X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week7.mdwn;h=af8c17e5f05c286b2fd01237449d82337df5e7fc;hp=2abf10d735c201942ed827a52a69bcd05d5aaffd;hb=386b3d90a12ae3c787bd6620194e6fe736cf8e0e;hpb=2ffca8b607c2198802f7227d899564d738ddcd6b diff --git a/week7.mdwn b/week7.mdwn index 2abf10d7..af8c17e5 100644 --- a/week7.mdwn +++ b/week7.mdwn @@ -1,8 +1,159 @@ [[!toc]] -Monads ------- +Towards Monads: Safe division +----------------------------- + +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 v with + None -> None + | Some 0 -> None + | Some y -> (match u with + None -> None + | Some x -> 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. + +For linguists: this is a complete theory of a particularly simply form +of presupposition projection (every predicate is a hole). + + + + +Monads in General +----------------- Start by (re)reading the discussion of monads in the lecture notes for week 6 [[Towards Monads]].