week9 tweak

[lambda.git] / week6.mdwn

diff --git a/week6.mdwn b/week6.mdwn
index 57158c6..2a4586a 100644 (file)
--- a/week6.mdwn
+++ b/week6.mdwn
@@ -1,16 +1,45 @@
 [[!toc]]
 
 [[!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.]
+
+<!--
+
+8.     The simply-typed lambda calculus<p>
+9.     Parametric polymorphism, System F, "type inference"<p>
+
+1.     Product or record types, e.g. pairs and triples
+2.     Sum or variant types; tagged or "disjoint" unions
+3.     Maybe/option types; representing "out-of-band" values
+10.    [Phil/ling application] inner/outer domain semantics for positive free logic
+       <http://philosophy.ucdavis.edu/antonelli/papers/pegasus-JPL.pdf>
+11.    [Phil/ling application] King vs Schiffer in King 2007, pp 103ff. [which paper?](http://rci.rutgers.edu/~jeffreck/pub.php)
+12. [Phil/ling application] King and Pryor on that clauses, predicates vs singular property-designators
+       Russell On Denoting / Kaplan on plexy
+13.    Possible excursion: [Frege's "On Concept and Object"](http://www.persiangig.com/pages/download/?dl=http://sahmir.persiangig.com/document/Frege%27s%20Articles/On%20Concept%20And%20object%20%28Jstore%29.pdf)<p>
+
+6.     Inductive types (numbers, lists)
+
+5.     Unit type
+4.     Zero/bottom types
+7.     "Pattern-matching" or type unpacking<p>
+
+-->
+
+
+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.
 
 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;;
 
 
     # let f x = x + 3;;
 

-The system replies with 
+The system replies with

 
     val f : int -> int = <fun>
 
 
     val f : int -> int = <fun>
 


@@ -32,7 +61,7 @@ element:
     # (3) = 3;;
     - : bool = true
 
     # (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;;
 functional objects may be identical:
 
     # (f) = f;;


@@ -40,13 +69,27 @@ functional objects may be identical:
 
 Oh well.
 
 
 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
 ----------------------------------------------
 
 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;;
 accomplished as follows:
 
     # if true then 1 else 2;;


@@ -65,7 +108,7 @@ That is,
     # match true with true -> 1 | false -> 2;;
     - : int = 1
 
     # match true with true -> 1 | false -> 2;;
     - : int = 1
 

-Compare with 
+Compare with

 
     # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
     - : int = 9
 
     # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
     - : int = 9


@@ -73,7 +116,7 @@ Compare with
 Unit and thunks
 ---------------
 
 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;;
 
     # 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.
 
 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
 
     # ();;
 object in this type, written `()`.  So
 
     # ();;


@@ -110,25 +153,45 @@ correct type is the unit:
     # f ();;
     - : int = 3
 
     # 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 = <fun>
     # f 5;;
     - : int = 120
 
 We can't define a function that is exactly analogous to our &omega;.
     val f : int -> int = <fun>
     # f 5;;
     - : int = 120
 
 We can't define a function that is exactly analogous to our &omega;.

-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?
 
 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:
 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.)
 
 
 (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.

 
 

-    # (fun x -> true) omega;;
+We can try our usual tricks:
+
+    # (fun x -> true) blackhole;;

     - : bool = true
 
     - : 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:
 
 reverse the order of the arguments:
 

-    # omega (fun x -> true);;
+    # blackhole (fun x -> true);;

 
 Infinite loop.
 
 Now consider the following variations in behavior:
 
 
 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 ();;
     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
+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").
 
 *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 &sigma; and term `M` has type &tau;, then 
-the abstract `\xM` has type &sigma; `-->` &tau;.
-
-If a term `M` has type &sigma; `-->` &tau;, and a term `N` has type
-&sigma;, then the application `MN` has type &tau;.
-
-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):
-
-<pre>
-Axiom: ---------
-        A |- A
-
-Structural Rules:
-
-          &Gamma;, A, B, &Delta; |- C
-Exchange: ---------------------------
-          &Gamma;, B, A, &Delta; |- C
-
-             &Gamma;, A, A |- B
-Contraction: -------------------
-             &Gamma;, A |- B
-
-           &Gamma; |- B
-Weakening: -----------------
-           &Gamma;, 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

 
 

-         &Gamma;, A |- B
---> I:   -------------------
-         &Gamma; |- A --> B  
+terminate?

 
 

-         &Gamma; |- A --> B         &Gamma; |- A
---> E:   -----------------------------------
-         &Gamma; |- B
-</pre>
+Bottom type, divergence
+-----------------------

 
 

-`A`, `B`, etc. are variables over formulas.  
-&Gamma;, &Delta;, etc. are variables over (possibly empty) sequences
-of formulas.  &Gamma; `|- A` is a sequent, and is interpreted as
-claiming that if each of the formulas in &Gamma; 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;;

 
 

-<pre>
--------  Id
-A |- A
----------- Weak
-A, B |- A
-------------- --> I
-A |- B --> A
------------------ --> I
-|- A --> B --> A
-</pre>
+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;;

 
 

-<pre>
-Axiom: -----------
-       x:A |- x:A
+       fun x y : int -> y;;

 
 

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

 
 

-Exchange: &Gamma;, x:A, y:B, &Delta; |- R:C
-          -------------------------------
-          &Gamma;, y:B, x:A, &Delta; |- R:C
+       let rec blackhole x = blackhole x in blackhole 1;;

 
 

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

 
 

-Weakening: &Gamma; |- R:B
-           --------------------- 
-           &Gamma;, 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;;

 
 

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

 
 

---> E:   &Gamma; |- f:(A --> B)      &Gamma; |- x:A
-         -------------------------------------
-         &Gamma; |- (fx):B
-</pre>
+By the way, what's the type of this:

 
 

-In these labeling rules, if a sequence &Gamma; 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.

 
 

-<pre>
-------------  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
-</pre>

 
 

-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.