[[!toc]]
-Types, OCaml
-------------
+Polymorphic Types and System F
+------------------------------
+
+[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.
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
----------------------------------------------
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
+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
--------------
-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. As a mnemonic aid, we'll append a `'` to the end of our new divide function.
-
-<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). To continue our mnemonic association, we'll put a `'` after the name "bind" as well.
-
-<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]].
+