From the documentation of the validation
package:
The AccValidation
data type is isomorphic to Either
, but has an instance of Applicative
that accumulates on the error side. That is to say, if two (or more) errors are encountered, they are appended using a Semigroup
operation.
As a consequence of this Applicative
instance, there is no corresponding Bind
or Monad
instance. AccValidation
is an example of, "An applicative functor that is not a monad."
It isn't evident to me why this is a consequence. I can imagine a Monad
instance for AccValidation
that behaves like Either
- What would make this unlawful?
Mechanically, the Either
-ish Monad
instance for AccValidation
would be
-- The (Monoid err) context is not used for anything,
-- it's just there to satisfy the Applicative super-instance
instance (Monoid err) => Monad (AccValidation err) where
return = AccSuccess
AccFailure err >>= f = AccFailure err
AccSuccess x >>= f = f x
which would mean we have
AccFailure err1 <*> AccFailure err2 = AccFailure (err1 <> err2)
AccFailure err1 `ap` AccFailure err2 = AccFailure err1
which breaks the monad law of <*> = ap
.
Intuitively, it can't be made a monad because in a monad, the effect (i.e. the validation failures) of a computation can depend on previously bound results. But in the case of a failure, there is no result. So Either
has no choice than to short-circuit to failure in that case, since there is nothing to feed to the subsequent functions on right-hand sides of (>>=)
s.
This is in stark contrast to applicative functors, where the effect (in this case, the validation failures) cannot depend on other results, which is why we can get all validation failures without having to feed results (where would they come from?) from one computation to the other.
The (<*>) = ap
exigence can be stated explicitly in terms of (>>=)
:
u <*> v = u >>= \f -> fmap f v -- [1]
Now, given the Functor
and Applicative
instances for AccValidation
, we have:
fmap _ (AccFailure e) = AccFailure e -- [2]
AccFailure e1 <*> AccFailure e2 = AccFailure (e1 <> e2) -- [3]
If we make u = AccFailure e1
and v = AccFailure e2
in [1], we get:
AccFailure e1 <*> AccFailure e2 = AccFailure e1 >>= \f -> fmap f (AccFailure e2)
Substituting [2] and [3] into that leads us to:
AccFailure (e1 <> e2) = AccFailure e1 >>= \_ -> AccFailure e2 -- [4]
The problem is that it is impossible to write a (>>=)
such that [4] holds. The left-hand side depends on an e2
value which must originate, on the right-hand side, from applying \_ -> AccFailure e2 :: Semigroup e => a -> AccValidation e b
. However, there is nothing to which it can be applied -- in particular, e1
has the wrong type. (See the final two paragraphs of Cactus' answer for further discussion of this point.) Therefore, there is no way of giving AccValidation
a Monad
instance which is consistent with its Applicative
one.