+[[!toc levels=2]]
+
# System F and recursive types
In the simply-typed lambda calculus, we write types like <code>σ
continuations.)
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 "<code>x:α</code>" represents an
-expression `x` whose type is <code>α</code>.
+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:α</code>" represents an expression `x`
+whose type is <code>α</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):
types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
-In the definition of the types, "`c`" is a type constant (e.g., `e` or
-`t`, or in arithmetic contexts, `N` or `Int`). "`'a`" is a type
-variable (the tick mark just indicates that the variable ranges over
-types rather than over values). "`τ1 -> τ2`" is the type of a
-function from expressions of type `τ1` to expressions of type `τ2`.
-And "`∀'a. τ`" is called a universal type, since it universally
-quantifies over the type variable `'a`. (You can expect that in
-`∀'a. τ`, the type `τ` will usually have at least one free occurrence
-of `'a` somewhere inside of it.)
+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`. "`'a`" 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
+(α, β, 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 "`∀'a. τ`" is called a
+universal type, since it universally quantifies over the type variable
+`'a`. You can expect that in `∀'a. τ`, the type `τ` will usually
+have at least one free occurrence of `'a` 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
<code>Λ 'a (λ x:'a . x)</code>
the <code>Λ</code> binds the type variable `'a` that occurs in
-the <code>λ</code> 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 to, say, a variable of type boolean, just do
-this:
+the <code>λ</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>(Λ 'a (λ x:'a . x)) [t]</code>
This 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
+values) specifies the value of the type variable `'a`. Not
surprisingly, the type of this type application is a function from
Booleans to Booleans:
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
+of type `'a`. In general, then, the type of the uninstantiated
(polymorphic) identity function is
<code>(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)</code>
----------------
We saw that the predecessor function couldn't be expressed in the
-simply-typed lambda calculus. It can be expressed in System F,
+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
-part you want is called "fullpoly"):
+relevant evaluator is called "fullpoly"):
N = All X . (X->X)->X->X;
- Pair = All X . (N -> N -> X) -> X;
+ Pair = (N -> N -> N) -> N;
let zero = lambda X . lambda s:X->X . lambda z:X. z in
+ let fst = lambda x:N . lambda y:N . x in
let snd = lambda x:N . lambda y:N . y in
- let pair = lambda x:N . lambda y:N . lambda X . lambda z:N->N->X . z x y in
+ let pair = lambda x:N . lambda y:N . lambda z:N->N->N . z x y in
let suc = lambda n:N . lambda X . lambda s:X->X . lambda z:X . s (n [X] s z) in
- let shift = lambda p:Pair . p [Pair] (lambda a:N . lambda b:N . pair (suc a) a) in
- let pre = lambda n:N . n [Pair] shift (pair zero zero) [N] snd in
+ let shift = lambda p:Pair . pair (suc (p fst)) (p fst) in
+ let pre = lambda n:N . n [Pair] shift (pair zero zero) snd in
pre (suc (suc (suc zero)));
lower case (for ordinary lambda) or upper case (for type-level
lambda).
-The key to the extra flexibility provided by System F is that we can
-instantiate the `pair` function to return a number, as in the
-definition of `pre`, or we can instantiate it to return an ordered
-pair, as in the definition of the `shift` function. Because we don't
-have to choose a single type for all uses of the pair-building
-function, we aren't forced into a infinite regress of types.
-
+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 `All X . (X->X)->X->X`. 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, `(X -> X) -> X -> X`. And then we'd have to
+replace each of these `X`'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, pp. 350--353, for `tail` for lists in System F.]
+Press, chapter 23.]
Typing ω
--------------
-In fact, it is even possible to give a type for &omeage; in System F.
+In fact, unlike in the simply-typed lambda calculus,
+it is even possible to give a type for ω in System F.
- omega = lambda x:(All X. X->X) . x [All X . X->X] x in
- omega;
+<code>ω = lambda x:(All X. X->X) . x [All X . X->X] x</code>
-Each time the internal application is performed, the type of the head
-is chosen anew. And each time, we choose the same type as before, the
-type of a function that takes an argument of any type and returns a
-result of the same type...
+In order to see how this works, we'll apply ω to the identity
+function.
+<code>ω id ==</code>
+
+ (lambda x:(All X . X->X) . x [All X . X->X] x) (lambda X . lambda x:X . x)
+
+Since the type of the identity function is `(All X . X->X)`, 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 `X` to the universal type `All X . X->X`. 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. Furthermore, it turns out that no Turing complete system can
+be strongly normalizing, from which it follows that System F is not
+Turing complete.
+
+
+#Types in OCaml
-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.