On Fri, Jul 9, 2010 at 8:46 PM, Kevin Quick <quick@sparq.org> wrote:
On Fri, 09 Jul 2010 16:26:13 -0700, Ivan Lazar Miljenovic <ivan.miljenovic@gmail.com> wrote:

"Kevin Quick" <quick@sparq.org> writes:

I would think that only mutually recursive default methods would
require respecification and that there could be any number of default
methods that were reasonable as is.  Since it's probably quite
difficult for the Haskell compiler to analytically detect
non-terminating v.s. terminating mutual recursion it may be useful to
define an explicit comment flag for this case.

For example:

  class Show a where
     shows = showsPrec 5
     showsPrec _ = shows
     {-# REDEFINE_ONE: shows showsPrec #-}

This would fairly simply allow a warning to be generated for an
instance which did not redefine one of the identified methods; it
would capture that requirement in the same place the recursive
definition was defined, it would avoid false warnings, and it would be
backward compatible (and it might be Haddock-able as well).

This should be generalised IMO, since there might be cases where you
have to redefine either (foo && bar) || baz; of course, that makes the
syntax specification, etc. of the pragma more difficult...

I'm having trouble envisioning a restriction case such as you describe.  Can you provide an example?

Examples: 

class Bifunctor f where
    bimap :: (a -> b) -> (c -> d) -> f a c -> f b d
    first :: (a -> b) -> f a c -> f b c
    second :: (a -> b) -> f c a -> f c b
    first f = bimap f id
    second = bimap id
    bimap f g = second g . first f
    {-# MUTUAL = first second | bimap #-}

The existing definition of Arrow is somewhat unsatisfying because its product bifunctor definition (given by first, second and (***)) is asymmetric. They choose to require you to define first, but could very well use the same trick. (I am not advocating changing the well documented historical definition of Arrow, just providing another example in the same vein.)

class Category a => Arrow a where
        arr :: (b -> c) -> a b c
        first :: a b c -> a (b,d) (c,d)
        second :: a b c -> a (d,b) (d,c)
        (***) :: a b c -> a b' c' -> a (b,b') (c,c')
        (&&&) :: a b c -> a b c' -> a b (c,c')

        first = (*** id)

        second = (id ***)

        f *** g = first f >>> second g


        f &&& g = arr (\b -> (b,b)) >>> f *** g
        {-# MUTUAL first second | (***) #-}
An example that almost works would be Monad/Comonad where you can define in terms of return/fmap/join or return/bind. However, the definition of fmap is in another class, but if it wasn't:

class Comonad w where
    liftW :: (a -> b) -> w a -> w b
    extract :: w a -> a
    extend :: (w a -> b) -> w a -> w b
    duplicate :: w a -> w (w a)
    extend = fmap f . duplicate
    duplicate = extend id

    {-# MUTUAL liftW duplicate | extend #-}
 
The comment can't dictate that the resulting redefined method isn't still mutually recursive, but the warning for the lack of any override should provide enough of a trigger for the developer to read the docs/code and write an appropriate method.  If foo, bar, and baz are all interrelated it seems to me that an appropriate override of any of them could provide the necessary exit from recursion.

It turns out to be fairly tricky to pull off the definition in such a way that you can define any one combinator in turn of the others in a big long cycle. Foldable does this for instance in such a way that foldMap and foldr are defined cyclically. 

That's probably an interesting assertion that one of the category theorists around here could prove or disprove.  ;-)
 
I hope the above demonstrate that there are at least some fairly reasonable (and, given your request, appropriately category theoretic!) examples where one would want the ability to specify that there is more than one member of a minimal mutual definition. =)

-Edward Kmett