+++ /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").
-