+++ /dev/null
-[[!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.
-
-<pre>
-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
-*)
-</pre>
-
-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:
-
-<pre>
-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 = <fun>
-# 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
-*)
-</pre>
-
-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:
-
-<pre>
-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);;
-</pre>
-
-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:
-
-<pre>
-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 = <fun>
-# add' (Some 12) (Some 4);;
-- : int option = Some 16
-# add' (div' (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-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.
-
-<pre>
-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
-*)
-</pre>
-
-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.]
-
-