How exactly does the `(<*>) = ap` Applicative/Monad law relate the two classes?

Question

ap doesn't have a documented spec, and reads with a comment pointing out it could be <*>, but isn't for practical reasons:

ap                :: (Monad m) => m (a -> b) -> m a -> m b
ap m1 m2          = do { x1 <- m1; x2 <- m2; return (x1 x2) }
-- Since many Applicative instances define (<*>) = ap, we
-- cannot define ap = (<*>)

So I assume the ap in the (<*>) = ap law is shorthand for "right-hand side of ap" and the law actually expresses a relationship between >>= , return and <*> right? Otherwise the law is meaningless.

The context is me thinking about Validation and how unsatisfying it is that it can't seem to have a lawful Monad instance. I'm also thinking about ApplicativeDo and how that transformation sort of lets us recover from the practical effects of a Monad instance for Validation; what I most often want to do is accumulate errors as far as possible, but still be able to use bind when necessary. We actually export a bindV function which we need to use just about everywhere, it's all kind of absurd. The only practical consequence I can think of the lawlessness is that we accumulate different or fewer errors depending on what sort of composition we use (or how our program might theoretically be transformed by rewrite rules, though I'm not sure why applicative composition would ever get converted to monadic).

EDIT: The documentation for the same laws in Monad is more extensive:

Furthermore, the Monad and Applicative operations should relate as follows:

pure = return
(<*>) = ap

The above laws imply:

fmap f xs  =  xs >>= return . f
(>>) = (*>)

"The above laws imply"... so is the idea here that these are the real laws we care about?

But now I'm left trying to understand these in the context of Validation. The first law would hold. The second could obviously be made to hold if we just define (>>) = (*>).

But the documentation for Monad surprisingly says nothing at all (unless I'm just missing it) about how >> should relate. Presumably we want that

a >> b = a >>= \_ -> b

...and (>>) is included in the class so that it can be overridden for efficiency, and this just never quite made it into the docs.

So if that's the case, then I guess the way Monad and Applicative relate is actually something like:

return = pure
xs >>= return . f = fmap f xs
a >>= \_ -> b = fmap (const id) a <*> b

Answer 1

Every Monad gives rise to an Applicative, and for that induced Applicative, <*> = ap will hold definitionally. But given two structures - Monad m and Applicative m - there is no guarantee that these structures agree without the two laws <*> = ap and pure = return. For example, take the 'regular' Monad instance for lists, and the zip-list Applicative instance. While there is nothing fundamentally 'wrong' about a Monad and Applicative instance disagreeing, it would probably be confusing to most users, and so it's prohibited by the Monad laws.

tl;dr The laws in question serve to ensure that Monad and Applicative agree in an intuitively obvious way.

Answer 2

So I assume the ap in the (<*>) = ap law is shorthand for "right-hand side of ap" and the law actually expresses a relationship between >>=, return and <*> right?

It seems to me (<*>) = ap doesn't strictly imply anything (at least post-AMP). Presumably it's trying to express some relationship between <*> and the right-hand side of ap. Maybe I'm being pedantic.

Speaking pedantically, I'd say the opposite: because ap is definitionally equal to its right-hand side, saying (<*>) = ap is exactly the same as saying m1 <*> m2 = do { x1 <- m1; x2 <- m2; return (x1 x2) }. It's just the normal first step of dealing with equalities like that: expanding the definitions.

Reply to the comment:

Right, but the definition is free to change.

Then the law would change or be removed too. Just as when/if join is added to Monad the current definition will become a law instead.

it wouldn't have been possible to define it literally as ap = <*>

Do you mean it would be impossible to define ap or the law in this way?

If ap, then you are correct: it would have the wrong type. But stating the law like this would be fine.