`x:α`

" represents an expression `x`
whose type is `α`

.
Then System F can be specified as follows:
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. In
various discussions, type variables are distinguished by using letters
from the greek alphabet (α, β, etc.), as we do here, or by
using capital roman letters (X, Y, etc.), or by adding a tick mark
(`'a`, `'b`, etc.), as in OCaml. "`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 `α`. 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 `Λ`

is a capital `λ`

: just
like the lower-case `λ`

, `Λ`

binds
variables in its body, except that unlike `λ`

,
`Λ`

binds type variables instead of expression
variables. So in the expression
`Λ α (λ x:α. x)`

the `Λ`

binds the type variable `α` that occurs in
the `λ`

abstract.
This expression 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:
`(Λ α (λ x:α. x)) [t]`

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 the expression that results from this type
application is a function from Booleans to Booleans:
`((Λα (λ x:α . 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`:
`((Λα (λ x:α. x)) [e]): (e->e)`

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
`(Λα (λx:α . x)): (∀α. α->α)`

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:
let N = ∀α.(α->α)->α->α in
let Pair = (N->N->N)->N in
let zero = Λα. 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 succ = λn:N. Λα. λs:α->α. λz:α. s (n [α] s z) in
let shift = λp:Pair. pair (succ (p fst)) (p fst) in
let pred = λn:N. n [Pair] shift (pair zero zero) snd in
pre (suc (suc (suc zero)));
[If you want to run this code 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", and you'll need to
truncate the names of "suc(c)" and "pre(d)", since those are
reserved words in Pierce's system.]
Exercise: convince yourself that `zero` has type `N`.
The key to the extra expressive power provided by System F is evident
in the typing imposed by the definition of `pred`. The variable `n`
is typed as a Church number, i.e., as `N ≡ ∀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 certain
pair-manipulating function, which is the heart of the strategy for
this version of computing the predecessor function.
Could we try to accommodate the needs of the predecessor function by
building 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 `N` 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, which we're imagining is `(Pair -> Pair) ->
Pair -> Pair`. And then we'd have to replace each of these `Pairs`'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 ω
--------------
In fact, unlike in the simply-typed lambda calculus,
it is even possible to give a type for ω in System F.
`ω = λx:(∀α.α->α). x [∀α.α->α] x`

In order to see how this works, we'll apply ω to the identity
function.
`ω id ≡ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)`

Since the type of the identity function is `∀α.α->α`, it's the
right type to serve as the argument to ω. The definition of
ω 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 ω, there is no general type for Ω ≡ ω
ω. In fact, 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 = λl:t. λr:t. l r false
and = Λα.Λβ.λl:α->β.λr:α->β.λx:α. and [β] (l x) (r x)
The idea is that the basic *and* (the one defined in the first line)
conjoins expressions of type `t`. But when *and* conjoins functional
types (the definition in the second line), it builds a function that
distributes its argument across the two conjuncts and conjoins the two
results. The intention is that `Ann left and slept` will evaluate to
`(\x.and(left x)(slept x)) ann`. Following the terminology of Partee
and Rooth, this 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. We know how to handle some
cases of using a function name inside of its own definition in the
untyped lambda calculus, but System F does not have
recursion. [Exercise: convince yourself that the fixed-point
combinator `Y` can't be typed in System F.]
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, so that
"`and [β]`

" evaluates to the kind of *and* that
coordinates expressions of type β. But fully instantiating the
definition as given requires type application to a *pair* of types,
not to just to a single type. We want to somehow guarantee that β
will always itself be a complex type. This goes beyond the expressive
power of System F.
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.
There is some work using System F to express generalizations about
natural language: Ponvert, Elias. 2005. Polymorphism in English Logical
Grammar. In *Lambda Calculus Type Theory and Natural Language*: 47--60.
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 =