free-3.3.1: Monads for free

Portability MPTCs, fundeps provisional Edward Kmett None

Control.Monad.Free

Description

Monads for free

Synopsis

# Documentation

class Monad m => MonadFree f m | m -> f whereSource

Monads provide substitution (`fmap`) and renormalization (`join`):

`m `>>=` f = `join` . `fmap` f m`

A free `Monad` is one that does no work during the normalization step beyond simply grafting the two monadic values together.

`[]` is not a free `Monad` (in this sense) because `join [[a]]` smashes the lists flat.

On the other hand, consider:

``` data Tree a = Bin (Tree a) (Tree a) | Tip a
```
``` instance `Monad` Tree where
`return` = Tip
Tip a `>>=` f = f a
Bin l r `>>=` f = Bin (l `>>=` f) (r `>>=` f)
```

This `Monad` is the free `Monad` of Pair:

``` data Pair a = Pair a a
```

And we could make an instance of `MonadFree` for it directly:

``` instance `MonadFree` Pair Tree where
`wrap` (Pair l r) = Bin l r
```

Or we could choose to program with `Free Pair` instead of `Tree` and thereby avoid having to define our own `Monad` instance.

Moreover, the `kan-extensions` package provides `MonadFree` instances that can improve the asymptotic complexity of code that constructors free monads by effectively reassociating the use of (`>>=`).

See `Free` for a more formal definition of the free `Monad` for a `Functor`.

Methods

wrap :: f (m a) -> m aSource

Add a layer.

Instances

 (Monad (ListT m), Functor f, MonadFree f m) => MonadFree f (ListT m) (Monad (IdentityT m), Functor f, MonadFree f m) => MonadFree f (IdentityT m) (Monad (MaybeT m), Functor f, MonadFree f m) => MonadFree f (MaybeT m) (Monad (Free f), Functor f) => MonadFree f (Free f) (Monad (Free f), Functor f) => MonadFree f (Free f) (Monad (F f), Functor f) => MonadFree f (F f) (Monad (ErrorT e m), Functor f, MonadFree f m, Error e) => MonadFree f (ErrorT e m) (Monad (WriterT w m), Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) (Monad (WriterT w m), Functor f, MonadFree f m, Monoid w) => MonadFree f (WriterT w m) (Monad (StateT s m), Functor f, MonadFree f m) => MonadFree f (StateT s m) (Monad (StateT s m), Functor f, MonadFree f m) => MonadFree f (StateT s m) (Monad (ReaderT e m), Functor f, MonadFree f m) => MonadFree f (ReaderT e m) (Monad (FreeT f m), Functor f, Monad m) => MonadFree f (FreeT f m) (Monad (RWST r w s m), Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m) (Monad (RWST r w s m), Functor f, MonadFree f m, Monoid w) => MonadFree f (RWST r w s m)

data Free f a Source

The `Free` `Monad` for a `Functor` `f`.

Formally

A `Monad` `n` is a free `Monad` for `f` if every monad homomorphism from `n` to another monad `m` is equivalent to a natural transformation from `f` to `m`.

Why Free?

Every "free" functor is left adjoint to some "forgetful" functor.

If we define a forgetful functor `U` from the category of monads to the category of functors that just forgets the `Monad`, leaving only the `Functor`. i.e.

`U (M,`return`,`join`) = M`

then `Free` is the left adjoint to `U`.

Being `Free` being left adjoint to `U` means that there is an isomorphism between

`Free f -> m` in the category of monads and `f -> U m` in the category of functors.

Morphisms in the category of monads are `Monad` homomorphisms (natural transformations that respect `return` and `join`).

Morphisms in the category of functors are `Functor` homomorphisms (natural transformations).

Given this isomorphism, every monad homomorphism from `Free f` to `m` is equivalent to a natural transformation from `f` to `m`

Showing that this isomorphism holds is left as an exercise.

In practice, you can just view a `Free f a` as many layers of `f` wrapped around values of type `a`, where `(>>=)` performs substitution and grafts new layers of `f` in for each of the free variables.

This can be very useful for modeling domain specific languages, trees, or other constructs.

This instance of `MonadFree` is fairly naive about the encoding. For more efficient free monad implementations that require additional extensions and thus aren't included here, you may want to look at the `kan-extensions` package.

A number of common monads arise as free monads,

• Given `data Empty a`, `Free Empty` is isomorphic to the `Identity` monad.
• `Free Maybe` can be used to model a partiality monad where each layer represents running the computation for a while longer.

Constructors

 Pure a Free (f (Free f a))

Instances

 MonadTrans Free This is not a true monad transformer. It is only a monad transformer "up to `retract`". (Monad (Free m), Functor m, MonadError e m) => MonadError e (Free m) (Monad (Free m), Functor m, MonadReader e m) => MonadReader e (Free m) (Monad (Free m), Functor m, MonadState s m) => MonadState s (Free m) (Monoid e, Monad (Free m), Functor m, MonadWriter e m) => MonadWriter e (Free m) (Monad (Free f), Functor f) => MonadFree f (Free f) Functor f => Monad (Free f) Functor f => Functor (Free f) Typeable1 f => Typeable1 (Free f) (Monad (Free v), Functor v, MonadPlus v) => MonadPlus (Free v) This violates the MonadPlus laws, handle with care. (Functor (Free f), Functor f) => Applicative (Free f) Foldable f => Foldable (Free f) (Functor (Free f), Foldable (Free f), Traversable f) => Traversable (Free f) (Applicative (Free v), Alternative v) => Alternative (Free v) This violates the Alternative laws, handle with care. (Monad (Free m), Functor m, MonadCont m) => MonadCont (Free m) (Foldable1 (Free f), Traversable (Free f), Traversable1 f) => Traversable1 (Free f) (Foldable (Free f), Foldable1 f) => Foldable1 (Free f) (Functor (Free f), Functor f) => Apply (Free f) (Apply (Free f), Functor f) => Bind (Free f) (Eq (f (Free f a)), Eq a) => Eq (Free f a) (Typeable (Free f a), Typeable1 f, Typeable a, Data a, Data (f (Free f a))) => Data (Free f a) (Eq (Free f a), Ord (f (Free f a)), Ord a) => Ord (Free f a) (Read (f (Free f a)), Read a) => Read (Free f a) (Show (f (Free f a)), Show a) => Show (Free f a)

retract :: Monad f => Free f a -> f aSource

`retract` is the left inverse of `lift` and `liftF`

``` `retract` . `lift` = `id`
`retract` . `liftF` = `id`
```

liftF :: Functor f => f a -> Free f aSource

A version of `lift` that can be used with just a `Functor` for `f`.

iter :: Functor f => (f a -> a) -> Free f a -> aSource

Tear down a `Free` `Monad` using iteration.

hoistFree :: Functor g => (forall a. f a -> g a) -> Free f b -> Free g bSource

Lift a natural transformation from `f` to `g` into a natural transformation from `FreeT f` to `FreeT g`.