Selective applicative functors

This is a study of selective applicative functors , an abstraction between Applicative and Monad :

class Applicative f => Selective f where
    select :: f (Either a b) -> f (a -> b) -> f b

-- | An operator alias for 'select'.
(<*?) :: Selective f => f (Either a b) -> f (a -> b) -> f b
(<*?) = select

infixl 4 <*?

Think of select as a selective function application : you must apply the function of type a -> b when given a value of type Left a , but you may skip the function and associated effects, and simply return b when given Right b .

Note that you can write a function with this type signature using Applicative functors, but it will always execute the effects associated with the second argument, hence being potentially less efficient:

selectA :: Applicative f => f (Either a b) -> f (a -> b) -> f b
selectA x f = (\e f -> either f id e) <$> x <*> f

Selective is more powerful than Applicative : you can recover the application operator <*> as follows (I'll use the suffix S to denote Selective equivalents of commonly known functions).

apS :: Selective f => f (a -> b) -> f a -> f b
apS f x = select (Left <$> f) (flip ($) <$> x)

Here we wrap a given function a -> b into Left and turn the value a into a function ($a) , which simply feeds itself to the function a -> b yielding b as desired. Note: apS is a perfectly legal application operator <*> , i.e. it satisfies the laws dictated by the Applicative type class as long asof the Selective type class hold.

The branch function is a natural generalisation of select : instead of skipping an unnecessary effect, it chooses which of the two given effectful functions to apply to a given argument; the other effect is unnecessary. It is possible to implement branch in terms of select , which is a good puzzle (give it a try!).

branch :: Selective f => f (Either a b) -> f (a -> c) -> f (b -> c) -> f c
branch = ... -- Try to figure out the implementation!

Finally, any Monad is Selective :

selectM :: Monad f => f (Either a b) -> f (a -> b) -> f b
selectM mx mf = do
    x <- mx
    case x of
        Left  a -> fmap ($a) mf
        Right b -> pure b

Selective functors are sufficient for implementing many conditional constructs, which traditionally require the (more powerful) Monad type class. For example:

-- | Branch on a Boolean value, skipping unnecessary effects.
ifS :: Selective f => f Bool -> f a -> f a -> f a
ifS i t e = branch (bool (Right ()) (Left ()) <$> i) (const <$> t) (const <$> e)

-- | Conditionally perform an effect.
whenS :: Selective f => f Bool -> f () -> f ()
whenS x act = ifS x act (pure ())

-- | Keep checking an effectful condition while it holds.
whileS :: Selective f => f Bool -> f ()
whileS act = whenS act (whileS act)

-- | A lifted version of lazy Boolean OR.
(<||>) :: Selective f => f Bool -> f Bool -> f Bool
(<||>) a b = ifS a (pure True) b

-- | A lifted version of 'any'. Retains the short-circuiting behaviour.
anyS :: Selective f => (a -> f Bool) -> [a] -> f Bool
anyS p = foldr ((<||>) . p) (pure False)

-- | Return the first @Right@ value. If both are @Left@'s, accumulate errors.
orElse :: (Selective f, Semigroup e) => f (Either e a) -> f (Either e a) -> f (Either e a)
orElse x = select (Right <$> x) . fmap (\y e -> first (e <>) y)

See more examples in src/Control/Selective.hs .

Laws

Instances of the Selective type class must satisfy a few laws to make it possible to refactor selective computations. These laws also allow us to establish a formal relation with the Applicative and Monad type classes. The laws are complex, but I couldn't figure out how to simplify them. Please let me know if you find an improvement.

(F1) Apply a pure function to the result:

f <$> select x y = select (second f <$> x) ((f .) <$> y)

(F2) Apply a pure function to the Left case of the first argument:
```
select (first f <$> x) y = select x ((. f) <$> y)
```

(F3) Apply a pure function to the second argument:

select x (f <$> y) = select (first (flip f) <$> x) (flip ($) <$> y)

(P1) Selective application of a pure function:
```
select x (pure y) = either y id <$> x
```

(A1) Associativity:

select x (select y z) = select (select (f <$> x) (g <$> y)) (h <$> z)
  where
    f x = Right <$> x
    g y = \a -> bimap (,a) ($a) y
    h z = uncurry z

-- or in operator form:

x <*? (y <*? z) = (f <$> x) <*? (g <$> y) <*? (h <$> z)

Note that there are no laws for selective application of a function to a pure Left or Right value, i.e. we do not require that the following laws hold:

select (pure (Left  x)) y = y <*> pure x -- P2
select (pure (Right x)) y =       pure x -- P3

In particular, the following is allowed too:

select (pure (Left  x)) y = pure ()       -- when y :: f (a -> ())
select (pure (Right x)) y = const x <$> y

We therefore allow select to be selective about effects in these cases, which in practice allows to under- or over-approximate possible effects in static analysis using instances like Under and Over .

