[[!toc]] Towards Monads: Safe division ----------------------------- [This section used to be near the end of the lecture notes for week 6] We begin by reasoning about what should happen when someone tries to divide by zero. This will lead us to a general programming technique called a *monad*, which we'll see in many guises in the weeks to come. Integer division presupposes that its second argument (the divisor) is not zero, upon pain of presupposition failure. Here's what my OCaml interpreter says: # 12/0;; Exception: Division_by_zero. So we want to explicitly allow for the possibility that division will return something other than a number. We'll use OCaml's `option` type, which works like this: # type 'a option = None | Some of 'a;; # None;; - : 'a option = None # Some 3;; - : int option = Some 3 So if a division is normal, we return some number, but if the divisor is zero, we return `None`. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
let div' (x:int) (y:int) =
  match y with
	  0 -> None
    | _ -> Some (x / y);;

(*
val div' : int -> int -> int option = fun
# div' 12 2;;
- : int option = Some 6
# div' 12 0;;
- : int option = None
# div' (div' 12 2) 3;;
Characters 4-14:
  div' (div' 12 2) 3;;
        ^^^^^^^^^^
Error: This expression has type int option
       but an expression was expected of type int
*)
This starts off well: dividing 12 by 2, no problem; dividing 12 by 0, just the behavior we were hoping for. But we want to be able to use the output of the safe-division function as input for further division operations. So we have to jack up the types of the inputs:
let div' (u:int option) (v:int option) =
  match u with
	  None -> None
	| Some x -> (match v with
				  Some 0 -> None
				| Some y -> Some (x / y));;

(*
val div' : int option -> int option -> int option = 
# div' (Some 12) (Some 2);;
- : int option = Some 6
# div' (Some 12) (Some 0);;
- : int option = None
# div' (div' (Some 12) (Some 0)) (Some 3);;
- : int option = None
*)
Beautiful, just what we need: now we can try to divide by anything we want, without fear that we're going to trigger any system errors. I prefer to line up the `match` alternatives by using OCaml's built-in tuple type:
let div' (u:int option) (v:int option) =
  match (u, v) with
	  (None, _) -> None
    | (_, None) -> None
    | (_, Some 0) -> None
	| (Some x, Some y) -> Some (x / y);;
So far so good. But what if we want to combine division with other arithmetic operations? We need to make those other operations aware of the possibility that one of their arguments has triggered a presupposition failure:
let add' (u:int option) (v:int option) =
  match (u, v) with
	  (None, _) -> None
    | (_, None) -> None
    | (Some x, Some y) -> Some (x + y);;

(*
val add' : int option -> int option -> int option = 
# add' (Some 12) (Some 4);;
- : int option = Some 16
# add' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
This works, but is somewhat disappointing: the `add'` operation doesn't trigger any presupposition of its own, so it is a shame that it needs to be adjusted because someone else might make trouble. But we can automate the adjustment. The standard way in OCaml, Haskell, etc., is to define a `bind` operator (the name `bind` is not well chosen to resonate with linguists, but what can you do). To continue our mnemonic association, we'll put a `'` after the name "bind" as well.
let bind' (u: int option) (f: int -> (int option)) =
  match u with
	  None -> None
    | Some x -> f x;;

let add' (u: int option) (v: int option)  =
  bind' u (fun x -> bind' v (fun y -> Some (x + y)));;

let div' (u: int option) (v: int option) =
  bind' u (fun x -> bind' v (fun y -> if (0 = y) then None else Some (x / y)));;

(*
#  div' (div' (Some 12) (Some 2)) (Some 3);;
- : int option = Some 2
#  div' (div' (Some 12) (Some 0)) (Some 3);;
- : int option = None
# add' (div' (Some 12) (Some 0)) (Some 3);;
- : int option = None
*)
Compare the new definitions of `add'` and `div'` closely: the definition for `add'` shows what it looks like to equip an ordinary operation to survive in dangerous presupposition-filled world. Note that the new definition of `add'` does not need to test whether its arguments are None objects or real numbers---those details are hidden inside of the `bind'` function. The definition of `div'` shows exactly what extra needs to be said in order to trigger the no-division-by-zero presupposition. [Linguitics note: Dividing by zero is supposed to feel like a kind of presupposition failure. If we wanted to adapt this approach to building a simple account of presupposition projection, we would have to do several things. First, we would have to make use of the polymorphism of the `option` type. In the arithmetic example, we only made use of `int option`s, but when we're composing natural language expression meanings, we'll need to use types like `N option`, `Det option`, `VP option`, and so on. But that works automatically, because we can use any type for the `'a` in `'a option`. Ultimately, we'd want to have a theory of accommodation, and a theory of the situations in which material within the sentence can satisfy presuppositions for other material that otherwise would trigger a presupposition violation; but, not surprisingly, these refinements will require some more sophisticated techniques than the super-simple Option monad.]