X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=week6.mdwn;h=2a4586a01c9b36c546e3aeacbb599a14c3e581d8;hp=f26285f3b9cb92768b509c3e93768166a3729a6a;hb=671e314f91abff10acfa31eab2211bb7eeb17881;hpb=0bdafef1c31c5372b1b7d5a960e2a016754d5388 diff --git a/week6.mdwn b/week6.mdwn index f26285f3..2a4586a0 100644 --- a/week6.mdwn +++ b/week6.mdwn @@ -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 +69,27 @@ 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.) -Booleans in OCAML, and simple pattern matching + # (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 +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;; @@ -65,7 +108,7 @@ 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 @@ -73,7 +116,7 @@ Compare with 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;; @@ -87,7 +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 # ();; @@ -110,25 +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 rec f n = if (0 = n) then 1 else (n * (f (n - 1)));; +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 = # 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: @@ -140,176 +203,101 @@ 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 = + +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. - # (fun x -> true) omega;; +We can try our usual tricks: + + # (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;; + - # 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"). -Curry-Howard, take 1 --------------------- - -We will return to the Curry-Howard correspondence a number of times -during this course. It expresses a deep connection between logic, -types, and computation. Today we'll discuss how the simply-typed -lambda calculus corresponds to intuitionistic logic. This naturally -give rise to the question of what sort of computation classical logic -corresponds to---as we'll see later, the answer involves continuations. - -So at this point we have the simply-typed lambda calculus: a set of -ground types, a set of functional types, and some typing rules, given -roughly as follows: - -If a variable `x` has type σ and term `M` has type τ, then -the abstract `\xM` has type σ `-->` τ. - -If a term `M` has type σ `-->` τ, and a term `N` has type -σ, then the application `MN` has type τ. - -These rules are clearly obverses of one another: the functional types -that abstract builds up are taken apart by application. - -The next step in making sense out of the Curry-Howard corresponence is -to present a logic. It will be a part of intuitionistic logic. We'll -start with the implicational fragment (that is, the part of -intuitionistic logic that only involves axioms and implications): - -
```-Axiom: ---------
-        A |- A
-
-Structural Rules:
-
-          Γ, A, B, Δ |- C
-Exchange: ---------------------------
-          Γ, B, A, Δ |- C
-
-             Γ, A, A |- B
-Contraction: -------------------
-             Γ, A |- B
-
-           Γ |- B
-Weakening: -----------------
-           Γ, A |- B
+Question: why do thunks work? We know that `blackhole ()` doesn't terminate, so why do expressions like:

-Logical Rules:
+	let f = fun () -> blackhole ()
+	in true

-         Γ, A |- B
---> I:   -------------------
-         Γ |- A --> B
+terminate?

-         Γ |- A --> B         Γ |- A
---> E:   -----------------------------------
-         Γ |- B
-```
+Bottom type, divergence +----------------------- -`A`, `B`, etc. are variables over formulas. -Γ, Δ, etc. are variables over (possibly empty) sequences -of formulas. Γ `|- A` is a sequent, and is interpreted as -claiming that if each of the formulas in Γ is true, then `A` -must also be true. +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: -This logic allows derivations of theorems like the following: + type 'a option = None | Some of 'a;; + type 'a option = None | Some of 'a | bottom;; -
```--------  Id
-A |- A
----------- Weak
-A, B |- A
-------------- --> I
-A |- B --> A
------------------ --> I
-|- A --> B --> A
-```
+Here are some exercises that may help better understand this. Figure out what is the type of each of the following: -Should remind you of simple types. (What was `A --> B --> A` the type -of again?) + fun x y -> y;; -The easy way to grasp the Curry-Howard correspondence is to *label* -the proofs. Since we wish to establish a correspondence between this -logic and the lambda calculus, the labels will all be terms from the -simply-typed lambda calculus. Here are the labeling rules: + fun x (y:int) -> y;; -
```-Axiom: -----------
-       x:A |- x:A
+	fun x y : int -> y;;

-Structural Rules:
+	let rec blackhole x = blackhole x in blackhole;;

-          Γ, x:A, y:B, Δ |- R:C
-Exchange: -------------------------------
-          Γ, y:B, x:A, Δ |- R:C
+	let rec blackhole x = blackhole x in blackhole 1;;

-             Γ, x:A, x:A |- R:B
-Contraction: --------------------------
-             Γ, x:A |- R:B
+	let rec blackhole x = blackhole x in fun (y:int) -> blackhole y y y;;

-           Γ |- R:B
-Weakening: ---------------------
-           Γ, x:A |- R:B     [x chosen fresh]
+	let rec blackhole x = blackhole x in (blackhole 1) + 2;;

-Logical Rules:
+	let rec blackhole x = blackhole x in (blackhole 1) || false;;

-         Γ, x:A |- R:B
---> I:   -------------------------
-         Γ |- \xM:A --> B
+	let rec blackhole x = blackhole x in 2 :: (blackhole 1);;

-         Γ |- f:(A --> B)      Γ |- x:A
---> E:   -------------------------------------
-         Γ |- (fx):B
-```
+By the way, what's the type of this: -In these labeling rules, if a sequence Γ in a premise contains -labeled formulas, those labels remain unchanged in the conclusion. + let rec blackhole (x:'a) : 'a = blackhole x in blackhole -What is means for a variable `x` to be chosen *fresh* is that -`x` must be distinct from any other variable in any of the labels -used in the proof. -Using these labeling rules, we can label the proof -just given: +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. -
```-------------  Id
-x:A |- x:A
----------------- Weak
-x:A, y:B |- x:A
-------------------------- --> I
-x:A |- (\y.x):(B --> A)
----------------------------- --> I
-|- (\x y. x):A --> B --> A
-```
-We have derived the *K* combinator, and typed it at the same time! +Towards Monads +-------------- -Need a proof that involves application, and a proof with cut that will -show beta reduction, so "normal" proof. - -[To do: add pairs and destructors; unit and negation...] +This has now been moved to the start of [[week7]]. -Excercise: construct a proof whose labeling is the combinator S.