X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=topics%2F_week8_division_by_zero.mdwn;fp=topics%2F_week8_division_by_zero.mdwn;h=0000000000000000000000000000000000000000;hp=4bbf60b9e16fcc66f1a8bfc47214a4fe16bd371c;hb=3707173d39189e802943a1ef97a751677ee6e4a6;hpb=656e541d83b0697e9677bb8b0edad218a8bd8497 diff --git a/topics/_week8_division_by_zero.mdwn b/topics/_week8_division_by_zero.mdwn deleted file mode 100644 index 4bbf60b9..00000000 --- a/topics/_week8_division_by_zero.mdwn +++ /dev/null @@ -1,169 +0,0 @@ -[[!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 =- -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: - --# 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 -*) -
-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 =- -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. - --# add' (Some 12) (Some 4);; -- : int option = Some 16 -# add' (div' (Some 12) (Some 0)) (Some 4);; -- : int option = None -*) -
-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.] - -