Abstract Types

The type 'a stack above is abstract: the Stack module type says that there is a type name 'a stack in any module that implements the module type, but it does not say what that type is defined to be. Once we add the : Stack module type annotation to ListStack, its 'a stack type also becomes abstract. Outside of the module, no one is allowed to know that 'a stack and 'a list are synonyms.

A module that implements a module type must specify concrete types for the abstract types in the signature and define all the names declared in the signature. Only declarations in the signature are accessible outside of the module. For example, functions defined in the module's structure but not in the module type's signature are not accessible. We say that the structure is sealed by the signature: nothing except what is revealed in the signature may be accessed.

Here is another implementation of the Stack module type:

module MyStack : Stack = struct
  type 'a stack = 
  | Empty 
  | Entry of 'a * 'a stack

  let empty = Empty
  let is_empty s = s = Empty
  let push x s = Entry (x, s)
  let peek = function
    | Empty -> failwith "Empty"
    | Entry(x,_) -> x
  let pop = function
    | Empty -> failwith "Empty"
    | Entry(_,s) -> s
end

In that implementation, we provide our own custom variant for the representation type. Of course, that custom variant is more or less the same as the built-in list type: it has two constructors, one the carries no data, and the other that carries a pair of an element and (recursively) the same variant type.

Because 'a stack is abstract in the Stack module type, no client of this data structure will be able to discern whether stacks are being implemented with the built-in list type or the custom one we just used. Clients may only access the stack in the ways that are defined by the Stack interface, which nowhere mentions list or Empty or Entry.

You can even observe that abstraction in utop. Observe what happens when utop displays the value that results from this expression:

# MyStack.push 1 MyStack.empty;;
- : int MyStack.stack = <abstr>

The value has type int MyStack.stack, which is to say, it is the MyStack.stack type constructor applied to int. And the value is...well, utop won't tell us! It simply prints <abstr> to indicate that the value has been abstracted.

Notice how verbose the type int MyStack.stack is. The module name already tells us that the value is related to MyStack; the word stack following that isn't particularly helpful. For that reason, it is idiomatic OCaml to name the primary representation type of a data structure simply t. Here's the Stack module type rewritten that way:

module type Stack = sig
  type 'a t
  val empty    : 'a t
  val is_empty : 'a t -> bool
  val push     : 'a -> 'a t -> 'a t
  val peek     : 'a t -> 'a
  val pop      : 'a t -> 'a t
end

Given that renaming, here's what the toplevel would display as the type:

# MyStack.push 1 MyStack.empty;;
- : int MyStack.t = <abstr>

And now by convention we would usually pronounce that type as "int MyStack", simply ignoring the t, though it does technically have to be there to be legal OCaml code.

Custom Printers

It is possible to install custom printers so that the toplevel will convert a value of an abstract type to a string and print it instead of <abstr>. This doesn't violate abstraction, because programmers still can't access the value. It just allows the toplevel to provide better pretty printing. Here's an example utop session, based on code that appears below:

# #install_printer ListStack.format;;

# open ListStack;;

# empty |> push 1 |> push 2;;
- : int stack = [2; 1; ]

Notice how the value of the stack is helpfully printed. The code that makes this happen is in ListStack.format:

module type Stack = sig
  type 'a stack
  (* ... all the usual operations ... *)
  val format : (Format.formatter -> 'a -> unit) -> Format.formatter -> 'a stack -> unit
end

module ListStack : Stack = struct
  type 'a stack = 'a list
  (* ... all the usual operations ... *)
  let format fmt_elt fmt s =
    Format.fprintf fmt "[";
    List.iter (fun elt -> Format.fprintf fmt "%a; " fmt_elt elt) s; 
    Format.fprintf fmt "]"
end

For more information, see the toplevel manual (search for #install_printer), and the Format module, as well as this patch in the OCaml Bug Tracker.

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