Safe Haskell	Trustworthy
Language	Haskell2010

Data.Fix

Contents

Fix
Mu - least fixed point
Nu - greatest fixed point
Refolding
Monadic variants
Deprecated aliases

Description

Fixed points of a functor.

Type f should be a Functor if you want to use simple recursion schemes or Traversable if you want to use monadic recursion schemes. This style allows you to express recursive functions in non-recursive manner. You can imagine that a non-recursive function holds values of the previous iteration.

An example:

First we define a base functor. The arguments b are recursion points.

>>> data ListF a b = Nil | Cons a b deriving (Show, Functor)

The list is then a fixed point of ListF

>>> type List a = Fix (ListF a)

We can write length function. Note that the function we give to foldFix is not recursive. Instead the results of recursive calls are in b positions, and we need to deal only with one layer of the structure.

>>> :{
let length :: List a -> Int
    length = foldFix $ \x -> case x of
        Nil      -> 0
        Cons _ n -> n + 1
:}

If you already have recursive type, like '[Int]', you can first convert it to `Fix (ListF a)` and then foldFix. Alternatively you can use recursion-schemes combinators which work directly on recursive types.

Synopsis

newtype Fix f = Fix {
- unFix :: f (Fix f)
}
hoistFix :: Functor f => (forall a. f a -> g a) -> Fix f -> Fix g
hoistFix' :: Functor g => (forall a. f a -> g a) -> Fix f -> Fix g
foldFix :: Functor f => (f a -> a) -> Fix f -> a
unfoldFix :: Functor f => (a -> f a) -> a -> Fix f
wrapFix :: f (Fix f) -> Fix f
unwrapFix :: Fix f -> f (Fix f)
newtype Mu f = Mu {
- unMu :: forall a. (f a -> a) -> a
}
hoistMu :: (forall a. f a -> g a) -> Mu f -> Mu g
foldMu :: (f a -> a) -> Mu f -> a
unfoldMu :: Functor f => (a -> f a) -> a -> Mu f
wrapMu :: Functor f => f (Mu f) -> Mu f
unwrapMu :: Functor f => Mu f -> f (Mu f)
data Nu f = forall a. Nu (a -> f a) a
hoistNu :: (forall a. f a -> g a) -> Nu f -> Nu g
foldNu :: Functor f => (f a -> a) -> Nu f -> a
unfoldNu :: (a -> f a) -> a -> Nu f
wrapNu :: Functor f => f (Nu f) -> Nu f
unwrapNu :: Functor f => Nu f -> f (Nu f)
refold :: Functor f => (f b -> b) -> (a -> f a) -> a -> b
foldFixM :: (Monad m, Traversable t) => (t a -> m a) -> Fix t -> m a
unfoldFixM :: (Monad m, Traversable t) => (a -> m (t a)) -> a -> m (Fix t)
refoldM :: (Monad m, Traversable t) => (t b -> m b) -> (a -> m (t a)) -> a -> m b
cata :: Functor f => (f a -> a) -> Fix f -> a
ana :: Functor f => (a -> f a) -> a -> Fix f
hylo :: Functor f => (f b -> b) -> (a -> f a) -> a -> b
cataM :: (Monad m, Traversable t) => (t a -> m a) -> Fix t -> m a
anaM :: (Monad m, Traversable t) => (a -> m (t a)) -> a -> m (Fix t)
hyloM :: (Monad m, Traversable t) => (t b -> m b) -> (a -> m (t a)) -> a -> m b

Fix

newtype Fix f Source #

A fix-point type.

