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):
+Then System F can be specified as follows:
System F:
---------
- types τ ::= c | 'a | τ1 -> τ2 | ∀'a. τ
- expressions e ::= x | λx:τ. e | e1 e2 | Λ'a. e | e [τ]
+ 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`. "`'a`" is a type variable. The
+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
+than over values; in various discussion below and later, type variables
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
+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 `∀'a. τ`, the type `τ` will usually
-have at least one free occurrence of `'a` somewhere inside of it.
+`'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
+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 called a *type
+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>Λ</code> is a capital <code>λ</code>: just
like the lower-case <code>λ</code>, <code>Λ</code> binds
<code>Λ</code> binds type variables instead of expression
variables. So in the expression
-<code>Λ α (λ x:α . x)</code>
+<code>Λ α (λ x:α. x)</code>
-the <code>Λ</code> binds the type variable `'a` that occurs in
+the <code>Λ</code> binds the type variable `α` that occurs in
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
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>
+<code>(Λ α (λ 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 `'a`. Not
+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>((Λ 'a (λ x:'a . x)) [t]): (b -> b)</code>
+<code>((Λα (λ 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>((Λ 'a (λ x:'a . x)) [e]): (e -> e)</code>
+<code>((Λα (λ x:α. x)) [e]): (e->e)</code>
-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 uninstantiated
+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>(Λ 'a (λ x:'a . x)): (∀ 'a . 'a -> 'a)</code>
+<code>(Λα (λx:α . x)): (∀α. α-α)</code>
Pred in System F
----------------
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 = (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 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 . pair (suc (p fst)) (p fst) in
- let pre = lambda n:N . n [Pair] shift (pair zero zero) snd in
+ N = ∀α.(α->α)->α->α;
+ Pair = (N->N->N)->N;
+
+ 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 suc = λn:N. Λα. λs: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)));
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
+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
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
+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.
In fact, unlike in the simply-typed lambda calculus,
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>
+<code>ω = λx:(∀α.α->α). x [∀α.α->α] x</code>
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)
+ (λx:(∀α.α->α). x [∀α.α->α] x) (Λα.λx:α.x)
-Since the type of the identity function is `(All X . X->X)`, it's the
+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 `X` to the universal type `All X . X->X`. Instantiating the
+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.
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 'a . lambda 'b .
- lambda l:'a->'b . lambda r:'a->'b .
- lambda x:'a . and:'b (l x) (r x)
+ 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* conjoins expressions of type `t`, and
when *and* conjoins functional types, it builds a function that
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 'b will always itself be a complex type.
+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