[Haskell-cafe] Why can't we make an instance declaration on a type synonym?

[Brandon Simmons]

On Mon, Jan 4, 2010 at 12:11 PM, Reid Barton <rwbarton at math.harvard.edu> wrote:
> On Mon, Jan 04, 2010 at 11:50:57AM -0500, Brandon Simmons wrote:
>> On Sun, Jan 3, 2010 at 2:22 AM, Ivan Lazar Miljenovic
>> <ivan.miljenovic at gmail.com> wrote:
>> > jberryman <brandon.m.simmons at gmail.com> writes:
>> >
>> >> This may be a dumb question, but why can we not declare a Monad
>> >> instance of a type synonym? This question came to me while working
>> >> with the State monad recently and feeling that the requirement that we
>> >> wrap our functions in the State constructor is a bit... kludgy.
>> >>
>> >
>> > Because type defines an _alias_.  If you define "type Foo = Maybe Int",
>> > then everywhere you have a "Foo" the compiler should be able to replace
>> > it with "Maybe Int".
>> >
>> > As such, if you have a custom instance on your type synonym (say a
>> > custom Show instance for Foo), then which instance will the compiler
>> > use?
>>
>>
>> Thanks. I guess what I'm really asking is if there is any way to
>> redefine the monad instance for (->) such that we can have a State
>> monad without the data constructor wrapper.
>
> It would be a nightmare for type inference.  Consider:
>
> return "foo" :: String -> (String, String)    -- \x -> ("foo", x)
>
> which would be valid in a language where we can do what you suggest.
> `return` has type `a -> m a`, so `a` must be `String`, and now we need
> to unify `String -> m String` with `String -> String -> (String, String)`.
> In Haskell, these just don't unify because there are no type-level lambdas.
> Even if there were, how is the typechecker supposed to know that we want
> the solution `m a = String -> (a, String)` and not `m a = a -> (a, a)`
> or many other possibilites?
>
> The purpose of the newtype State is to have something to unify `m` with:
>
> return "foo" :: State String String
>
> We can unify `String -> m String` with `String -> State String String`
> by setting `m` to `State String`.
>
> Regards,
> Reid Barton
>

Thanks, that makes a lot of sense.

Brandon