manip trees tweaks

[lambda.git] / week6.mdwn

diff --git a/week6.mdwn b/week6.mdwn
index b97a09f..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.)
+
+       # (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` 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'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 = <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:


@@ -140,175 +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.
+
+We can try our usual tricks:

 
 

-    # (fun x -> true) omega;;
+    # (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").
 

+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
 --------------
 
 Towards Monads
 --------------
 

-So the integer division operation presupposes that its second argument
-(the divisor) is not zero, upon pain of presupposition failure.
-Here's what my OCAML interpreter says:
-
-    # 12/0;;
-    Exception: Division_by_zero.
-
-So we want to explicitly allow for the possibility that 
-division will return something other than a number.
-We'll use OCAML's option type, which works like this:
-
-    # type 'a option = None | Some of 'a;;
-    # None;;
-    - : 'a option = None
-    # Some 3;;
-    - : int option = Some 3
-
-So if a division is normal, we return some number, but if the divisor is
-zero, we return None:
-
-<pre>
-let div (x:int) (y:int) = 
-  match y with 0 -> None |
-               _ -> Some (x / y);;
-
-(*
-val div : int -> int -> int option = fun
-# div 12 3;;
-- : int option = Some 4
-# div 12 0;;
-- : int option = None
-# div (div 12 3) 2;;
-Characters 4-14:
-  div (div 12 3) 2;;
-      ^^^^^^^^^^
-Error: This expression has type int option
-       but an expression was expected of type int
-*)
-</pre>
-
-This starts off well: dividing 12 by 3, no problem; dividing 12 by 0,
-just the behavior we were hoping for.  But we want to be able to use
-the output of the safe-division function as input for further division
-operations.  So we have to jack up the types of the inputs:
-
-<pre>
-let div (x:int option) (y:int option) = 
-  match y with None -> None |
-               Some 0 -> None |
-               Some n -> (match x with None -> None |
-                                       Some m -> Some (m / n));;
-
-(*
-val div : int option -> int option -> int option = <fun>
-# div (Some 12) (Some 4);;
-- : int option = Some 3
-# div (Some 12) (Some 0);;
-- : int option = None
-# div (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-Beautiful, just what we need: now we can try to divide by anything we
-want, without fear that we're going to trigger any system errors.
-
-I prefer to line up the `match` alternatives by using OCAML's 
-built-in tuple type:
-
-<pre>
-let div (x:int option) (y:int option) = 
-  match (x, y) with (None, _) -> None |
-                    (_, None) -> None |
-                    (_, Some 0) -> None |
-                    (Some m, Some n) -> Some (m / n);;
-</pre>
-
-So far so good.  But what if we want to combine division with 
-other arithmetic operations?  We need to make those other operations 
-aware of the possibility that one of their arguments will trigger a
-presupposition failure:
-
-<pre>
-let add (x:int option) (y:int option) = 
-  match (x, y) with (None, _) -> None |
-                    (_, None) -> None |
-                    (Some m, Some n) -> Some (m + n);;
-
-(*
-val add : int option -> int option -> int option = <fun>
-# add (Some 12) (Some 4);;
-- : int option = Some 16
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-This works, but is somewhat disappointing: the `add` operation
-doesn't trigger any presupposition of its own, so it is a shame that
-it needs to be adjusted because someone else might make trouble.
-
-But we can automate the adjustment.  The standard way in OCAML,
-Haskell, etc., is to define a `bind` operator (the name `bind` is not
-well chosen to resonate with linguists, but what can you do):
-
-<pre>
-let bind (x: int option) (f: int -> (int option)) = 
-  match x with None -> None | 
-               Some n -> f n;;
-
-let add (x: int option) (y: int option)  =
-  bind x (fun x -> bind y (fun y -> Some (x + y)));;
-
-let div (x: int option) (y: int option) =
-  bind x (fun x -> bind y (fun y -> if (0 = y) then None else Some (x / y)));;
-
-(*
-#  div (div (Some 12) (Some 2)) (Some 4);;
-- : int option = Some 1
-#  div (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-# add (div (Some 12) (Some 0)) (Some 4);;
-- : int option = None
-*)
-</pre>
-
-Compare the new definitions of `add` and `div` closely: the definition
-for `add` shows what it looks like to equip an ordinary operation to
-survive in dangerous presupposition-filled world.  Note that the new
-definition of `add` does not need to test whether its arguments are
-None objects or real numbers---those details are hidden inside of the
-`bind` function.
-
-The definition of `div` shows exactly what extra needs to be said in
-order to trigger the no-division-by-zero presupposition.
-
-For linguists: this is a complete theory of a particularly simply form
-of presupposition projection (every predicate is a hole).
+This has now been moved to the start of [[week7]].
+