[Haskell-cafe] Re: Different choice operations in a continuation monad

[Heinrich Apfelmus]

Sebastian Fischer wrote:
> Edward Kmett wrote:
>> Sebastian Fischer wrote:
>>> Heinrich Apfelmus wrote:
>>> newtype CMaybe a = CMaybe (forall r. (a -> Maybe r) -> Maybe r)
>> Yes, with this type `orElse` has the same type as `mplus`, which is
>> very nice.
>>
>> This type is the same as "Codensity Maybe" using category-extras which
>> is a 'bit bigger than Maybe'. (To see why, figure out how Codensity
>> Reader is isomorphic to State!) This is the wiggle room you're using
>> to get the distributive operator.
> 
> I encounter the Codensity type constructor every now and then. I used it
> to Reinvent Haskell Backtracking, learned about implementing a state
> monad with a reader monad wrapped in Codensity when implementing  Lazy
> Nondeterministic Programming and Janis Voigtländer also used it to
> improve the asymptotics of free monads.
> 
> I wonder whether for every monad `m` and `a :: Codensity m a`
> 
>     getCodensity a f  =  getCodensity a return >>= f
> 
> Is this true? Why (not)?

It's not true.

  a = Codensity $ \x -> Just 42
  f = return . (+1)

    getCodensity a f            = Just 42
  ≠ getCodensity a return >>= f = Just 42 >>= f = Just 43

It probably is true if  a  is only built from  >>= , return  and actions
from the original monad, though.


Regards,
Heinrich Apfelmus

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