On 10 Dec 2007, at 11:33 AM, Dan Weston wrote:
Questioning apfelmus definitely gives me pause, but...
> id :: a -> a -- "arity" 1
> id = ($) :: (a -> b) -> (a -> b) -- "arity" 2
I agree with the arities given above (but without quotes) and see
no ill-definedness to arity.
But these are two different classes of functions. There are
arguments of the first function that cannot be applied to the
second (e.g. 5). The fact that the two have different type
signatures shows that Haskell can distinguish them (e.g. in the
instantiation of a type class)
Not really. The types of id and ($) can't be instances of a type
class, since an instance of a type class has to be a monomorphic
type. So the decision as to which instance to use has to be made
based on the particular monomorphic type id or ($) is used at. But
that monomorphic type may still contain free type variables; those
type variables themselves represent some single monomorphic type,
which may or may not be a function type. So we still don't know what
the arity of an arbitrary expression is. (We don't know what its
type is, even the way we know the type of id or ($), if it or any of
its free variables is lambda-bound).
The difficulties of Haskell's type system in the presence/
intersection of ad hoc/parametric polymorphism is an orthogonal
issue. I think that every function application must have a unique
monomorphic type at the call site of the "arity" function (assisted
or not by type annotation), and this type is known to converge in a
Template Haskell construction.
> We have to specialize the type of id before
> supplying it to wrap . For example,
>
> wrap (id :: Int -> Int)
>
> works just fine.
The necessity of type annotation/restriction is an orthogonal issue
to the above.
Am I missing something more fundamental?
apfelmus wrote:
Luke Palmer wrote:
Hmm, this still seems ill-defined to me.
compose :: (Int -> Int -> Int) -> (Int -> Int) -> Int -> Int -> Int
Is a valid expression given that definition (with a,b = Int and c
= Int -> Int),
but now the arity is 4.
That's correct, the arity of a function is not well-defined due to
polymorphism. The simplest example is probably
id :: a -> a -- "arity" 1
id = ($) :: (a -> b) -> (a -> b) -- "arity" 2
Therefore, the polymorphic expression
wrap id
is problematic. It roughly has the type
wrap id ~~ [String] -> a
But it's clearly ambiguous: do we have
wrap id (x:_) = read x
or
wrap id (f:x:_) = wrap ($) (f:x:_) = read f (read x)
or what? (assuming a read instance for function types)
GHCi gives it a type
> :type wrap id
wrap id :: (FunWrap (a -> a) y) => [String] -> y
but trying to use it like in
> let x = wrap id ["1"] :: Int
yields lots of type errors. We have to specialize the type of id
before supplying it to wrap . For example,
wrap (id :: Int -> Int)
works just fine.
I don't like this behavior of wrap since it violates the nice
property of polymorphic expressions that it's unimportant when a
type variable is instantiated, like in
map ((+1) :: Int -> Int) [1..5]
= map (+1) ([1..5] :: [Int])
= (map (+1) [1..5]) :: [Int]
Regards,
apfelmus
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