[OCaml MOOC] week4: HIGHER ORDER FUNCTIONS

[TOC]

1. FUNCTIONAL EXPRESSIONS

syntax for functional expr: function some_identifier -> some_expr
the type of the functional expr is t1 -> t2 where t1 is the type of some_identifier, t2 is type of some_expr

ex.

function x -> x + 1;;  
(function x -> 2*x) 5;; (*annonymous function*)   

the previous way of defining function:

let f x = e,

is just an abbreviation for common let-binding:

let f = function x -> e

In fact, the most general form of a function definitoin contains a seq of pattern matching:

function   
  | pattern_1 -> expr_1  
  | pattern_2 -> expr_2   

When only 1 pattern, the | is omitted.

example

let rc length = function   
  | [] -> 0  
  | _::r -> 1 + length r;;  
type expr =   
  | Int of int  
  | Add of expr * expr;;  
let rec eval = function   
  | Int n -> n  
  | Add(e1, e2) -> (eval e1) + (eval e2);;   

2. FUNCTIONS AS FIRST-CLASS VALUES

functional types are just values of a particular type, thus this allows the uniform way of naming a value let x = ...

Types govern function application. We can apply e1 to e2 when:

  • e1 has type t1 -> t2
  • t1 matchs the type of e2

let f1 = [(function x -> x+1); (function x -> 2*x)];;  
(List.hd fl) 17;;  

let apply_twice f x = f (f x);;  
apply_twice (function x -> 2*x) 1;;  
let rec apply_n_times f n x =   
  if n<=0 then x  
  else apply_n_times f (n-1) (f x);;  
apply_n_times (function x -> 2*x) 10 1;;  

let compose f g = (function x -> f (g x));;  
let fg = compose (function x->x+1) (function x->2*x) in  
fg 10;;   

function pitfalls

function apply from left to right (function application associates to the left):

exp1 exp2 exp3
is equivalent to: (exp1 exp2) exp3

let double = function x -> 2*x;;  
double double 5;;  
double (double 5);;   

3. FUNCTIONS WITH MULTIPLE ARGUMENTS

anonymous function with several arguments: use keyword fun:
fun p1 ... pn -> exp

unlike function keyword, fun only admits one case/branch

remark: funs with several arguments are just abbrevations for single-argument functions that returns a function:

let f1 = function n -> (function x -> n+x);;  
(f1 17) 73;;  
f1 17 73;;  
let f2 = fun n x -> n+x;;  
f2 17 73;;  
(f2 17) 73;;   

in fact, fun x1 ... xn -> e is just abbreviation for: function x1 -> (..(function x2 -> ... -> (function xn -> e)..)

4. PARTIAL FUNCTION APPLICATION

let f = fun x y -> exp is equivalent to: let f = function x -> (function y -> exp)

⇒ partially apply f (ie, f x) will give a function.

let f1 = fun x y x -> x + y *z;;  
let f2 = f1 1;;  
let f3 = f2 2;;  
f3 4;;   

what happens at func-application:
when applying f = function x->e to a:
→ evaluate e in the context x=a
→ the arrow -> will block any evaluation


let f = fun x y -> (x / 0) + y;;  
let f2 = f1 17;; (*error will not happen here, as the `->` blocks the evaluation!*)  
f2 42;;   

partial evaluation

sometimes we can do part of a calculation as soon as we have the frist few arguments
⇒ extract that part of calculation before the arrow !

let egal l1 l2 =   
  List.sort compare l1 = List.sort compare l2;;  
let egalp l1 =   
  let l1sorted = List.sort compare l1 in (*sort the 1st argument before going to the next functional abstraction*)  
  function l2 -> l1sorted = List.sort compare l2;   

5. MAPPING FUNCTIONS ON LISTS

many useful functions in the List module, either open List at beginning, or with pointed notation (List.hd)

implementation

let rec map f = function  
  | [] -> []  
  | h::r -> (f h):(map f r);;  
map (function x -> x*x) [1;2;3;4;5];;  

let map2 f l1 l2 = match (l1, l2) with  
  | [],[] -> []  
  | h1::r1, h2::r2 -> (f h1 h2)::(map2 f r1 r2)  
  | _ -> raise (Invalid_argument "List.map2");;  
map2 (fun x y -> x+y) [1;2;3] [10;20;30];;   

examples

example1: int vectors/matrices

row vector: int list
matrix: list of row vectors
turn infix operators into functions : using parentheses (+) (/) ( * ) ...(note: spaces are necessary for *, otherwise this turns into comments...

let vsum  = List.map2 (+);; (* use partial application *)  
vsum [1;2;3] [10;20;30];;  
let msum = List.map2 (List.map2 (+));; (* nested partial application *)  
msum [[1;2;]; [3;4]] [[10;20]; [30;40]];;   

example2: get all sublists of a list
type: a' list -> 'a list list
write using induction:

let rec sublists = function   
  | [] -> [ [] ]  
  | h::r ->   
        let rp = sublists r  
        let appendh = function lst -> h::lst in  
        rp @ (map appendh rp)   

6. FOLDING FUNCTIONS ON LISTS

  • map: apping a unary function on list, all elements considered isolated.
  • folding: combining all elements of a list using a binary operator.
  • 2 different ways of folding: fold-left/fold-right

fold_right

fold_right: ('a -> 'b -> 'b) -> 'a list -> 'b -> 'b

implementation and usage:

let rec fold_right f l b = match l with   
  | [] -> b  
  | h::r ->  f h (fold_right f r b);;  
fold_right (+) [1;2;3;4] 0;;  
let concat = fold_right (fun x y -> x::y);; (partial application of fold_right)  
concat [1;2;3] [3;4;5];;   

fold_left

fold_left: ('a -> 'b -> 'a) -> 'a -> 'b list -> 'a
note: the default value's position is different from that of fold_right !

let rec fold_left f a l = match l with  
  | [] -> a  
  | h::r -> fold_left f (f a h) r;;  
fold_left (+) 0 [1;2;3];;  
let reverse = fold_left (fun x y -> y::x) [];; (* partial application *)  
reverse [1;2;3;4];;   

examples

example1: Inner product of int vectors
first get pairwise product, then sum up.

let product v1 v2 = List.fold_left (+) 0 (List.map2 ( * ) v1 v2);;  
product [1;2;3] [4;5;6];;   

example2: countif

let countif p l = List.fold_left   
  (fun acc elem -> if p elem then acc+1 else acc) 0 l;;  
countif (fun x-> x>0) [3;1;-2;0;4];;   
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