Extensible Haskell Type Classes

Question

I am reading a paper on dependently-typed programming and came across the following quote:

"[...] in contrast to Haskell's type classes, the data type [...] is closed", in the sense that one cannot add new types to the universe without extending the data type.

My newbie question is: in what sense are Haskell type classes open? How are they extensible? Also, what are the type-theoretical consequences of having this property (open vs closed)?

Thank you!

Answer 1

Given a type class like:

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

... it is (basically) implemented under the hood as a dictionary type:

data Monoid m = Monoid
    { mempty  :: m
    , mappend :: m -> m -> m
    }

Instances like:

instance Monoid [a] where
    mempty  = []
    mappend = (++)

... get translated to dictionaries:

listIsAMonoid :: Monoid [a]
listIsAMonoid = Monoid
    { mempty  = []
    , mappend = (++)
    }

... and the compiler consults above dictionary whenever you use lists in their capacity as Monoids.

This brings us to your questions:

in what sense are Haskell type classes open? How are they extensible?

They are open in the same sense that polymorphic values are open. We have some polymorphic data type:

data Monoid m = ...

... and we can instantiate the polymorphic m type variable to any type where we can provide suitable values for the mempty and mappend fields.