If f is also a Monad , we require that select = selectM , from which one can prove apS = <*> , and furthermore the above two laws P2-P3 now hold.

Static analysis of selective functors

Like applicative functors, selective functors can be analysed statically. We can make the Const functor an instance of Selective as follows.

instance Monoid m => Selective (Const m) where
    select = selectA

Although we don't need the function Const m (a -> b) (note that Const m (Either a b) holds no values of type a ), we choose to accumulate the effects associated with it. This allows us to extract the static structure of any selective computation very similarly to how this is done with applicative computations.

The Validation instance is perhaps a bit more interesting.

data Validation e a = Failure e | Success a deriving (Functor, Show)

instance Semigroup e => Applicative (Validation e) where
    pure = Success
    Failure e1 <*> Failure e2 = Failure (e1 <> e2)
    Failure e1 <*> Success _  = Failure e1
    Success _  <*> Failure e2 = Failure e2
    Success f  <*> Success a  = Success (f a)

instance Semigroup e => Selective (Validation e) where
    select (Success (Right b)) _ = Success b
    select (Success (Left  a)) f = Success ($a) <*> f
    select (Failure e        ) _ = Failure e

Here, the last line is particularly interesting: unlike the Const instance, we choose to actually skip the function effect in case of Failure . This allows us not to report any validation errors which are hidden behind a failed conditional.

Let's clarify this with an example. Here we define a function to construct a Shape (a circle or a rectangle) given a choice of the shape s and the shape's parameters ( r , w , h ) in a selective context f .

type Radius = Int
type Width  = Int
type Height = Int

data Shape = Circle Radius | Rectangle Width Height deriving Show

shape :: Selective f => f Bool -> f Radius -> f Width -> f Height -> f Shape
shape s r w h = ifS s (Circle <$> r) (Rectangle <$> w <*> h)

We choose f = Validation [String] to report the errors that occurred when parsing a value. Let's see how it works.

> shape (Success True) (Success 10) (Failure ["no width"]) (Failure ["no height"])
Success (Circle 10)

> shape (Success False) (Failure ["no radius"]) (Success 20) (Success 30)
Success (Rectangle 20 30)

> shape (Success False) (Failure ["no radius"]) (Success 20) (Failure ["no height"])
Failure ["no height"]

> shape (Success False) (Failure ["no radius"]) (Failure ["no width"]) (Failure ["no height"])
Failure ["no width","no height"]

> shape (Failure ["no choice"]) (Failure ["no radius"]) (Success 20) (Failure ["no height"])
Failure ["no choice"]

In the last example, since we failed to parse which shape has been chosen, we do not report any subsequent errors. But it doesn't mean we are short-circuiting the validation. We will continue accumulating errors as soon as we get out of the opaque conditional, as demonstrated below.

twoShapes :: Selective f => f Shape -> f Shape -> f (Shape, Shape)
twoShapes s1 s2 = (,) <$> s1 <*> s2

> s1 = shape (Failure ["no choice 1"]) (Failure ["no radius 1"]) (Success 20) (Failure ["no height 1"])
> s2 = shape (Success False) (Failure ["no radius 2"]) (Success 20) (Failure ["no height 2"])
> twoShapes s1 s2
Failure ["no choice 1","no height 2"]

Alternative formulations

There are other ways of expressing selective functors in Haskell and most of them are compositions of applicative functors and the Either monad. Below I list a few examples:

-- Alternative type classes for selective functors. They all come with an
-- additional requirement that we run effects from left to right.

-- Composition of Applicative and Either monad
class Applicative f => SelectiveA f where
    (|*|) :: f (Either e (a -> b)) -> f (Either e a) -> f (Either e b)

-- Composition of Starry and Either monad
-- See: https://duplode.github.io/posts/applicative-archery.html
class Applicative f => SelectiveS f where
    (|.|) :: f (Either e (b -> c)) -> f (Either e (a -> b)) -> f (Either e (a -> c))

-- Composition of Monoidal and Either monad
-- See: http://blog.ezyang.com/2012/08/applicative-functors/
class Applicative f => SelectiveM f where
    (|**|) :: f (Either e a) -> f (Either e b) -> f (Either e (a, b))

I believe these formulations are equivalent to Selective defined above, but I have not proved the equivalence formally. I chose the most minimalistic definition of the type class, but the above alternatives are worth consideration too. In particular, SelectiveS has a much nicer associativity law, which is just (x |.| y) |.| z = x |.| (y |.| z) .

Do we still need monads?

Yes! Here is what selective functors cannot do: join :: Selective f => f (f a) -> f a .

Selective applicative functors