Booleans in OCAML, and simple pattern matching
----------------------------------------------
-Where we would write `true 1 2` and expect it to evaluate to `1`, in
-OCAML boolean types are not functions (equivalently, are functions
-that take zero arguments). Choices are made as follows:
+Where we would write `true 1 2` in our pure lambda calculus and expect
+it to evaluate to `1`, in OCAML boolean types are not functions
+(equivalently, are functions that take zero arguments). Selection is
+accomplished as follows:
# if true then 1 else 2;;
- : int = 1
# match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
- : int = 9
-Unit
-----
+Unit and thunks
+---------------
All functions in OCAML take exactly one argument. Even this one:
# (fun x -> x);;
- : 'a -> 'a = <fun>
# (fun x -> x) true;;
- - : book = true
+ - : bool = true
(But `(fun x -> x x)` still won't work.)
Infinite loop.
-Now consider the following differences:
+Now consider the following variations in behavior:
# let test = omega omega;;
[Infinite loop, need to control c out]
execution. In Scheme parlance, functions on the unit type are called
*thunks* (which I've always assumed was a blend of "think" and "chunk").
+Towards Monads
+--------------
+
+So the integer division operation 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:
+
+<pre>
+let div (x:int) (y:int) =
+ match y with 0 -> None |
+ _ -> Some (x / y);;
+
+(*
+val div : int -> int -> int option = \<fun\>
+# div 12 3;;
+- : int option = Some 4
+# div 12 0;;
+- : int option = None
+# div (div 12 3) 2;;
+Characters 4-14:
+ div (div 12 3) 2;;
+ ^^^^^^^^^^
+Error: This expression has type int option
+ but an expression was expected of type int
+*)
+</pre>
+
+This starts off well: dividing 12 by 3, 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 (x:int option) (y:int option) =
+ match y with None -> None |
+ Some 0 -> None |
+ Some n -> (match x with None -> None |
+ Some m -> Some (m / n));;
+
+(*
+val div : int option -> int option -> int option = <fun>
+# div (Some 12) (Some 4);;
+- : int option = Some 3
+# div (Some 12) (Some 0);;
+- : int option = None
+# div (div (Some 12) (Some 0)) (Some 4);;
+- : 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 (x:int option) (y:int option) =
+ match (x, y) with (None, _) -> None |
+ (_, None) -> None |
+ (_, Some 0) -> None |
+ (Some m, Some n) -> Some (m / n);;
+</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 will trigger a
+presupposition failure:
+
+<pre>
+let add (x:int option) (y:int option) =
+ match (x, y) with (None, _) -> None |
+ (_, None) -> None |
+ (Some m, Some n) -> Some (m + n);;
+
+(*
+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` prediction
+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):
+
+<pre>
+let bind (x: int option) (f: int -> (int option)) =
+ match x with None -> None | Some n -> f n;;
+
+let add (x: int option) (y: int option) =
+ bind x (fun x -> bind y (fun y -> Some (x + y)));;
+
+let div (x: int option) (y: int option) =
+ bind x (fun x -> bind y (fun y -> if (0 = y) then None else Some (x / y)));;
+
+(*
+# div (div (Some 12) (Some 2)) (Some 4);;
+- : int option = Some 1
+# div (div (Some 12) (Some 0)) (Some 4);;
+- : int option = None
+# add (div (Some 12) (Some 0)) (Some 4);;
+- : 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 a presupposition-filled world, and the definition of `div`
+shows exactly what extra needs to be added in order to trigger the
+no-division-by-zero presupposition.
+