[Haskell-cafe] Extensible Type Unification

[Leon Smith]

I've been toying with some type-level programming ideas I just can't quite
make work,   and it seems what I really want is a certain kind of type
unification.

Basically,  I'd like to introduce two new kind operators:

kind Set as   -- a finite set of ground type terms of kind as

kind Range as = Range (Set as) (Set as)      -- a greatest lower bound and
a least upper bound


A type expression of kind (Set as)  would either be a type variable of kind
(Set as),  or set of *ground* type terms of kind (as).      A type
expression of kind (Range as) would be a type variable
of kind (Range as),  or two type expressions of kind (Set as).   To unify
ground terms of these types,  one would compute as follows:


unifySets xs ys
   |  xs == ys  = Just xs
   |  otherwise = Nothing

unifyRanges (Range glb0 lub0) (Range glb1 lub1)
    | glb' `subset` lub' = Just (Range glb' lub')
    | otherwise          = Nothing
  where
    glb' = glb0 `union` glb1
    lub' = lub0 `isect` lub1


I say sets of ground types,  because I have no idea what unification
between sets of non-ground types would mean,  and I really don't need it.

Among applications that came to mind,   one could use this to create a
restricted IO abstraction that could tell you which kinds of actions might
be taken  (read from mouse,  talk to network,  etc),   and be able to run
only those scripts that are restricted to certain resources.     Or,  one
could define a singular GADT that represents messages/events decorated by
various categories,   and then define functions that
only operate on selected categories of messages.    E.G. for something
vaguely IRC-ish,  one could write something like:

data EventCategory
    = Channel
    | Presence
    | Information
    | Conversation

data Event :: Range EventCategory -> *  where
    ChanCreate     ::  ...  -> Event (Range (Set '[Channel])      a)
    ChanJoin       ::  ...  -> Event (Range (Set '[Channel])      a)
    ChanLeave      ::  ...  -> Event (Range (Set '[Channel])      a)
    PresAvailable  ::  ...  -> Event (Range (Set '[Presence])     a)
    PresAway       ::  ...  -> Event (Range (Set '[Presence])     a)
    WhoisQuery     ::  ...  -> Event (Range (Set '[Information])  a)
    WhoisResponse  ::  ...  -> Event (Range (Set '[Information])  a)
    Message        ::  ...  -> Event (Range (Set '[Conversation]) a)

And then be able to write functions such as


dispatch :: Event (Range a (Set '[Channel, Conversation]))  -> IO ()
dispatch e =
    case e of
        ChanCreate{..} -> ...
        ChanJoin{..} -> ...
        ChanLeave{..} -> ...
        Message{..} -> ...


In this case,  the case analysis tool would be able to warn me if I'm
missing any possible events in this dispatcher,  or  if I have extraneous
events that I can't be passed (according to my type.)

Anyway,  I've been trying to see if I can't come up with something similar
using existing type-level functionality in 7.6,   with little success.
(Though I'm not very experienced with type-level programming.)     If not,
 might it be possible to add some kind of extensible unification mechanism,
 in furtherance of type-level programming?

Best,
Leon
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