Control.Selective.Multi

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: https://dl.acm.org/doi/10.1145/3341694.

This module defines multi-way selective functors, which are more efficient when selecting from a large number of options. They also fully subsume the Applicative type class because they allow to express the notion of independet effects.

This definition is inspired by the following construction by Daniel Peebles, with the main difference being the added Enumerable constraint: https://gist.github.com/copumpkin/d5bdbc7afda54ff04049b6bdbcffb67e

Generalised sum types

data Sigma t where

A generalised sum type where t stands for the type of constructor "tags". Each tag has a type parameter x which determines the type of the payload. A Sigma t value therefore contains a payload whose type is not visible externally but is revealed when pattern-matching on the tag.

See Two, eitherToSigma and sigmaToEither for an example.

Constructors

Sigma :: t x -> x -> Sigma t

inject :: t x -> x -> Sigma t

An injection into a generalised sum. An alias for Sigma.

data Zero a

A data type defining no tags. Similar to Void but parameterised.

Instances

Enumerable Zero

Defined in Control.Selective.Multi

Methods

enumerate :: [Some Zero]

data One a b where

A data type with a single tag. This data type is commonly known as Refl, see Data.Type.Equality.

Constructors

One :: One a a

Instances

Enumerable (One a)

Defined in Control.Selective.Multi

Methods

enumerate :: [Some (One a)]

data Two a b c where

A data type with two tags A and B that allows us to encode the good old Either as Sigma Two, where the tags A and B correspond to Left and Right, respectively. See eitherToSigma and sigmaToEither that witness the isomorphism between Either a b and Sigma (Two a b).

Constructors

A :: Two a b a

B :: Two a b b

Instances

Enumerable (Two a b)

Defined in Control.Selective.Multi

Methods

enumerate :: [Some (Two a b)]

data Many a b where

A potentially uncountable collection of tags for the same unit () payload.

Constructors

Many :: a -> Many a ()

Instances

Enum a => Enumerable (Many a)

Defined in Control.Selective.Multi

Methods

enumerate :: [Some (Many a)]

many :: a -> Sigma (Many a)

matchPure :: Sigma t -> (forall x. t x -> x -> a) -> a

Generalised pattern matching on a Sigma type using a Pi type to describe how to handle each case.

This is a specialisation of matchCases for f = Identity. We could also have also given it the following type:

matchPure :: Sigma t -> (t ~> Case Identity a) -> a

We chose to simplify it by inlining ~>, Case and Identity.

eitherToSigma :: Either a b -> Sigma (Two a b)

Encode Either into a generalised sum type.

sigmaToEither :: Sigma (Two a b) -> Either a b

Decode Either from a generalised sum type.

Selective functors

data Some t where

Hide the type of the payload a tag.

There is a whole library dedicated to this nice little GADT: http://hackage.haskell.org/package/some.

Constructors

Some :: t a -> Some t

class Enumerable t where

A class of tags that can be enumerated.

A valid instance must list every tag in the resulting list exactly once.

Methods

enumerate :: [Some t]

Instances

Enumerable Zero

Defined in Control.Selective.Multi

Methods

enumerate :: [Some Zero]

Enum a => Enumerable (Many a)

Defined in Control.Selective.Multi

Methods

enumerate :: [Some (Many a)]

Enumerable (One a)

Defined in Control.Selective.Multi

Methods

enumerate :: [Some (One a)]

Enumerable (Two a b)

Defined in Control.Selective.Multi

Methods

enumerate :: [Some (Two a b)]

class Applicative f => Selective f where

Multi-way selective functors. Given a computation that produces a value of a sum type, we can match it to the corresponding computation in a given product type.

For greater similarity with matchCases, we could have given the following type to match:

match :: f (Sigma t) -> (t ~> Case f a) -> f a

We chose to simplify it by inlining ~> and Case.

Methods

match :: Enumerable t => f (Sigma t) -> (forall x. t x -> f (x -> a)) -> f a

Instances

Monoid m => Selective (Over m)

Defined in Control.Selective.Multi

Methods

match :: Enumerable t => Over m (Sigma t) -> (forall x. t x -> Over m (x -> a)) -> Over m a

Monoid m => Selective (Under m)

Defined in Control.Selective.Multi

Methods

match :: Enumerable t => Under m (Sigma t) -> (forall x. t x -> Under m (x -> a)) -> Under m a

newtype Over m a

Static analysis of selective functors with over-approximation.

Constructors

Over

Fields

• getOver :: m

Instances

Monoid m => Applicative (Over m)

Defined in Control.Selective.Multi

Methods

pure :: a -> Over m a

(<*>) :: Over m (a -> b) -> Over m a -> Over m b

liftA2 :: (a -> b -> c) -> Over m a -> Over m b -> Over m c

(*>) :: Over m a -> Over m b -> Over m b

(<*) :: Over m a -> Over m b -> Over m a

Functor (Over m)

Defined in Control.Selective.Multi

Methods

fmap :: (a -> b) -> Over m a -> Over m b

(<$) :: a -> Over m b -> Over m a

Monoid m => Selective (Over m)

Defined in Control.Selective.Multi

Methods

match :: Enumerable t => Over m (Sigma t) -> (forall x. t x -> Over m (x -> a)) -> Over m a

Show m => Show (Over m a)

Defined in Control.Selective.Multi

Methods

showsPrec :: Int -> Over m a -> ShowS

show :: Over m a -> String

showList :: [Over m a] -> ShowS

Eq m => Eq (Over m a)

Defined in Control.Selective.Multi

Methods

(==) :: Over m a -> Over m a -> Bool

(/=) :: Over m a -> Over m a -> Bool

Ord m => Ord (Over m a)

Defined in Control.Selective.Multi

Methods

compare :: Over m a -> Over m a -> Ordering

(<) :: Over m a -> Over m a -> Bool

(<=) :: Over m a -> Over m a -> Bool

(>) :: Over m a -> Over m a -> Bool

(>=) :: Over m a -> Over m a -> Bool

max :: Over m a -> Over m a -> Over m a

min :: Over m a -> Over m a -> Over m a

newtype Under m a

Static analysis of selective functors with under-approximation.

