+++ /dev/null
-[[!toc]]
-
-Types, OCaml
-------------
-
-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
-
- # let f x = x + 3;;
-
-The system replies with
-
- val f : int -> int = <fun>
-
-Since `+` is only defined on integers, it has type
-
- # (+);;
- - : int -> int -> int = <fun>
-
-The parentheses are there to turn off the trick that allows the two
-arguments of `+` to surround it in infix (for linguists, SOV) argument
-order. That is,
-
- # 3 + 4 = (+) 3 4;;
- - : bool = true
-
-In general, tuples with one element are identical to their one
-element:
-
- # (3) = 3;;
- - : bool = true
-
-though OCaml, like many systems, refuses to try to prove whether two
-functional objects may be identical:
-
- # (f) = f;;
- Exception: Invalid_argument "equal: functional value".
-
-Oh well.
-
-
-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
-(equivalently, are functions that take zero arguments). Selection is
-accomplished as follows:
-
- # if true then 1 else 2;;
- - : int = 1
-
-The types of the `then` clause and of the `else` clause must be the
-same.
-
-The `if` construction can be re-expressed by means of the following
-pattern-matching expression:
-
- match <bool expression> with true -> <expression1> | false -> <expression2>
-
-That is,
-
- # match true with true -> 1 | false -> 2;;
- - : int = 1
-
-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:
-
- # let f x y = x + y;;
- # f 2 3;;
- - : int = 5
-
-Here's how to tell that `f` has been curry'd:
-
- # f 2;;
- - : int -> int = <fun>
-
-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
-object in this type, written `()`. So
-
- # ();;
- - : unit = ()
-
-Just as you can define functions that take constants for arguments
-
- # let f 2 = 3;;
- # f 2;;
- - : int = 3;;
-
-you can also define functions that take the unit as its argument, thus
-
- # let f () = 3;;
- val f : unit -> int = <fun>
-
-Then the only argument you can possibly apply `f` to that is of the
-correct type is the unit:
-
- # f ();;
- - : int = 3
-
-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)));;
- val f : int -> int = <fun>
- # f 5;;
- - : int = 120
-
-We can't define a function that is exactly analogous to our ω.
-We could try `let rec omega x = x x;;` what happens? However, we can
-do this:
-
- # let rec omega x = omega x;;
-
-By the way, what's the type of this function?
-If you then apply this omega to an argument,
-
- # omega 3;;
-
-the interpreter goes into an infinite loop, and you have to control-C
-to break the loop.
-
-Oh, one more thing: lambda expressions look like this:
-
- # (fun x -> x);;
- - : 'a -> 'a = <fun>
- # (fun x -> x) true;;
- - : bool = true
-
-(But `(fun x -> x x)` still won't work.)
-
-So we can try our usual tricks:
-
- # (fun x -> true) omega;;
- - : bool = true
-
-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:
-
- # omega (fun x -> true);;
-
-Infinite loop.
-
-Now consider the following variations in behavior:
-
- # let test = omega omega;;
- [Infinite loop, need to control c out]
-
- # let test () = omega omega;;
- val test : unit -> 'a = <fun>
-
- # test;;
- - : unit -> 'a = <fun>
-
- # test ();;
- [Infinite loop, need to control c out]
-
-We can use functions that take arguments of type unit to control
-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` 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):
-
-<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 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).