Polymorphic Types and System F

[Notes still to be added. Hope you paid attention during seminar.]

Types in 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 accept (f) == f even though it doesn't accept (f) = f. However, don't expect it to figure out in general when two functions are equivalent. (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


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);;

By the way, what's the type of this:

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 the start of week7.