Constructors

Under

Fields

• getUnder :: m

Instances

Monoid m => Applicative (Under m)

Defined in Control.Selective.Multi

Methods

pure :: a -> Under m a

(<*>) :: Under m (a -> b) -> Under m a -> Under m b

liftA2 :: (a -> b -> c) -> Under m a -> Under m b -> Under m c

(*>) :: Under m a -> Under m b -> Under m b

(<*) :: Under m a -> Under m b -> Under m a

Functor (Under m)

Defined in Control.Selective.Multi

Methods

fmap :: (a -> b) -> Under m a -> Under m b

(<$) :: a -> Under m b -> Under m a

Monoid m => Selective (Under m)

Defined in Control.Selective.Multi

Methods

match :: Enumerable t => Under m (Sigma t) -> (forall x. t x -> Under m (x -> a)) -> Under m a

Show m => Show (Under m a)

Defined in Control.Selective.Multi

Methods

showsPrec :: Int -> Under m a -> ShowS

show :: Under m a -> String

showList :: [Under m a] -> ShowS

Eq m => Eq (Under m a)

Defined in Control.Selective.Multi

Methods

(==) :: Under m a -> Under m a -> Bool

(/=) :: Under m a -> Under m a -> Bool

Ord m => Ord (Under m a)

Defined in Control.Selective.Multi

Methods

compare :: Under m a -> Under m a -> Ordering

(<) :: Under m a -> Under m a -> Bool

(<=) :: Under m a -> Under m a -> Bool

(>) :: Under m a -> Under m a -> Bool

(>=) :: Under m a -> Under m a -> Bool

max :: Under m a -> Under m a -> Under m a

min :: Under m a -> Under m a -> Under m a

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

The basic "if-then" selection primitive from Control.Selective.

branch :: Selective f => f (Either a b) -> f (a -> c) -> f (b -> c) -> f c

Choose a matching effect with Either.

apS :: Selective f => f a -> f (a -> b) -> f b

Recover the application operator <*> from match.

bindS :: (Enum a, Selective f) => f a -> (a -> f b) -> f b

A restricted version of monadic bind.

Applicative functors

class Functor f => ApplicativeS f where

An alternative definition of applicative functors, as witnessed by ap and matchOne. This class is almost like Selective but has a strict constraint on t.

Methods

pureA :: a -> f a

matchOne :: t ~ One x => f (Sigma t) -> (forall x. t x -> f (x -> a)) -> f a

ap :: ApplicativeS f => f a -> f (a -> b) -> f b

Recover the application operator <*> from matchOne.

matchA :: (Applicative f, t ~ One x) => f (Sigma t) -> (forall x. t x -> f (x -> a)) -> f a

Every Applicative is also an ApplicativeS.

Monads

class Applicative f => MonadS f where

An alternative definition of monads, as witnessed by bind and matchM. This class is almost like Selective but has no the constraint on t.

Methods

matchUnconstrained :: f (Sigma t) -> (forall x. t x -> f (x -> a)) -> f a

bind :: MonadS f => f a -> (a -> f b) -> f b

Monadic bind.

matchM :: Monad f => f (Sigma t) -> (forall x. t x -> f (x -> a)) -> f a

Every monad is a multi-way selective functor.

Generalised products and various combinators

type (~>) t u = forall x. t x -> u x infixl 4

A generalised product type (Pi), which holds an appropriately tagged payload u x for every possible tag t x.

Note that this looks different than the standard formulation of Pi types. Maybe it's just all wrong!

See Two, pairToPi and piToPair for an example.

type Pi t = t ~> Identity

A product type where the payload has the type specified with the tag.

project :: t a -> Pi t -> a

A projection from a generalised product.

identity :: t ~> t

A trivial product type that stores nothing and simply returns the given tag as the result.

compose :: (u ~> v) -> (t ~> u) -> t ~> v

As it turns out, one can compose such generalised products. Why not: given a tag, get the payload of the first product and then pass it as input to the second. This feels too trivial to be useful but is still somewhat cute.

apply :: (t ~> u) -> Sigma t -> Sigma u

Update a generalised sum given a generalised product that takes care of all possible cases.

toSigma :: a -> Sigma (One a)

Encode a value into a generalised sum type that has a single tag One.

fromSigma :: Sigma (One a) -> a

Decode a value from a generalised sum type that has a single tag One.

toPi :: a -> Pi (One a)

Encode a value into a generalised product type that has a single tag One.

fromPi :: Pi (One a) -> a

Decode a value from a generalised product type that has a single tag One.

pairToPi :: (a, b) -> Pi (Two a b)

Encode (a, b) into a generalised product type.

piToPair :: Pi (Two a b) -> (a, b)

Decode (a, b) from a generalised product type.

newtype Case f a x

Handler of a single case in a generalised pattern matching matchCases.

Constructors

Case

Fields

• handleCase :: f (x -> a)

matchCases :: Functor f => Sigma t -> (t ~> Case f a) -> f a

Generalised pattern matching on a Sigma type using a Pi type to describe how to handle each case.