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, are functions that take zero arguments). Selection is
+(equivalently, they're functions that take zero arguments). Instead, selection is
accomplished as follows:
# if true then 1 else 2;;
# f ();;
- : int = 3
-Let's have some fn: think of `rec` as our `Y` combinator. Then
+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>
- : 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:
+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.]
+
- # let rec omega x = omega x;;
+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 omega to an argument,
- # omega 3;;
+If you then apply this `blackhole` function to an argument,
+
+ # blackhole 3;;
-the interpreter goes into an infinite loop, and you have to control-C
+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:
(But `(fun x -> x x)` still won't work.)
-So we can try our usual tricks:
+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) omega;;
+ # (fun x -> true) blackhole;;
- : 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
+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:
- # omega (fun x -> true);;
+ # blackhole (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 = blackhole blackhole;;
+ <Infinite loop, need to control-c to interrupt>
- # let test () = omega omega;;
+ # let test () = blackhole blackhole;;
val test : unit -> 'a = <fun>
# test;;
- : unit -> 'a = <fun>
# test ();;
- [Infinite loop, need to control c out]
+ <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.
+
+
+
Dividing by zero: Towards Monads
--------------------------------