Overlapping families (specificity/degrees of freedom)

[Richard Eisenberg]

Hi Gabor,

I can't comment specifically on your code, because I'm afraid I don't understand it all. But, I think I can answer your question:

GHC will select a type instance branch to use in a given type family application if and only if the following 2 conditions hold:
  1. There is a substitution from the variables in the branch statement that makes the left-hand side of the branch coincide with the type family application.
  2. There exists *no* substitution from the variables in the branch statement ***and the variables in the type family application*** that make the left-hand side of the branch coincide with the (substituted) type family application.

Another way to state (2) is that the left-hand side of the branch must not *unify* with the type family application. In your example, this means that GHC must be sure that 'r' and 'one' are distinct types before selecting the fourth branch.

Why do this? It's a matter of type safety. Say GHC selected the fourth branch just because the third one doesn't apply at the moment, because the type variables in the application are distinct. The problem is that those variables might later be instantiated at the same value, and then the third branch would have applied. You can convince this sort of inconsistency to produce unsafeCoerce.

It gets worse. GHC has no internal notion of inequality, so it can't use previous, failed term-level GADT pattern matches to refine its type assumptions. For example:

> data G :: * -> * where
>   GInt  :: G Int
>   GBool :: G Bool
>
> type family Foo (a :: *) :: *
> type instance where
>   Foo Int = Char
>   Foo a   = Double
>
> bar :: G a -> Foo a
> bar GInt  = 'x'
> bar _     = 3.14

The last line will fail to typecheck, because GHC doesn't know that the type variable 'a' can't be Int here, even though it's obvious. The only general way I can see to fix this is to have inequality evidence introduced into GHC, and that's a big deal and I don't know if we have the motivation for such a change yet.

I hope this helps you understand why your code isn't working.

On the other hand, have you looked at the singletons library I wrote last year? The library takes normal code and, using Template Haskell, "singletonizes" it. For example:

> {-# LANGUAGE TemplateHaskell, DataKinds, PolyKinds, TypeFamilies, GADTs, 
>              UndecidableInstances, FlexibleContexts, RankNTypes #-}
> 
> import Data.Singletons
> 
> $(singletons [d|
>   member :: Eq a => a -> [a] -> Bool
>   member _ [] = False
>   member x (h : t) = x == h || member x t
> 
>   intersect :: Eq a => [a] -> [a] -> [a]
>   intersect []      _ = []
>   intersect (h : t) b = if member h b then h : (intersect t b) else intersect t b
>  |])

This works on GHC 7.6.1 and HEAD. Does it do what you want?

Richard

PS: You said Omega has a different policy about type family reduction. How does that deal with the problem I've discussed here?

On Mar 14, 2013, at 12:35 PM, Gabor Greif wrote:

> Hi Richard,
> 
> I have a question regarding type families with overlapping syntax.
> 
> In below example the last two lines are overlapped, so the first gets selected
> when I present type equality witness to GHC:
> 
>> help :: Sing (ls :: Inventory a) -> Sing (r :: Inventory a) -> Sing (l :: a) -> Sing (InterAppend ls r l)
>> help l@(Snoc _ _) (Snoc _ r) one | Just Eq <- isSame r one = Snoc l one    --- OKAY
> 
>> type family InterAppend (l' :: Inventory a) (r' :: Inventory a) (one' :: a) :: Inventory a
>> type instance where
>>  InterAppend Empty r one = Empty
>>  InterAppend (More ls l) Empty one = Empty
>>  InterAppend (More ls l) (More rs one) one = More (More ls l) one   --- 4 degrees of freedom
>>  InterAppend (More ls l) (More rs r) one = More ls l     --- 5 degrees of freedom
> 
> However I cannot manage to trigger the last line when *not* presenting
> the witness, e.g. in this case:
> 
>> help l@(Snoc _ _) (Snoc _ _) _ = l    --- DOES NOT WORK
> 
> IIRC, when implementing something like this in Omega, the type
> function evaluator considered the number of type variables and tried
> to match the more constrained leg (i.e. preferred less degrees of
> freedom).
> 
> Can you give me a hint how this is supposed to work in your implementation?
> 
> Must I present a type non-equality witness to trigger the fourth leg
> of InterAppend above?
> 
> What I am trying to implement is an intersection algorithm for
> singleton-typed list witnesses,
> roughly
> 
>> intersect :: SingEqual (t :: a) => Sing (lhs :: '[a]) -> Sing (rhs :: '[a]) -> Sing (Intersect lhs rhs)
> 
> ... but I am not there yet, obviously!
> 
> I'd be grateful for any hint,
> 
> cheers,
> 
>    Gabor