+++ /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.
-
-[Note: There is a limited way you can compare functions, using the
-`==` operator instead of the `=` operator. Later when we discuss mutation,
-we'll discuss the difference between these two equality operations.
-Scheme has a similar pair, which they name `eq?` and `equal?`. In Python,
-these are `is` and `==` respectively. It's unfortunate that OCaml uses `==` for the opposite operation that Python and many other languages use it for. In any case, OCaml will understand `(f) == f` even though it doesn't understand
-`(f) = f`. However, don't expect it to figure out in general when two functions
-are identical. (That question is not Turing computable.)
-
- # (f) == (fun x -> x + 3);;
- - : bool = false
-
-Here OCaml says (correctly) that the two functions don't stand in the `==` relation, which basically means they're not represented in the same chunk of memory. However as the programmer can see, the functions are extensionally equivalent. The meaning of `==` is rather weird.]
-
-
-
-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, they're functions that take zero arguments). Instead, 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
-
-Now why would that be useful?
-
-Let's have some fun: 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?
-
-[Note: if you want to learn more OCaml, you might come back here someday and try:
-
- # let id x = x;;
- val id : 'a -> 'a = <fun>
- # let unwrap (`Wrap a) = a;;
- val unwrap : [< `Wrap of 'a ] -> 'a = <fun>
- # let omega ((`Wrap x) as y) = x y;;
- val omega : [< `Wrap of [> `Wrap of 'a ] -> 'b as 'a ] -> 'b = <fun>
- # unwrap (omega (`Wrap id)) == id;;
- - : bool = true
- # unwrap (omega (`Wrap omega));;
- <Infinite loop, need to control-c to interrupt>
-
-But we won't try to explain this now.]
-
-
-Even if we can't (easily) express omega in OCaml, we can do this:
-
- # let rec blackhole x = blackhole x;;
-
-By the way, what's the type of this function?
-
-If you then apply this `blackhole` function to an argument,
-
- # blackhole 3;;
-
-the interpreter goes into an infinite loop, and you have to type 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.)
-
-You may also see this:
-
- # (function x -> x);;
- - : 'a -> 'a = <fun>
-
-This works the same as `fun` in simple cases like this, and slightly differently in more complex cases. If you learn more OCaml, you'll read about the difference between them.
-
-We can try our usual tricks:
-
- # (fun x -> true) blackhole;;
- - : bool = true
-
-OCaml declined to try to fully reduce the argument before applying the
-lambda function. Question: Why is that? Didn't we say that OCaml is a call-by-value/eager language?
-
-Remember that `blackhole` is a function too, so we can
-reverse the order of the arguments:
-
- # blackhole (fun x -> true);;
-
-Infinite loop.
-
-Now consider the following variations in behavior:
-
- # let test = blackhole blackhole;;
- <Infinite loop, need to control-c to interrupt>
-
- # let test () = blackhole blackhole;;
- val test : unit -> 'a = <fun>
-
- # test;;
- - : unit -> 'a = <fun>
-
- # test ();;
- <Infinite loop, need to control-c to interrupt>
-
-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").
-
-Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like:
-
- let f = fun () -> blackhole ()
- in true
-
-terminate?
-
-Bottom type, divergence
------------------------
-
-Expressions that don't terminate all belong to the **bottom type**. This is a subtype of every other type. That is, anything of bottom type belongs to every other type as well. More advanced type systems have more examples of subtyping: for example, they might make `int` a subtype of `real`. But the core type system of OCaml doesn't have any general subtyping relations. (Neither does System F.) Just this one: that expressions of the bottom type also belong to every other type. It's as if every type definition in OCaml, even the built in ones, had an implicit extra clause:
-
- type 'a option = None | Some of 'a;;
- type 'a option = None | Some of 'a | bottom;;
-
-Here are some exercises that may help better understand this. Figure out what is the type of each of the following:
-
- fun x y -> y;;
-
- fun x (y:int) -> y;;
-
- fun x y : int -> y;;
-
- let rec blackhole x = blackhole x in blackhole;;
-
- let rec blackhole x = blackhole x in blackhole 1;;
-
- let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;;
-
- let rec blackhole x = blackhole x in (blackhole 1) + 2;;
-
- let rec blackhole x = blackhole x in (blackhole 1) || false;;
-
- let rec blackhole x = blackhole x in 2 :: (blackhole 1);;
-
- let rec blackhole (x:'a) : 'a = blackhole x in blackhole
-
-
-Back to thunks: the reason you'd want to control evaluation with thunks is to
-manipulate when "effects" happen. In a strongly normalizing system, like the
-simply-typed lambda calculus or System F, there are no "effects." In Scheme and
-OCaml, on the other hand, we can write programs that have effects. One sort of
-effect is printing (think of the [[damn]] example at the start of term).
-Another sort of effect is mutation, which we'll be looking at soon.
-Continuations are yet another sort of effect. None of these are yet on the
-table though. The only sort of effect we've got so far is *divergence* or
-non-termination. So the only thing thunks are useful for yet is controlling
-whether an expression that would diverge if we tried to fully evaluate it does
-diverge. As we consider richer languages, thunks will become more useful.
-
-
-Towards Monads
---------------
-
-This has now been moved to [its own page](/towards_monads).
-