Philippa Cowderoy wrote:
I've done a bit more thinking about partial type annotations (as proposed
on the Haskell' list), and I have a somewhat more concrete proposals for
some of the extensions to them that perhaps also makes more sense of the
original basic idea as well. I'm sending it to the Cafe this time as it's
a bit early to consider this for standardisation.
I'd like to propose a new quantifier for type variables, which for now
I'll call unknown[1] - correspondingly I'll talk about "unknown-quantified
variables" and probably "unknown variables" where it's not ambiguous.
Unknown quantifiers will never be introduced by the typechecker without a
corresponding annotation - only propagated inwards. Whereas universal type
variables must not accumulate additional constraints during typechecking
(and in a traditional Hindley-Milner implementation only become
universally quantified during a generalisation step), unknown type
variables can - indeed this is their raison d'etre. Furthermore, they are
never propagated 'outwards' - either the variable is constrained
sufficiently that it can be replaced with a monotype or, having otherwise
finished typechecking the corresponding term, the unknown quantifier is
replaced with a forall.
The intention is that the unknown variable might eventually get a
concrete type, but you'd rather let the typechecker work it out?
I think in the language of the GHC typechecker you would say your
quantifier introduces a "wobbly" type variable, rather than the "rigid"
type variables of a forall.
For example:
add' :: unknown x. x -> x -> x
add' = (+)
add'' :: unknown x. x -> x -> x
add'' x y = (x::Int) + (y::Int)
will typecheck, resulting in these types when the identifiers are used:
add' :: forall x. Num x => x -> x -> x
add'' :: Int -> Int -> Int
It's probably also sensible to allow "wildcard" variables written _,
generated fresh and implicitly unknown-quantified much as universal
variables are now.
Type synonyms seem to present an interesting question though - it seems to
me most sensible to hang on to the quantification at top-level and
generalise it only as we finish type-checking a module, rather than
copying out the quantifier anew each time. Any comments?
I discussed something similar a while ago, in connection with the
discussions about alternate notation for class constraints, except I
suggested copying the quantifier around with the synonym. This lets you
write types like
type P a = some b . (b -> a, a)
or otherwise make a type synonym that asserts some part of your type is
an instance of a certain scheme without fixing concrete types or
requiring full parametricity.
More abstractly, type synonyms exist to name and reuse parts of type
signatures. If you introduce partial type signatures, it seems fitting
to extend type synonyms to naming and reusing part of partial type
signatures as well.
The amount of thinking I've done about interactions with rank-n variables
is limited - I guess we'd need to prohibit unifying with type variables
that're of smaller scope than the unknown variable? I'm don't think I can
see any other worrying issues there.
Unknown variables in method declarations seem... meaningless to me,
they're not proper existentials and I don't think there's a sane meaning
for them that isn't a kludge for associated types instead. I don't think
they belong in instances for similar reasons.
Incidentally, I think there's also a cute use case in .hs-boot files,
where an unknown-quantified variable could be used to tie some knots in a
manner similar to the way recursive bindings are checked now. If so, I
have an interesting use for this.
[1] Other names that have occurred to me are "solve" (as in "solve for
x"), and "meta" (by analogy to metavariables in the typechecker - because
by the time we've finished checking the annotated term we'll have removed
the quantifier), but unknown seems by far the strongest to me. No doubt
someone'll suggest a far better name shortly after I post this
Daan Leijen calls it "some".
It's implemented in Morrow, and described in his paper on existential
types. He translates type annotations to type-restricting functions,
where
e :: forall a . T
becomes
(ann :: (forall a . T) -> (forall a . T)) e
and
e :: some a . T
becomes
(ann :: forall a . T -> T) e
Writing type signatures for things unpacked from existentials is another
nice use for this quantifier. The some a. can unify with the skolem
constant from the unpacking. One of his examples:
type Key a = { val :: a, key :: a -> Int }
absInt :: exists a. Key a
absInt = { val = 1, key i = i }
one1 = (\x -> x.key x.val) (head [absInt])
one2 = let x :: some a. Key a
x = absInt
in x.key x.val
Brandon
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