Portability MPTCs, fundeps provisional Edward Kmett None

Description

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, Control.Monad.Free.Church provides a `MonadFree` instance that can improve the asymptotic complexity of code that constructs free monads by effectively reassociating the use of (`>>=`). You may also want to take a look at the `kan-extensions` package (http://hackage.haskell.org/package/kan-extensions).

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

Methods

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

``` wrap (fmap f x) ≡ wrap (fmap return x) >>= f
```

Instances

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 implementation see Control.Monad.Free.Church, in particular note the `improve` combinator. You may also want to take a look at the `kan-extensions` package (http://hackage.haskell.org/package/kan-extensions).

• 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`". (Functor m, MonadError e m) => MonadError e (Free m) (Functor m, MonadReader e m) => MonadReader e (Free m) (Functor m, MonadState s m) => MonadState s (Free m) (Functor m, MonadWriter e m) => MonadWriter e (Free m) Functor f => MonadFree f (Free f) Functor f => Monad (Free f) Functor f => Functor (Free f) Typeable1 f => Typeable1 (Free f) Functor f => MonadFix (Free f) (Functor v, MonadPlus v) => MonadPlus (Free v) This violates the MonadPlus laws, handle with care. Functor f => Applicative (Free f) Foldable f => Foldable (Free f) Traversable f => Traversable (Free f) Alternative v => Alternative (Free v) This violates the Alternative laws, handle with care. (Functor m, MonadCont m) => MonadCont (Free m) Traversable1 f => Traversable1 (Free f) Foldable1 f => Foldable1 (Free f) Functor f => Apply (Free f) Functor f => Bind (Free f) (Eq (f (Free f a)), Eq a) => Eq (Free f a) (Typeable1 f, Typeable a, Data a, Data (f (Free f a))) => Data (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, MonadFree f m) => f a -> m 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.

iterM :: (Monad m, Functor f) => (f (m a) -> m a) -> Free f a -> m aSource

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`.

toFreeT :: (Functor f, Monad m) => Free f a -> FreeT f m aSource

Convert a `Free` monad from Control.Monad.Free to a `FreeT` monad from Control.Monad.Trans.Free.

_Pure :: forall f m a p. (Choice p, Applicative m) => p a (m a) -> p (Free f a) (m (Free f a))Source

This is `Prism' (Free f a) a` in disguise

````>>> ````preview _Pure (Pure 3)
```Just 3
```
````>>> ````review _Pure 3 :: Free Maybe Int
```Pure 3
```

_Free :: forall f m a p. (Choice p, Applicative m) => p (f (Free f a)) (m (f (Free f a))) -> p (Free f a) (m (Free f a))Source

This is `Prism' (Free f a) (f (Free f a))` in disguise

````>>> ````preview _Free (review _Free (Just (Pure 3)))
```Just (Just (Pure 3))
```
````>>> ````review _Free (Just (Pure 3))
```Free (Just (Pure 3))
```