Selective applicative functors
This is a library for selective applicative functors, or just selective functors
for short, an abstraction between applicative functors and monads, introduced in
this paper.
What are selective functors?
While you're encouraged to read the paper, here is a brief description of
the main idea. Consider the following new type class introduced 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) <$> x <*> f
Any Applicative
instance can thus be given a corresponding Selective
instance simply by defining select = selectA
. The opposite is also true
in the sense that one can recover the operator <*>
from select
as
follows (I'll use the suffix S
to denote Selective
equivalents of
commonly known functions).
apS :: Selective f => f (a > b) > f a > f b
apS f x = select (Left <$> f) ((&) <$> x)
Here we wrap a given function a > b
into Left
and turn the value a
into a function ($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 as the laws of 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 shortcircuiting 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.
Code written using selective combinators can be both statically analysed
(by reporting all possible effects of a computation) and efficiently
executed (by skipping unnecessary effects).
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.

Identity:
x <*? pure id = either id id <$> x

Distributivity (note that y
and z
have the same type f (a > b)
):
pure x <*? (y *> z) = (pure x <*? y) *> (pure x <*? z)

Associativity:
x <*? (y <*? z) = (f <$> x) <*? (g <$> y) <*? (h <$> z)
where
f x = Right <$> x
g y = \a > bimap (,a) ($a) y
h z = uncurry z

Monadic select (for selective functors that are also monads):
select = selectM
There are also a few useful theorems:

Apply a pure function to the result:
f <$> select x y = select (fmap f <$> x) (fmap f <$> y)

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

Apply a pure function to the second argument:
select x (f <$> y) = select (first (flip f) <$> x) ((&) <$> y)

Generalised identity:
x <*? pure y = either y id <$> x

A selective functor is rigid if it satisfies <*> = apS
. The following
interchange law holds for rigid selective functors:
x *> (y <*? z) = (x *> y) <*? 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 = ($x) <$> y  PureLeft
select (pure (Right x)) y = pure x  PureRight
In particular, the following is allowed too:
select (pure (Left x)) y = pure ()  when y :: f (a > ())
select (pure (Right x)) y = const x <$> y
We therefore allow select
to be selective about effects in these cases, which
in practice allows to under or overapproximate possible effects in static
analysis using instances like Under
and Over
.
If f
is also a Monad
, we require that select = selectM
, from which one
can prove apS = <*>
, and furthermore the above PureLeft
and PureRight
properties now hold.
Static analysis of selective functors
Like applicative functors, selective functors can be analysed statically.
We can make the Const
functor an instance of Selective
as follows.
instance Monoid m => Selective (Const m) where
select = selectA
Although we don't need the function Const m (a > b)
(note that
Const m (Either a b)
holds no values of type a
), we choose to
accumulate the effects associated with it. This allows us to extract
the static structure of any selective computation very similarly
to how this is done with applicative computations.
The Validation
instance is perhaps a bit more interesting.
data Validation e a = Failure e  Success a deriving (Functor, Show)
instance Semigroup e => Applicative (Validation e) where
pure = Success
Failure e1 <*> Failure e2 = Failure (e1 <> e2)
Failure e1 <*> Success _ = Failure e1
Success _ <*> Failure e2 = Failure e2
Success f <*> Success a = Success (f a)
instance Semigroup e => Selective (Validation e) where
select (Success (Right b)) _ = Success b
select (Success (Left a)) f = Success ($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 shortcircuiting
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"]
Do we still need monads?
Yes! Here is what selective functors cannot do: join :: Selective f => f (f a) > f a
.
Further reading