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[[!toc levels=2]]

# System F and recursive types

In the simply-typed lambda calculus, we write types like <code>&sigma;
-> &tau;</code>.  This looks like logical implication.  We'll take
that resemblance seriously when we discuss the Curry-Howard
correspondence.  In the meantime, note that types respect modus
ponens: 

<pre>
Expression    Type      Implication
-----------------------------------
fn            &alpha; -> &beta;    &alpha; &sup; &beta;
arg           &alpha;         &alpha;
------        ------    --------
(fn arg)      &beta;         &beta;
</pre>

The implication in the right-hand column is modus ponens, of course.

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.)  

System F enhances the simply-typed lambda calculus with abstraction
over types.  Normal lambda abstraction abstracts (binds) an expression
(a term); type abstraction abstracts (binds) a type.

In order to state System F, we'll need to adopt the
notational convention (which will last throughout the rest of the
course) that "<code>x:&alpha;</code>" represents an expression `x`
whose type is <code>&alpha;</code>.

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       τ ::= c | α | τ1 -> τ2 | ∀α.τ
        expressions e ::= x | λx:τ.e | e1 e2 | Λα.e | e [τ]

In the definition of the types, "`c`" is a type constant.  Type
constants play the role in System F that base types play in the
simply-typed lambda calculus.  So in a lingusitics context, type
constants might include `e` and `t`.  "α" is a type variable.  The
tick mark just indicates that the variable ranges over types rather
than over values; in various discussion below and later, type variable
can be distinguished by using letters from the greek alphabet
(&alpha;, &beta;, etc.), or by using capital roman letters (X, Y,
etc.).  "`τ1 -> τ2`" is the type of a function from expressions of
type `τ1` to expressions of type `τ2`.  And "`∀α.τ`" is called a
universal type, since it universally quantifies over the type variable
`'a`.  You can expect that in `∀α.τ`, the type `τ` will usually
have at least one free occurrence of `α` somewhere inside of it.

In the definition of the expressions, we have variables "`x`" as usual.
Abstracts "`λx:τ. 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: "`Λα.e`" is called a *type
abstraction*, and "`e [τ]`" is called a *type application*.  The idea
is that <code>&Lambda;</code> is a capital <code>&lambda;</code>: just
like the lower-case <code>&lambda;</code>, <code>&Lambda;</code> binds
variables in its body, except that unlike <code>&lambda;</code>,
<code>&Lambda;</code> binds type variables instead of expression
variables.  So in the expression

<code>&Lambda; α (&lambda; x:α. x)</code>

the <code>&Lambda;</code> binds the type variable `α` that occurs in
the <code>&lambda;</code> abstract.  Of course, as long as type
variables are carefully distinguished from expression variables (by
tick marks, Grecification, or capitalization), there is no need to
distinguish expression abstraction from type abstraction by also
changing the shape of the lambda.

The expression immediately below is a polymorphic version of the
identity function.  It defines one general identity function that can
be adapted for use with expressions of any type. In order to get it
ready to apply this identity function to, say, a variable of type
boolean, just do this:

<code>(&Lambda; α (&lambda; x:α. x)) [t]</code>    

This type application (where `t` is a type constant for Boolean truth
values) specifies the value of the type variable `α`.  Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:

<code>((&Lambda;α (&lambda; x:α . x)) [t]): (b->b)</code>    

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`:

<code>((&Lambda;α (&lambda; x:α. x)) [e]): (e->e)</code>    

Clearly, for any choice of a type `α`, the identity function can be
instantiated as a function from expresions of type `α` to expressions
of type `α`.  In general, then, the type of the uninstantiated
(polymorphic) identity function is

<code>(&Lambda;α (&lambda;x:α . x)): (&forall;α. α-α)</code>

Pred in System F
----------------

We saw that the predecessor function couldn't be expressed in the
simply-typed lambda calculus.  It *can* be expressed in System F,
however.  Here is one way, coded in
[[Benjamin Pierce's type-checker and evaluator for
System F|http://www.cis.upenn.edu/~bcpierce/tapl/index.html]] (the
relevant evaluator is called "fullpoly"):

    N = ∀α. (α->α)->α->α;
    Pair = (N->N->N) -> N;
    let zero = α . λs:α->α . λz:α. z in 
    let fst = λx:N . λy:N . x in
    let snd = λx:N . λy:N . y in
    let pair = λx:N . λy:N . λz:N->N->N . z x y in
    let suc = λn:N . λα . λlambda s:α->α . λz:α . s (n [α] s z) in
    let shift = λp:Pair . pair (suc (p fst)) (p fst) in
    let pre = λn:N . n [Pair] shift (pair zero zero) snd in

    pre (suc (suc (suc zero)));

We've truncated the names of "suc(c)" and "pre(d)", since those are
reserved words in Pierce's system.  Note that in this code, there is
no typographic distinction between ordinary lambda and type-level
lambda, though the difference is encoded in whether the variables are
lower case (for ordinary lambda) or upper case (for type-level
lambda).

The key to the extra expressive power provided by System F is evident
in the typing imposed by the definition of `pre`.  The variable `n` is
typed as a Church number, i.e., as `∀ α . (α->α)->α->α`.  The type
application `n [Pair]` instantiates `n` in a way that allows it to
manipulate ordered pairs: `n [Pair]: (Pair->Pair)->Pair->Pair`.  In
other words, the instantiation turns a Church number into a
pair-manipulating function, which is the heart of the strategy for
this version of predecessor.  

