List.map has the signature ('a -> 'b) -> 'a list -> 'b list which in English is a function that takes a function (we'll call this the mapping function) from one type (namely 'a) to another type (namely 'b) and a list of the first type. The function returns a list of the second type where every element is the result of calling the mapping function on an element of the first list.

List.map string_of_int [ 1; 2; 3; 4 ]
#- [ "1"; "2"; "3"; "4" ] : string list

The types 'a and 'b don't have to be different. For example, we can map numbers to their squares just as easily.

let square x = x * x in
List.map square [ 1; 2; 3; 4 ]
#- [ 1; 4; 9; 16 ] : int list

The List.fold_left and List.fold_right functions are higher-order functions that implement the outer logic of list aggregation. Aggregating a list, sometimes also referred to as reducing a list, means computing a value derived from the sequential inspection of all items in that list.

The documentation of the List module states that

• List.fold_left f a [b1; ...; bn] is f (... (f (f a b1) b2) ...) bn.
• List.fold_right f [a1; ...; an] b is f a1 (f a2 (... (f an b) ...)). (This latter function is not tail-recursive.)

In plain English computing List.fold_left f a [b1; ...; bn] amounts to running through the list [b1; ...; bn] keeping track of an accumulator initially set to a: each time we see an item in the list, we use f to update the value of the accumulator, and when we are done, the accumulator is the final value of our computation. The List.fold_right function is similar.

Here are a few practical examples:

Compute the total sum of a list of numbers

List.fold_left ( + ) 0 lst

Compute the average of a list of floats

let average lst =
  let (sum, n) =
    List.fold_left (fun (sum, n) x -> (sum +. x, n + 1)) (0.0, 0) lst
  in
  sum /. (float_of_int n)

Re-implement basic list processing

The functions List.fold_left and List.fold_right are so general that they can be used to implement almost every other functions from the list module:

let list_length lst = (* Alternative implementation to List.length *)
  List.fold_left ( + ) 0 lst

let list_filter predicate lst = (* Alternative implementation to List.filter *)
  List.fold_right (fun a b -> if predicate a then a :: b else b) lst []

It is even possible to reimplement the List.iter function, remember that () is the global state of the program to interpret this code as a further example of list aggregation:

let list_iter f lst = (* Alternation implementation to List.iter *)
  List.fold_left (fun () b -> f b) () lst

These examples are meant to be learning material, these implementations have no virtue over the corresponding functions from the standard library.