[[!toc]]
-Types, OCAML
+Types, OCaml
------------
-OCAML has type inference: the system can often infer what the type of
+OCaml has type inference: the system can often infer what the type of
an expression must be, based on the type of other known expressions.
-For instance, if we type
+For instance, if we type
# let f x = x + 3;;
-The system replies with
+The system replies with
val f : int -> int = <fun>
# (3) = 3;;
- : bool = true
-though OCAML, like many systems, refuses to try to prove whether two
+though OCaml, like many systems, refuses to try to prove whether two
functional objects may be identical:
# (f) = f;;
Oh well.
-Booleans in OCAML, and simple pattern matching
+Booleans in OCaml, and simple pattern matching
----------------------------------------------
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
+it to evaluate to `1`, in OCaml boolean types are not functions
(equivalently, are functions that take zero arguments). Selection is
accomplished as follows:
# match true with true -> 1 | false -> 2;;
- : int = 1
-Compare with
+Compare with
# match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
- : int = 9
Unit and thunks
---------------
-All functions in OCAML take exactly one argument. Even this one:
+All functions in OCaml take exactly one argument. Even this one:
# let f x y = x + y;;
# f 2 3;;
After we've given our `f` one argument, it returns a function that is
still waiting for another argument.
-There is a special type in OCAML called `unit`. There is exactly one
+There is a special type in OCaml called `unit`. There is exactly one
object in this type, written `()`. So
# ();;
Let's have some fn: think of `rec` as our `Y` combinator. Then
- # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
+ # let rec f n = if (0 = n) then 1 else (n * (f (n - 1)));;
val f : int -> int = <fun>
# f 5;;
- : int = 120
# (fun x -> true) omega;;
- : bool = true
-OCAML declined to try to evaluate the argument before applying the
+OCaml declined to try to evaluate the argument before applying the
functor. But remember that `omega` is a function too, so we can
reverse the order of the arguments:
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:
+Here's what my OCaml interpreter says:
# 12/0;;
Exception: Division_by_zero.
-So we want to explicitly allow for the possibility that
+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:
+We'll use OCaml's option type, which works like this:
# type 'a option = None | Some of 'a;;
# None;;
- : int option = Some 3
So if a division is normal, we return some number, but if the divisor is
-zero, we return None:
+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) =
+let div' (x:int) (y:int) =
match y with 0 -> None |
_ -> Some (x / y);;
(*
-val div : int -> int -> int option = \<fun\>
-# div 12 3;;
+val div' : int -> int -> int option = fun
+# div' 12 3;;
- : int option = Some 4
-# div 12 0;;
+# div' 12 0;;
- : int option = None
-# div (div 12 3) 2;;
+# div' (div' 12 3) 2;;
Characters 4-14:
- div (div 12 3) 2;;
+ div' (div' 12 3) 2;;
^^^^^^^^^^
Error: This expression has type int option
but an expression was expected of type int
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
+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) =
+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);;
+val div' : int option -> int option -> int option = <fun>
+# div' (Some 12) (Some 4);;
- : int option = Some 3
-# div (Some 12) (Some 0);;
+# div' (Some 12) (Some 0);;
- : int option = None
-# div (div (Some 12) (Some 0)) (Some 4);;
+# 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
+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) =
+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
+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) =
+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);;
+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);;
+# add' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
*)
</pre>
-This works, but is somewhat disappointing: the `add` prediction
+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,
+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):
+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 (x: int option) (f: int -> (int option)) =
- match x with None -> None | Some n -> f n;;
+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 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)));;
+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);;
+# div' (div' (Some 12) (Some 2)) (Some 4);;
- : int option = Some 1
-# div (div (Some 12) (Some 0)) (Some 4);;
+# div' (div' (Some 12) (Some 0)) (Some 4);;
- : int option = None
-# add (div (Some 12) (Some 0)) (Some 4);;
+# 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.
+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).