Free Monad

data Free f a = Pure a | Free (f (Free f a))

instance Functor f => Monad (Free f) where
  return = Pure
  Free x >>= f = Free (fmap (>>= f) x)
  Pure x >>= f = f x

Coyoneda

data Coyoneda f a where
    Coyoneda :: (b -> a) -> (f b) -> Coyoneda f a

instance Functor (Coyoneda f) where
    fmap f (Coyoneda g fb) = Coyoneda (f . g) fb

抽象データ型からファンクタを作ることができる。Freeと組み合わせれば手軽にモナド(DSL)が手に入る。

data Proc a = Action1 a | Action2 a | Action3 a

act1 = Free $ Coyoneda id (Action1 (Pure ()))
act2 = Free $ Coyoneda id (Action2 (Pure ()))
act3 = Free $ Coyoneda id (Action3 (Pure ()))

runProc :: Free (Coyoneda Proc) () -> IO ()
runProc (Pure ()) = putStrLn "end"
runProc (Free (Coyoneda f act)) = case act of
    Action1 a -> putStrLn "act1" >> runProc (f a)
    Action2 a -> putStrLn "act2" >> runProc (f a)
    Action3 a -> putStrLn "act3" >> runProc (f a)

proc :: Free (Coyoneda Proc) ()
proc = do
    act1
    act2
    act3

main = runProc proc

newtype Natural f g = Natural (forall x. f x -> g x)

yoneda :: Functor f => f a -> Natural ((->) a) f
yoneda f = Natural $ \ax -> fmap ax f

https://twitter.com/fumieval/status/576349394724765697

Operational Monad

Ideal Monad

Cofree

EDSL

The Expression Problem

Tagless Final

Trees that grow

Free Arrow