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 a
-expression 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):
- System F:
+ 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`). "`'a`" is a type variable (the tick mark just indicates that
-the variable ranges over types rather than 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`.
-
-In the definition of the expressions, we have variables "`x`".
+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
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: "`Λ'a. e`" is a type
-abstraction, and "`e [τ]`" is a type application. The idea is that
-<code>Λ</code> is a capital <code>λ</code>. Just like
-the lower-case <code>λ</code>, <code>Λ</code> binds
-variables in its body; unlike <code>λ</code>,
-<code>Λ</code> binds type variables. So in the expression
+additional ways of forming expressions: "`Λ'a. e`" is called a *type
+abstraction*, and "`e [τ]`" is called a *type application*. The idea
+is that <code>Λ</code> is a capital <code>λ</code>: just
+like the lower-case <code>λ</code>, <code>Λ</code> binds
+variables in its body, except that unlike <code>λ</code>,
+<code>Λ</code> binds type variables instead of expression
+variables. So in the expression
<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 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:
+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.
+
+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:
<code>(Λ 'a (λ x:'a . x)) [t]</code>
-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
+This type application (where `t` is a type constant for Boolean truth
+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)): (\forall 'a . 'a -> 'a)
+<code>(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)</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 = All X . (X->X)->X->X;
+ Pair = All X . (N -> N -> X) -> X;
+ let zero = lambda X . lambda s:X->X . lambda z:X. z 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 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
+
+ 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 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.
+
+[See Benjamin C. Pierce. 2002. *Types and Programming Languages*, MIT
+Press, pp. 350--353, for `tail` for lists in System F.]
+
+Typing ω
+--------------
+
+In fact, it is even possible to give a type for ω in System F.
+
+<code>ω = lambda x:(All X. X->X) . x [All X . X->X] x</code>
+
+In order to see how this works, we'll apply ω to the identity
+function.
+
+<code>ω [All X . X -> X] 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 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 (in this case, the identity 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. (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