First, for clarity, note that
const id = flip const
Consider a (right-)occlusive functor. We immediately see that
liftA2 (flip const) m (pure x) = pure x
Using the Applicative laws, we can restate this:
x <$ m = pure x
We get the same sort of result for a left-occlusive effect.
So an occlusive effect can't have any *observable* side effects. It must be
"read only".
On Thu, Nov 12, 2020, 3:59 PM Olaf Klinke
First, instead of `const id` I will use `const`, that is, we shall prove
const = liftM2 const :: M a -> M b -> M a
which I believe should be equivalent to your property.
My belief was wrong, which is evident when using do-notation. Kim-Ee stated
do {_ <- b; x <- a; return x} = a
while I examined
do {x <- a; _ <- b; return x} = a
Since do {x <- a; return x} = a holds for any monad, Kim-Ee's property can be reduced to
do {_ <- b; a} = a or more concisely b >> a = a
which I called Kleisli-const in my previous post. As we seemed to have settled for "occlusive" I suggest calling b >> a = a "right-occlusive" or "occlusive from the right" because the right action occludes the side-effects of the left action, and do {x <- a; _ <- b; return x} = a should be called "left-occlusive" or "occlusive from the left" because the left action hides the effect of the right action. Under commutativity, both are the same but in general these might be different properties. I do not have a distinguishing counterexample, though, because all monads I come up with are commutative.
David Feuer hinted at the possibility to define occlusiveness more generally for Applicative functors. Commutativity might be stated for Applicatives as
liftA2 f a b = liftA2 (flip f) b a
So far I can only see two classes of occlusive monads: The reader-like (Identity ~ Reader ()) and the set-like monads.
Olaf
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