Constructors

Fix
Fields unFix :: f (Fix f)

Instances

Instances details

(Typeable f, Data (f (Fix f))) => Data (Fix f) Source #
Instance details Defined in Data.Fix Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Fix f -> c (Fix f) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Fix f) # toConstr :: Fix f -> Constr # dataTypeOf :: Fix f -> DataType # dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (Fix f)) # dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Fix f)) # gmapT :: (forall b. Data b => b -> b) -> Fix f -> Fix f # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Fix f -> r # gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Fix f -> r # gmapQ :: (forall d. Data d => d -> u) -> Fix f -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Fix f -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Fix f -> m (Fix f) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Fix f -> m (Fix f) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Fix f -> m (Fix f) #
Generic (Fix f) Source #
Instance details Defined in Data.Fix Associated Types type Rep (Fix f) :: Type -> Type # Methods from :: Fix f -> Rep (Fix f) x # to :: Rep (Fix f) x -> Fix f #
Read1 f => Read (Fix f) Source #
Instance details Defined in Data.Fix Methods readsPrec :: Int -> ReadS (Fix f) # readList :: ReadS [Fix f] # readPrec :: ReadPrec (Fix f) # readListPrec :: ReadPrec [Fix f] #
Show1 f => Show (Fix f) Source #
Instance details Defined in Data.Fix Methods showsPrec :: Int -> Fix f -> ShowS # show :: Fix f -> String # showList :: [Fix f] -> ShowS #
NFData1 f => NFData (Fix f) Source #
Instance details Defined in Data.Fix Methods rnf :: Fix f -> () #
Eq1 f => Eq (Fix f) Source #
Instance details Defined in Data.Fix Methods (==) :: Fix f -> Fix f -> Bool # (/=) :: Fix f -> Fix f -> Bool #
Ord1 f => Ord (Fix f) Source #
Instance details Defined in Data.Fix Methods compare :: Fix f -> Fix f -> Ordering # (<) :: Fix f -> Fix f -> Bool # (<=) :: Fix f -> Fix f -> Bool # (>) :: Fix f -> Fix f -> Bool # (>=) :: Fix f -> Fix f -> Bool # max :: Fix f -> Fix f -> Fix f # min :: Fix f -> Fix f -> Fix f #
Hashable1 f => Hashable (Fix f) Source #
Instance details Defined in Data.Fix Methods hashWithSalt :: Int -> Fix f -> Int # hash :: Fix f -> Int #
type Rep (Fix f) Source #
Instance details Defined in Data.Fix type Rep (Fix f) = D1 ('MetaData "Fix" "Data.Fix" "data-fix-0.3.3-1QdpNFZfaL0IO3DNDPBIWQ" 'True) (C1 ('MetaCons "Fix" 'PrefixI 'True) (S1 ('MetaSel ('Just "unFix") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 (f (Fix f)))))

hoistFix :: Functor f => (forall a. f a -> g a) -> Fix f -> Fix g Source #

Change base functor in Fix.

hoistFix' :: Functor g => (forall a. f a -> g a) -> Fix f -> Fix g Source #

Like hoistFix but fmapping over g.

foldFix :: Functor f => (f a -> a) -> Fix f -> a Source #

Fold Fix.

>>> let fp = unfoldFix (\i -> if i < 4 then Cons i (i + 1) else Nil) (0 :: Int)
>>> foldFix (elimListF 0 (+)) fp
6

unfoldFix :: Functor f => (a -> f a) -> a -> Fix f Source #

Unfold Fix.

>>> unfoldFix (\i -> if i < 4 then Cons i (i + 1) else Nil) (0 :: Int)
Fix (Cons 0 (Fix (Cons 1 (Fix (Cons 2 (Fix (Cons 3 (Fix Nil))))))))

wrapFix :: f (Fix f) -> Fix f Source #

Wrap Fix.

>>> let x = unfoldFix (\i -> if i < 3 then Cons i (i + 1) else Nil) (0 :: Int)
>>> wrapFix (Cons 10 x)
Fix (Cons 10 (Fix (Cons 0 (Fix (Cons 1 (Fix (Cons 2 (Fix Nil))))))))

Since: 0.3.2

unwrapFix :: Fix f -> f (Fix f) Source #

Unwrap Fix.

>>> let x = unfoldFix (\i -> if i < 3 then Cons i (i + 1) else Nil) (0 :: Int)
>>> unwrapFix x
Cons 0 (Fix (Cons 1 (Fix (Cons 2 (Fix Nil)))))

Since: 0.3.2

Mu - least fixed point

newtype Mu f Source #

Least fixed point. Efficient folding.

