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.
Abstract of the paper:
Applicative functors and monads have conquered the world of functional programming by providing general and powerful ways of describing effectful computations using pure functions. Applicative functors provide a way to compose independent effects that cannot depend on values produced by earlier computations, and all of which are declared statically. Monads extend the applicative interface by making it possible to compose dependent effects, where the value computed by one effect determines all subsequent effects, dynamically.
This paper introduces an intermediate abstraction called selective applicative functors that requires all effects to be declared statically, but provides a way to select which of the effects to execute dynamically. We demonstrate applications of the new abstraction on several examples, including two real-life case studies.
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
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 <*?
select as a selective function application: you must apply the function
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
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) <$> x <*> f
Applicative instance can thus be given a corresponding
instance simply by defining
select = selectA. The opposite is also true
in the sense that one can recover the operator
follows (I'll use the suffix
S to denote
Selective equivalents of
commonly known functions).
apS :: Selective f => f (a -> b) -> f a -> f b apS f x = select (Left <$> f) (flip ($) <$> x)
Here we wrap a given function
a -> b into
Left and turn the value
into a function
($a), which simply feeds itself to the function
a -> b
b as desired. Note:
apS is a perfectly legal
<*>, i.e. it satisfies the laws dictated by the
Applicative type class as long as the laws of the
type class hold.
branch function is a natural generalisation of
select: instead of
skipping an unnecessary effect, it chooses which of the two given effectful
functions to apply to a given argument; the other effect is unnecessary. It
is possible to implement
branch in terms of
select, which is a good
puzzle (give it a try!).
branch :: Selective f => f (Either a b) -> f (a -> c) -> f (b -> c) -> f c branch = ... -- Try to figure out the implementation!
selectM :: Monad f => f (Either a b) -> f (a -> b) -> f b selectM mx mf = do x <- mx case x of Left a -> fmap ($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.
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).
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
x <*? pure id = either id id <$> x
Distributivity (note that
zhave the same type
f (a -> b)):
pure x <*? (y *> z) = (pure x <*? y) *> (pure x <*? z)
x <*? (y <*? z) = (f <$> 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 <$> y)
Apply a pure function to the
Leftcase of the first argument:
select (first f <$> x) y = select x ((. f) <$> y)
Apply a pure function to the second argument:
select x (f <$> y) = select (first (flip f) <$> x) (flip ($) <$> y)
x <*? pure y = either y id <$> 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
Right value, i.e. we do not require that the following laws hold:
select (pure (Left x)) y = ($x) <$> y -- Pure-Left select (pure (Right x)) y = pure x -- Pure-Right
In particular, the following is allowed too:
select (pure (Left x)) y = pure () -- when y :: f (a -> ()) select (pure (Right x)) y = const x <$> y
We therefore allow
select to be selective about effects in these cases, which
in practice allows to under- or over-approximate possible effects in static
analysis using instances like
f is also a
Monad, we require that
select = selectM, from which one
apS = <*>, and furthermore the above
properties now hold.
Static analysis of selective functors
Like applicative functors, selective functors can be analysed statically.
We can make the
Const functor an instance of
Selective as follows.
instance Monoid m => Selective (Const m) where select = selectA
Although we don't need the function
Const m (a -> b) (note that
Const m (Either a b) holds no values of type
a), we choose to
accumulate the effects associated with it. This allows us to extract
the static structure of any selective computation very similarly
to how this is done with applicative computations.
Validation instance is perhaps a bit more interesting.
data Validation e a = Failure e | Success a deriving (Functor, Show) instance Semigroup e => Applicative (Validation e) where pure = Success Failure e1 <*> Failure e2 = Failure (e1 <> e2) Failure e1 <*> Success _ = Failure e1 Success _ <*> Failure e2 = Failure e2 Success f <*> Success a = Success (f a) instance Semigroup e => Selective (Validation e) where select (Success (Right b)) _ = Success b select (Success (Left a)) f = Success ($a) <*> f select (Failure e ) _ = Failure e
Here, the last line is particularly interesting: unlike the
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
Shape (a circle or a rectangle) given a choice of the
s and the shape's parameters (
h) in a selective
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)
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"]
Do we still need monads?
Yes! Here is what selective functors cannot do:
join :: Selective f => f (f a) -> f a.
- A paper introducing selective functors: https://www.staff.ncl.ac.uk/andrey.mokhov/selective-functors.pdf.
- An older blog post introducing selective functors: https://blogs.ncl.ac.uk/andreymokhov/selective.