X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=2a4586a01c9b36c546e3aeacbb599a14c3e581d8;hp=1839455a6542b7df4b9339222295d6f871b14358;hb=f13c3904992a39aa33db9c1c80a4a984dabc341b;hpb=80ad862c64373ac07b6e33236a47a50e98583d62
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@@ -1,16 +1,45 @@
[[!toc]]
-Types, OCAML
-------------
+Polymorphic Types and System F
+------------------------------
-OCAML has type inference: the system can often infer what the type of
+[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
+For instance, if we type
# let f x = x + 3;;
-The system replies with
+The system replies with
val f : int -> int =
@@ -32,7 +61,7 @@ element:
# (3) = 3;;
- : bool = true
-though OCAML, like many systems, refuses to try to prove whether two
+though OCaml, like many systems, refuses to try to prove whether two
functional objects may be identical:
# (f) = f;;
@@ -40,13 +69,28 @@ functional objects may be identical:
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
+
+Booleans in OCaml, and simple pattern matching
----------------------------------------------
-Where we would write `true 1 2` and expect it to evaluate to `1`, in
-OCAML boolean types are not functions (equivalently, are functions
-that take zero arguments). Choices are made as follows:
+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
@@ -64,15 +108,15 @@ That is,
# match true with true -> 1 | false -> 2;;
- : int = 1
-Compare with
+Compare with
# match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
- : int = 9
-Unit
-----
+Unit and thunks
+---------------
-All functions in OCAML take exactly one argument. Even this one:
+All functions in OCaml take exactly one argument. Even this one:
# let f x y = x + y;;
# f 2 3;;
@@ -86,7 +130,7 @@ Here's how to tell that `f` has been curry'd:
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
+There is a special type in OCaml called `unit`. There is exactly one
object in this type, written `()`. So
# ();;
@@ -109,25 +153,45 @@ correct type is the unit:
# 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)));;
+ # 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:
+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 =
+ # let unwrap (`Wrap a) = a;;
+ val unwrap : [< `Wrap of 'a ] -> 'a =
+ # let omega ((`Wrap x) as y) = x y;;
+ val omega : [< `Wrap of [> `Wrap of 'a ] -> 'b as 'a ] -> 'b =
+ # unwrap (omega (`Wrap id)) == id;;
+ - : bool = true
+ # unwrap (omega (`Wrap omega));;
+
+
+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:
@@ -135,38 +199,105 @@ Oh, one more thing: lambda expressions look like this:
# (fun x -> x);;
- : 'a -> 'a =
# (fun x -> x) true;;
- - : book = true
+ - : bool = true
(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 =
+
+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 differences:
+Now consider the following variations in behavior:
- # let test = omega omega;;
- [Infinite loop, need to control c out]
+ # let test = blackhole blackhole;;
+
- # let test () = omega omega;;
+ # let test () = blackhole blackhole;;
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
+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);;
+
+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]].
+