1: Add continuation argument to any function definition: • let f arg = e  let f arg k = e • Idea: Every function takes an extra parameter saying where the result goes • Step 2: A simple expression in tail position should be passed to a continuation instead of returned: • return a  k a • Assuming a is a constant or variable. • “Simple” = “No available function calls.”

CPS Transformation • Step 3: Pass the current continuation to every function call in tail position • return f arg  f arg k • The function “isn’t going to return,” so we need to tell it where to put the result. • Step 4: Each function call not in tail position needs to be built into a new continuation (containing the old continuation as appropriate) • return op (f arg)  f arg (fun r -> k(op r)) • op represents a primitive operation

Before: let rec add_list lst = match lst with [ ] -> 0 | 0 :: xs -> add_list xs | x :: xs -> (+) x (add_list xs);; After: let rec add_listk lst k = (* rule 1 *) match lst with | [ ] -> k 0 (* rule 2 *) | 0 :: xs -> add_listk xs k (* rule 3 *) | x :: xs -> add_listk xs (fun r -> k ((+) x r));; (* rule 4 *) Example

Record Types • Record types must be declared before they can be used in OCaml # typeperson = {name : string;ss: (int * int * int);age : int};; type person = { name : string; ss : int * int * int; age : int; } • person is the type being introduced • name, ss and age are the labels, or fields

Variants - Syntax (slightly simplified) • type name = C1[of ty1] | . . . | Cn[of tyn] • Introduce a type called name • (fun x -> Ci x) : ty1 -> name • Ci is called a constructor; if the optional type argument is omitted, it is called a constant • Constructors are the basis of almost all pattern matching

Recursive Data Types # type int_Bin_Tree = Leaf of int | Node of (int_Bin_Tree * int_Bin_Tree);; type int_Bin_Tree = Leaf of int | Node of (int_Bin_Tree * int_Bin_Tree)

Recursive Functions # let rec first_leaf_value tree = match tree with (Leaf n) -> n | Node (left_tree, right_tree) -> first_leaf_value left_tree;; val first_leaf_value : int_Bin_Tree -> int = <fun> # let left = first_leaf_value bin_tree;; val left : int = 3

Recursive Data Type Values bin_tree = Node Node Leaf (-7) Leaf 3 Leaf 6

Rules of Type Derivation

Understanding Partial Functions and Recursion in OCaml Programming