kirstin-rhys / free-applicative

Just a quick implementation of Application for the classic Free algebra

free-applicative

In two senses, there really isn't anything new here. The implementation of the Free Monad has been known for a long time. More recently, a half dozen implementations of the Free Applicative Functor have been published.

The construction of the data algebra for the Free Monad was chosen so it closely aligns to join, and the construction of the data algebras for the Free Applicative Functors were chosen so that they closely align to contextual application (<*> aka ⊛ aka ap).

But are these constructions for Free Applicative Functors necessary? Do they make an implementation of the Free Monad more efficient? Are they easier to understand?

Obviously, if we have a trivial implementation of the free monad, we could just define (<>) as ap === liftM2 id, but we should be able to define <> without reference to bind.

For some reason, I have not found any implementations or discussion of implementing <*> for the classic free monad. I suspect that that is because it's not easy to derive without computer assistance.

But, fortunately, with computer assistance, it's pretty straightforward to derive, and it's as efficient as the implentations for the data structures designed to mimic the <*> signature.

instance Functor φ ⇒ Applicative (Free φ) where
  pure = V
  (V x) ⊛ f = x <$> f
  (N g) ⊛ f = N $ (flip ($) <$> f ⊛) <$> g

For me at least, the proof of (N g) ⊛ f is not something I might easily figure out or stumble upon.