[[!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
*)
```