Could we try to build a system for doing Church arithmetic in which
the type for numbers always manipulated ordered pairs?  The problem is
that the ordered pairs we need here are pairs of numbers.  If we tried
to replace the type for Church numbers with a concrete (simple) type,
we would have to replace each `X` with the type for Pairs, `(N -> N ->
N) -> N`.  But then we'd have to replace each of these `N`'s with the
type for Church numbers, `(α -> α) -> α -> α`.  And then we'd have to
replace each of these `α`'s with... ad infinitum.  If we had to choose
a concrete type built entirely from explicit base types, we'd be
unable to proceed.
 
[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
Press, chapter 23.]

Typing &omega;
--------------

In fact, unlike in the simply-typed lambda calculus, 
it is even possible to give a type for &omega; in System F. 

<code>&omega; = lambda x:(∀ α. α->α) . x [∀ α . α->α] x</code>

In order to see how this works, we'll apply &omega; to the identity
function.  

<code>&omega; id ==</code>

    (lambda x:(∀ α . α->α) . x [∀ α . α->α] x) (lambda α . lambda x:α . x)

Since the type of the identity function is `(∀ α . α->α)`, it's the
right type to serve as the argument to &omega;.  The definition of
&omega; instantiates the identity function by binding the type
variable `α` to the universal type `∀ α . α->α`.  Instantiating the
identity function in this way results in an identity function whose
type is (in some sense, only accidentally) the same as the original
fully polymorphic identity function.

So in System F, unlike in the simply-typed lambda calculus, it *is*
possible for a function to apply to itself!

Does this mean that we can implement recursion in System F?  Not at
all.  In fact, despite its differences with the simply-typed lambda
calculus, one important property that System F shares with the
simply-typed lambda calculus is that they are both strongly
normalizing: *every* expression in either system reduces to a normal
form in a finite number of steps.  

Not only does a fixed-point combinator remain out of reach, we can't
even construct an infinite loop.  This means that although we found a
type for &omega;, there is no general type for &Omega; &equiv; &omega;
&omega;.  Furthermore, it turns out that no Turing complete system can
be strongly normalizing, from which it follows that System F is not
Turing complete.


## Polymorphism in natural language

Is the simply-typed lambda calclus enough for analyzing natural
language, or do we need polymorphic types? Or something even more expressive?

The classic case study motivating polymorphism in natural language
comes from coordination.  (The locus classicus is Partee and Rooth
1983.)

    Ann left and Bill left.
    Ann left and slept.
    Ann and Bill left.
    Ann read and reviewed the book.

In English (likewise, many other languages), *and* can coordinate
clauses, verb phrases, determiner phrases, transitive verbs, and many
other phrase types.  In a garden-variety simply-typed grammar, each
kind of conjunct has a different semantic type, and so we would need
an independent rule for each one.  Yet there is a strong intuition
that the contribution of *and* remains constant across all of these
uses.  Can we capture this using polymorphic types?

    Ann, Bill      e
    left, slept    e -> t    
    read, reviewed e -> e -> t

With these basic types, we want to say something like this:

    and:t->t->t = lambda l:t . lambda r:t . l r false
    and = lambda α . lambda β . 
            lambda l:α->β . lambda r:α->β . 
              lambda x:α . and:β (l x) (r x)

The idea is that the basic *and* conjoins expressions of type `t`, and
when *and* conjoins functional types, it builds a function that
distributes its argument across the two conjuncts and conjoins the two
results.  So `Ann left and slept` will evaluate to `(\x.and(left
x)(slept x)) ann`.  Following the terminology of Partee and Rooth, the
strategy of defining the coordination of expressions with complex
types in terms of the coordination of expressions with less complex
types is known as Generalized Coordination.

But the definitions just given are not well-formed expressions in
System F.  There are three problems.  The first is that we have two
definitions of the same word.  The intention is for one of the
definitions to be operative when the type of its arguments is type
`t`, but we have no way of conditioning evaluation on the *type* of an
argument.  The second is that for the polymorphic definition, the term
*and* occurs inside of the definition.  System F does not have
recursion.  

The third problem is more subtle.  The defintion as given takes two
types as parameters: the type of the first argument expected by each
conjunct, and the type of the result of applying each conjunct to an
argument of that type.  We would like to instantiate the recursive use
of *and* in the definition by using the result type.  But fully
instantiating the definition as given requires type application to a
pair of types, not to just a single type.  We want to somehow
guarantee that β will always itself be a complex type.

So conjunction and disjunction provide a compelling motivation for
polymorphism in natural language, but we don't yet have the ability to
build the polymorphism into a formal system.

And in fact, discussions of generalized coordination in the
linguistics literature are almost always left as a meta-level
generalizations over a basic simply-typed grammar.  For instance, in
Hendriks' 1992:74 dissertation, generalized coordination is
implemented as a method for generating a suitable set of translation
rules, which are in turn expressed in a simply-typed grammar.

Not incidentally, we're not aware of any programming language that
makes generalized coordination available, despite is naturalness and
ubiquity in natural language.  That is, coordination in programming
languages is always at the sentential level.  You might be able to
evaluate `(delete file1) and (delete file2)`, but never `delete (file1
and file2)`.

We'll return to thinking about generalized coordination as we get
deeper into types.  There will be an analysis in term of continuations
that will be particularly satisfying.


#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 = <fun>

Since `+` is only defined on integers, it has type

     # (+);;
     - : int -> int -> int = <fun>

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 <bool expression> with true -> <expression1> | false -> <expression2>

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 = <fun>

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 = <fun>

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 = <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?

[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.]


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>
    # (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 = <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) 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;;
    <Infinite loop, need to control-c to interrupt>

    # let test () = blackhole blackhole;;
    val test : unit -> 'a = <fun>

    # test;;
    - : unit -> 'a = <fun>

    # test ();;
    <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
*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.