Are GHC's Type Famlies An Example of System F-omega?

Question

I'm reading up about the Lambda-Cube, and I'm particularly interested in System F-omega, which allows for "type operators" i.e. types depending on types. This sounds a lot like GHC's type families. For example

type family Foo a
type instance Foo Int = Int
type instance Foo Float = ...
...

where the actual type depends on the type parameter a. Am I right in thinking that type families are an example of the type operators ala system F-omega? Or am I out in left field?

Answer 1

System F-omega allows universal quantification, abstraction and application at higher kinds, so not only over types (at kind *), but also at kinds k1 -> k2, where k1 and k2 are themselves kinds generated from * and ->. Hence, the type level itself becomes a simply typed lambda-calculus.

Haskell delivers slightly less than F-omega, in that the type system allows quantification and application at higher kinds, but not abstraction. Quantification at higher kinds is how we have types like

fmap :: forall f, s, t. Functor f => (s -> t) -> f s -> f t

with f :: * -> *. Correspondingly, variables like f can be instantiated with higher-kinded type expressions, such as Either String. The lack of abstraction makes it possible to solve unification problems in type expressions by the standard first-order techniques which underpin the Hindley-Milner type system.

However, type families are not really another means to introduce higher-kinded types, nor a replacement for the missing type-level lambda. Crucially, they must be fully applied. So your example,

type family Foo a
type instance Foo Int = Int
type instance Foo Float = ...
....

should not be considered as introducing some

Foo :: * -> * -- this is not what's happening

because Foo on its own is not a meaningful type expression. We have only the weaker rule that Foo t :: * whenever t :: *.

Type families do, however, act as a distinct type-level computation mechanism beyond F-omega, in that they introduce equations between type expressions. The extension of System F with equations is what gives us the "System Fc" which GHC uses today. Equations s ~ t between type expressions of kind * induce coercions transporting values from s to t. Computation is done by deducing equations from the rules you give when you define type families.

Moreover, you can give type families a higher-kinded return type, as in

type family Hoo a
type instance Hoo Int = Maybe
type instance Hoo Float = IO
...

so that Hoo t :: * -> * whenever t :: *, but still we cannot let Hoo stand alone.

The trick we sometimes use to get around this restriction is newtype wrapping:

newtype Noo i = TheNoo {theNoo :: Foo i}

which does indeed give us

Noo :: * -> *

but means that we have to apply the projection to make computation happen, so Noo Int and Int are provably distinct types, but

theNoo :: Noo Int -> Int

So it's a bit clunky, but we can kind of compensate for the fact that type families do not directly correspond to type operators in the F-omega sense.