haskell - Why should Applicative be a superclass of Monad?

Question

Given:

Applicative m, Monad m => mf :: m (a -> b), ma :: m a

it seems to be considered a law that:

mf <*> ma === do { f <- mf; a <- ma; return (f a) }

or more concisely:

(<*>) === ap

The documentation for Control.Applicative says that <*> is "sequential application," and that suggests that (<*>) = ap. This means that <*> must evaluate effects sequentially from left to right, for consistency with >>=... But that feels wrong. McBride and Paterson's original paper seems to imply that the left-to-right sequencing is arbitrary:

The IO monad, and indeed any Monad, can be made Applicative by taking pure = return and <*> = ap. We could alternatively use the variant of ap that performs the computations in the opposite order, but we shall keep to the left-to-right order in this paper.

So there are two lawful, non-trivial derivations for <*> that follow from >>= and return, with distinct behavior. And in some cases, neither of these two derivations are desirable.

For example, the (<*>) === ap law forces Data.Validation to define two distinct data types: Validation and AccValidation. The former has a Monad instance similar to ExceptT, and a consistent Applicative instance which is of limited utility, since it stops after the first error. The latter, on the other hand, doesn't define a Monad instance, and is therefore free to implement an Applicative that, much more usefully, accumulates errors.

There's been some discussion about this previously on StackOverflow, but I don't think it really got to the meat of the question:

Why should this be a law?

The other laws for functors, applicatives and monads—such as identity, associativity, etc.—express some fundamental, mathematical properties of those structures. We can implement various optimizations using these laws and prove things about our own code using them. In contrast, it feels to me like the (<*>) === ap law imposes an arbitrary constraint with no corresponding benefit.

For what it's worth, I'd prefer to ditch the law in favor of something like this:

newtype LeftA m a = LeftA (m a)
instance Monad m => Applicative (LeftA m) where
  pure = return
  mf <*> ma = do { f <- mf; a <- ma; return (f a) }

newtype RightA m a = RightA (m a)
instance Monad m => Applicative (RightA m) where
  pure = return
  mf <*> ma = do { a <- ma; f <- mf; return (f a) }

I think that correctly captures the relationship between the two, without unduly constraining either.

So, a few angles to approach the question from:

• Are there any other laws relating Monad and Applicative?
• Is there any inherent mathematical reason for effects to sequence for Applicative in the same way that they do for Monad?
• Does GHC or any other tool perform code transformations that assume/require this law to be true?
• Why is the Functor-Applicative-Monad proposal considered such an overwhelmingly good thing? (Citations would be much appreciated here).

And one bonus question:

• How do Alternative and MonadPlus fit in to all this?

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Note: major edit to clarify the meat of the question. Answer posted by @duplode quotes an earlier version.

Answer 1

Well, I'm not terribly satisfied with the answers given so far, but I think the comments attached to them are a bit more compelling. So I'll summarize here:

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I think there's only one sensible Functor instance that follows from Applicative:

fmap f fa = pure f <*> fa

Assuming that's unique, it makes sense that Functor should be a superclass of Applicative, with that law. Likewise, I think there's only one sensible Functor instance that follows from Monad:

fmap f fa = fa >>= return . f

So again, it makes sense that Functor should be a superclass of Monad. The objection I had (and, really, still have) is that there are two sensible Applicative instances that follow from Monad and, in some specific instances, even more that are lawful; so why mandate one?

pigworker (first author on the original Applicative paper) writes:

"Of course it doesn't follow. It's a choice."

(on twitter): "do-notation is unjust punishment for working in a monad; we deserve applicative notation"

duplode similarly writes:

"... it is fair to say that pure === return and (<*>) === ap aren't laws in the strong sense that e.g. the monad laws are so ..."

"On the LeftA/RightA idea: there are comparable cases elsewhere in the standard libraries (e.g. Sum and Product in Data.Monoid). The problem of doing the same with Applicative is that the power-to-weight relation is too low to justify the extra precision/flexibility. The newtypes would make applicative style a lot less pleasant to use."

So, I'm happy to see that choice stated explicitly, justified by the simple reasoning that it makes the most common cases easier.