-Expression Type Implication ----------------------------------- -fn α -> β α ⊃ β -arg α α ------- ------ -------- -fn arg β β +Expression Type Implication +----------------------------------- +fn α -> β α ⊃ β +arg α α +------ ------ -------- +(fn arg) β βThe implication in the right-hand column is modus ponens, of course. -System F is usually attributed to Girard, but was independently -proposed around the same time by Reynolds. It enhances the -simply-typed lambda calculus with quantification over types. In -System F, you can say things like +System F was discovered by Girard (the same guy who invented Linear +Logic), but it was independently proposed around the same time by +Reynolds, who called his version the *polymorphic lambda calculus*. +(Reynolds was also an early player in the development of +continuations.) -

`Λ α (\x.x):(α -> α)`

+System F enhances the simply-typed lambda calculus with abstraction
+over types. In order to state System F, we'll need to adopt the
+notational convention that "`x:α`

" represents a
+expression whose type is `α`

.
-This says that the identity function maps arguments of type α to
-results of type α, for any choice of α. So the Λ is
-a universal quantifier over types.
+Then System F can be specified as follows (choosing notation that will
+match up with usage in O'Caml, whose type system is based on System F):
+ System F:
+ types Ï ::= c | 'a | Ï1 -> Ï2 | â'a. Ï
+ expressions e ::= x | Î»x:Ï. e | e1 e2 | Î'a. e | e [Ï]
+
+In the definition of the types, "`c`" is a type constant (e.g., `e` or
+`t`). "`'a`" is a type variable (the tick mark just indicates that
+the variable ranges over types rather than values). "`Ï1 -> Ï2`" is
+the type of a function from expressions of type `Ï1` to expressions of
+type `Ï2`. And "`â'a. Ï`" is called a universal type, since it
+universally quantifies over the type variable `'a`.
+
+In the definition of the expressions, we have variables "`x`".
+Abstracts "`Î»x:Ï. e`" are similar to abstracts in the simply-typed lambda
+calculus, except that they have their shrug variable annotated with a
+type. Applications "`e1 e2`" are just like in the simply-typed lambda calculus.
+In addition to variables, abstracts, and applications, we have two
+additional ways of forming expressions: "`Î'a. e`" is a type
+abstraction, and "`e [Ï]`" is a type application. The idea is that
+`Λ`

is a capital `λ`

. Just like
+the lower-case `λ`

, `Λ`

binds
+variables in its body; unlike `λ`

,
+`Λ`

binds type variables. So in the expression
+
+`Λ 'a (λ x:'a . x)`

+
+the `Λ`

binds the type variable `'a` that occurs in
+the `λ`

abstract. This expression is a polymorphic
+version of the identity function. It says that this one general
+identity function can be adapted for use with expressions of any
+type. In order to get it ready to apply to, say, a variable of type
+boolean, just do this:
+
+`(Λ 'a (λ x:'a . x)) [t]`

+
+The type application (where `t` is a type constant for Boolean truth
+values) specifies the value of the type variable `α`, which is
+the type of the variable bound in the `λ` expression. Not
+surprisingly, the type of this type application is a function from
+Booleans to Booleans:
+
+`((Λ 'a (λ x:'a . x)) [t]): (b -> b)`

+
+Likewise, if we had instantiated the type variable as an entity (base
+type `e`), the resulting identity function would have been a function
+of type `e -> e`:
+
+`((Λ 'a (λ x:'a . x)) [e]): (e -> e)`

+
+Clearly, for any choice of a type `'a`, the identity function can be
+instantiated as a function from expresions of type `'a` to expressions
+of type `'a`. In general, then, the type of the unapplied
+(polymorphic) identity function is
+
+```
(Λ 'a (λ x:'a . x)): (\forall 'a . 'a -> 'a)
+
+
+
+
+##
+
+
+
+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
+
+ # let f x = x + 3;;
+
+The system replies with
+
+ val f : int -> int =
```
+
+Since `+` is only defined on integers, it has type
+
+ # (+);;
+ - : int -> int -> int =
+
+The parentheses are there to turn off the trick that allows the two
+arguments of `+` to surround it in infix (for linguists, SOV) argument
+order. That is,
+
+ # 3 + 4 = (+) 3 4;;
+ - : bool = true
+
+In general, tuples with one element are identical to their one
+element:
+
+ # (3) = 3;;
+ - : bool = true
+
+though OCaml, like many systems, refuses to try to prove whether two
+functional objects may be identical:
+
+ # (f) = f;;
+ Exception: Invalid_argument "equal: functional value".
+
+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, they're functions that take zero arguments). Instead, selection is
+accomplished as follows:
+
+ # if true then 1 else 2;;
+ - : int = 1
+
+The types of the `then` clause and of the `else` clause must be the
+same.
+
+The `if` construction can be re-expressed by means of the following
+pattern-matching expression:
+
+ match with true -> | false ->
+
+That is,
+
+ # match true with true -> 1 | false -> 2;;
+ - : int = 1
+
+Compare with
+
+ # match 3 with 1 -> 1 | 2 -> 4 | 3 -> 9;;
+ - : int = 9
+
+Unit and thunks
+---------------
+
+All functions in OCaml take exactly one argument. Even this one:
+
+ # let f x y = x + y;;
+ # f 2 3;;
+ - : int = 5
+
+Here's how to tell that `f` has been curry'd:
+
+ # f 2;;
+ - : int -> int =
+
+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
+object in this type, written `()`. So
+
+ # ();;
+ - : unit = ()
+
+Just as you can define functions that take constants for arguments
+
+ # let f 2 = 3;;
+ # f 2;;
+ - : int = 3;;
+
+you can also define functions that take the unit as its argument, thus
+
+ # let f () = 3;;
+ val f : unit -> int =
+
+Then the only argument you can possibly apply `f` to that is of the
+correct type is the unit:
+
+ # f ();;
+ - : int = 3
+
+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 =
+ # 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?
+
+[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.]
+
+
+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 `blackhole` function to an argument,
+
+ # blackhole 3;;
+
+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:
+
+ # (fun x -> x);;
+ - : 'a -> 'a =
+ # (fun x -> x) true;;
+ - : bool = true
+
+(But `(fun x -> x x)` still won't work.)
+
+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) blackhole;;
+ - : bool = true
+
+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:
+
+ # blackhole (fun x -> true);;
+
+Infinite loop.
+
+Now consider the following variations in behavior:
+
+ # let test = blackhole blackhole;;
+
+
+ # let test () = blackhole blackhole;;
+ val test : unit -> 'a =
+
+ # test;;
+ - : unit -> 'a =
+
+ # test ();;
+
+
+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.
+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.