The most recent proposal for Rank-N types is Don's linked FPH paper. In my opinion it's also the nicest of the bunch. The main goal of all these systems is to require as few type annotations as possible. The problem is that when going from Hindley/Milner to System F we lose principal types and type inference becomes undecidable – hence the need for type annotations.

The basic idea of the "boxy types" work is to propagate type annotations as far as possible. The type checker switches between type checking and type inference mode and hopefully no more annotations are required. The downside here is that whether or not a type annotation is required is hard to explain because it depends on implementation details.

Remy's MLF system is so far the nicest proposal; it requires the least amount of type annotations and is stable under many code transformations. The problem is that it extends the type system. The following standard example illustrates this:

choose :: forall a. a -> a -> a id :: forall b. b -> b choose id :: forall c. (c -> c) -> (c -> c) choose id :: (forall c. c -> c) -> (forall c. c -> c)

Both the above types are admissable in System F. The first one is the standard Hindley/Milner type and uses predicative instantiation, the second one uses impredicative instantiation. Neither type is more general than the other, so type inference would have to guess which type the user wants, and that is usually a bad idea.

MLF instead extends System F with bounded quantification. The principal (= most general) type for the above example would be:

choose id :: forall (a < forall b. b -> b). a -> a

You can read this as " choose id has type a to a where a must be an instance of forall b. b -> b ".

Interestingly, this alone is no more powerful than standard Hindley/Milner. MLF therefore also allows rigid quantification. The following two types are equivalent:

(forall b. b -> b) -> (forall b. b -> b) forall (a = forall b. b -> b). a -> a

Rigid quantification is introduced by type annotations and the technical details are indeed quite complicated. The upside is that MLF only needs very few type annotations and there is a simple rule for when they are needed. The downsides are:

Types can become harder to read, because the right hand side of '<' can contain further nested quantifications.

Until recently no explicitly typed variant of MLF existed. This is important for typed compiler transformations (like GHC does). Part 3 of Boris Yakobowski's PhD thesis has a first attempt at such a variant. (Parts 1 & 2 are also interesting; they describe a more intuitive representation of MLF via "Graphical Types".)

Coming back to FPH, its basic idea is to use MLF techniques internally, but to require type annotations on let bindings. If you only want the Hindley/Milner type, then no annotations are necessary. If you want a higher-rank type, you need to specify the requested type, but only at the let (or top-level) binding.