I don't understand the final part of the question but here are some comments for the first part. I don't like the phrase:
the more powerfull a class is, the more fleixblility you have for combining them to complex programs
powerfull, more flexibility, complex programs -- are not so precise terms.
A => B
means that B can do everything that A can do and more (methods that are
specific to B). So if type is in B we can use all A's methods with it. Does
it make B more powerful or more flexible? Is Applicative less powerful than
a Monad? It depends on the program. If we don't ever need the B's specific
operations they will confuse us all the time. We are going to end up with
more complex program but not a better one. there are cases when Applicative
code is much better than a monadic one.
Anton
2013/5/28 Johannes Gerer
Dear Haskellers,
While trying to understand the interconnection and hierarchy behind the important typeclasses, I stumbled upon the following question that I am not able to answer:
There seems to be a hierachy of the type classes, in the sense, that the more powerfull a class is, the more fleixblility you have for combining them to complex programs. (Functor -> Applicative -> Arrow[Choice,Plus,Apply,..] -> Monad). It was nice to read in the Typeclassopedia, that ArrowApply and Monad are equivalent, which is shown by deriving two instances from each other:
instance Monad m => ArrowApply (Kleisli m) instance ArrowApply a => Monad (a anyType)
The logic seems to be, that if I can derive from every instance of class A an instance of class B, then A is more powerfull than B and (in general) it is easier to be of class B than of class A (e.g. more types can be made Applicatives, than Monads)
So far, I think I can follow. But what really hit me was the Cokleisli type. Using it and the logic from above, I can show that ANY type class is more (or equally) powerfull than the Monad:
instance AnyClass m => Monad (Cokleilsi m anyType)
I know this makes no sense, but where is the fallacy? Why even bother with the above derivation, if any type class can be made into a monad?
I can see, that the Monad instance from above does not really transform the type "a", but instead simply fix its first argument. But then on the other hand, the ArrowApply Instance does transform the "m" type (in a way similar to Cokleisli). If attention needs to be paid to the details, then what are they and why did they not matter above?
Thanks,
Johannes
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