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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 =
Since `+` is only defined on integers, it has type
# (+);;
- : int -> int -> int =
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 with true -> | false ->
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 =
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 =
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 =
# 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 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 =
# test;;
- : unit -> 'a =
# 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").