Haskell – Why No Type-Level Lambda Abstractions?

I actually think that return type polymorphism is one of the best features of type classes. After having used it for a while, it is sometimes hard for me to go back to OOP style modeling where I don't have it.

Consider the encoding of algebra. In Haskell we have a type class Monoid (ignoring mconcat)

class Monoid a where
   mempty :: a
   mappend :: a -> a -> a

How could we encode this as an interface in an OO language? The short answer is we can't. That's because the type of mempty is (Monoid a) => a aka, return type polymorphism. Having the ability to model algebra is incredibly useful IMO.*

You start your post with the complaint about "Referential Transparency." This raises an important point: Haskell is a value oriented language. So expressions like read 3 don't have to be understood as things that compute values, they can also be understood as values. What this means is that the real issue is not return type polymorphism: it is values with polymorphic type ([] and Nothing). If the language should have these, then it really has to have polymorphic return types for consistency.

Should we be able to say [] is of type forall a. [a]? I think so. These features are very useful, and they make the language much simpler.

If Haskell had subtype polymorphism [] could be a subtype for all [a]. The problem is, that I don't know of a way of encoding that without having the type of the empty list be polymorphic. Consider how it would be done in Scala (it is shorter than doing it in the canonical statically typed OOP language, Java)

abstract class List[A]
case class Nil[A] extends List[A]
case class Cons[A](h: A. t: List[A]) extends List[A]

Even here, Nil() is an object of type Nil[A] **

Another advantage of return type polymorphism is that it makes the Curry-Howard embedding much simpler.

Consider the following logical theorems:

 t1 = forall P. forall Q. P -> P or Q
 t2 = forall P. forall Q. P -> Q or P

We can trivially capture these as theorems in Haskell:

data Either a b = Left a | Right b
t1 :: a -> Either a b
t1 = Left
t2 :: a -> Either b a
t2 = Right

To sum up: I like return type polymorphism, and only think it breaks referential transparency if you have a limited notion of values (although this is less compelling in the ad hoc type class case). On the other hand, I do find your points about MR and type defaulting compelling.

*. In the comments ysdx points out this isn't strictly true: we could re-implement type classes by modeling the algebra as another type. Like the java:

abstract class Monoid<M>{
   abstract M empty();
   abstract M append(M m1, M m2);
}

You then have to pass objects of this type around with you. Scala has a notion of implicit parameters which avoids some, but in my experience not all, of the overhead of explicitly managing these things. Putting your utility methods (factory methods, binary methods, etc) on a separate F-bounded type turns out to be an incredibly nice way of managing things in an OO language that has support for generics. That said, I'm not sure I would have groked this pattern if I didn't have experience modeling things with typeclasses, and I'm not sure other people will.

It also has limitations, out of the box there is no way to get an object that implements the typeclass for an arbitrary type. You have to either pass the values explicitly, use something like Scala's implicits, or use some form of dependency injection technology. Life gets ugly. On the other hand, it is nice that you can have multiple implementations for the same type. Something can be a Monoid in multiple ways. Also, carrying around these structures separately has IMO a more mathematically modern, constructive, feel to it. So, although I still generally prefer the Haskell way of doing this, I probably overstated my case.

Typeclasses with return type polymorphism makes this kind of thing easy to handle. That doesn't meen it is the best way to do it.

**. Jörg W Mittag points out this isn't really the canonical way of doing this in Scala. Instead, we would follow the standard library with something more like:

abstract class List[+A] ...  
case class Cons[A](head: A, tail: List[A]) extends List[A] ...
case object Nil extends List[Nothing] ...

This takes advantage of Scala's support for bottom types, as well as covariant type paramaters. So, Nil is of type Nil not Nil[A]. At this point we are pretty far from Haskell, but it is interesting to note how Haskell represents the bottom type

undefined :: forall a. a

That is, it isn't the subtype of all types, it is polymorphically(sp) a member of all types.
Yet more return type polymorphism.

Yes, the evaluation strategy as described leads to strict semantics, and the examples are spectacularly badly chosen to conceal the difference between the two semantics. I think it goes something like this:

id (id (λz. id z))  # strict means we evaluate the right hand side
→ id (λz. id z)     # RHS has been reduced (id (λz. id z)) → (λz. id z) by inner id 
→ λz. id z          # now we have called the outer id to obtain the final value

id (id (λz. id z))  # normal form means we call the outer id and pass RHS as a closure
→ id (λz. id z)     # outer id just returned its argument unevaluated → id (λz. id z) 
→ λz. id z          # now same thing is repeated with inner id

So the derivation steps look syntactically the same, but different things are happening. Under the lazy evaluation scheme, id doesn't actually force evaluation of its argument: it simply returns that argument itself. So not only do id x and x yield the same value, but they are actually equivalent: id x really yields x itself, and then x yields its value later when actually needed. So likewise, id (id (λz. id z)) simply yields the unevaluated right hand side (id (λz. id z)).

What's confusing in the example is that it's based on nesting the same function, which is just id, such that two different reductions both yield id (λz. id z). In one this is just a copy of the inner id expression, and in the other, it's the value of the inner expression, being passed an argument to the outer id.