Tillmann Rendel wrote:
wren ng thornton wrote:
[1] In System F the capital-lambda binder is used for the term-level
abstraction of passing type representations. So for example we have,
id :: forall a. a -> a
id = /\a. \(x::a). x
Thus, the forall keyword is serving as the type-level abstraction.
Perhaps this is suboptimal syntax, but it is the standard. We could,
of course, have both a term-level /\ and a type-level /\ where the
latter is the type of the former (since the namespaces are separate)
though that's also dubious. Capital-pi is the canonical type-level
abstraction, though that evokes the idea of dependent types which are
much more complex.
What do you mean by "type-level abstraction" here?
I mean an abstraction, as in a lambda-abstraction (aka a
lambda-expression), at the type level.
In a language with type functions and polymorphism, we need three
different lambda binders:
(1) abstraction over terms in terms (to construct functions)
(2) abstraction over types in terms (to construct polymorphic values)
(3) abstraction over types in types (to construct type functions)
I think only (2) should be written as upper-case lambdas, while (1) and
(3) should both be written as lower-case lambdas. Since (1) and (3)
belong to different syntactic categories, they can not be confused, and
we can reuse the lower-case lambda at the type-level.
I'm sure that's fine. I was merely pointing out precedent.
The syntax of #3 could also be conflated with the syntax of #2, for the
same reason: they are in different syntactic categories. I pointed this
out because the capital-lambda was the syntax others in the thread were
using. Also, it makes sense to me to have #2 and #3 ("abstraction over
types in _") paired together, rather than #1 and #3 ("abstraction over X
in X"). Pairing #2/#3 also gives term/type symmetry as we have for other
built-ins like [], (,), and -> (though the symmetry is imperfect for ->).
Furthermore, we need three function types / kinds to describe the three
lambdas one level higher:
(4) the type of functions
(5) the type of polymorphic values
(6) the kind of type functions
In ghc, we already have "forall" for (5), and arrows for (4) and (6).
I would say that (3) is the "type-level abstraction", not (5).
I'm not sure I follow what you mean. Given the relationship between /\
and forall as demonstrated in the definition of id above, I don't see
such a difference between #3 and #5. That is, given the need for #2 the
need for #5 follows; from there #3 follows by extending the wording of
#5. (Though #6 is desirable from a theoretical perspective I'm not sure
whether the language needs to be able to express it. There's much else
at the kind-layer we cannot express.)
In other words, just because forall is the type of /\ doesn't mean that
that's all it is. All of these are effectively identical:
-- Just a type-level lambda
five :: (forall a. a) Int
five = 5
-- Making the term-level match the type exactly
five :: (forall a. a) Int
five = (/\a. 5::a) @Int
-- Hiding a CAF behind a constant type
-- (somewhat like how numeric constants really work)
five :: Int
five = (/\a. 5::a) @Int
-- Boring
five :: Int
five = 5
--
Live well,
~wren
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