Constructors

Mu
Fields unMu :: forall a. (f a -> a) -> a

Instances

Instances details

(Functor f, Read1 f) => Read (Mu f) Source #
Instance details Defined in Data.Fix Methods readsPrec :: Int -> ReadS (Mu f) # readList :: ReadS [Mu f] # readPrec :: ReadPrec (Mu f) # readListPrec :: ReadPrec [Mu f] #
(Functor f, Show1 f) => Show (Mu f) Source #
Instance details Defined in Data.Fix Methods showsPrec :: Int -> Mu f -> ShowS # show :: Mu f -> String # showList :: [Mu f] -> ShowS #
(Functor f, Eq1 f) => Eq (Mu f) Source #
Instance details Defined in Data.Fix Methods (==) :: Mu f -> Mu f -> Bool # (/=) :: Mu f -> Mu f -> Bool #
(Functor f, Ord1 f) => Ord (Mu f) Source #
Instance details Defined in Data.Fix Methods compare :: Mu f -> Mu f -> Ordering # (<) :: Mu f -> Mu f -> Bool # (<=) :: Mu f -> Mu f -> Bool # (>) :: Mu f -> Mu f -> Bool # (>=) :: Mu f -> Mu f -> Bool # max :: Mu f -> Mu f -> Mu f # min :: Mu f -> Mu f -> Mu f #

hoistMu :: (forall a. f a -> g a) -> Mu f -> Mu g Source #

Change base functor in Mu.

foldMu :: (f a -> a) -> Mu f -> a Source #

Fold Mu.

>>> let mu = unfoldMu (\i -> if i < 4 then Cons i (i + 1) else Nil) (0 :: Int)
>>> foldMu (elimListF 0 (+)) mu
6

unfoldMu :: Functor f => (a -> f a) -> a -> Mu f Source #

Unfold Mu.

>>> unfoldMu (\i -> if i < 4 then Cons i (i + 1) else Nil) (0 :: Int)
unfoldMu unFix (Fix (Cons 0 (Fix (Cons 1 (Fix (Cons 2 (Fix (Cons 3 (Fix Nil)))))))))

wrapMu :: Functor f => f (Mu f) -> Mu f Source #

Wrap Mu.

>>> let x = unfoldMu (\i -> if i < 3 then Cons i (i + 1) else Nil) (0 :: Int)
>>> wrapMu (Cons 10 x)
unfoldMu unFix (Fix (Cons 10 (Fix (Cons 0 (Fix (Cons 1 (Fix (Cons 2 (Fix Nil)))))))))

Since: 0.3.2

unwrapMu :: Functor f => Mu f -> f (Mu f) Source #

Unwrap Mu.

>>> let x = unfoldMu (\i -> if i < 3 then Cons i (i + 1) else Nil) (0 :: Int)
>>> unwrapMu x
Cons 0 (unfoldMu unFix (Fix (Cons 1 (Fix (Cons 2 (Fix Nil))))))

Since: 0.3.2

Nu - greatest fixed point

data Nu f Source #

Greatest fixed point. Efficient unfolding.

Constructors

forall a. Nu (a -> f a) a

Instances

Instances details

(Functor f, Read1 f) => Read (Nu f) Source #
Instance details Defined in Data.Fix Methods readsPrec :: Int -> ReadS (Nu f) # readList :: ReadS [Nu f] # readPrec :: ReadPrec (Nu f) # readListPrec :: ReadPrec [Nu f] #
(Functor f, Show1 f) => Show (Nu f) Source #
Instance details Defined in Data.Fix Methods showsPrec :: Int -> Nu f -> ShowS # show :: Nu f -> String # showList :: [Nu f] -> ShowS #
(Functor f, Eq1 f) => Eq (Nu f) Source #
Instance details Defined in Data.Fix Methods (==) :: Nu f -> Nu f -> Bool # (/=) :: Nu f -> Nu f -> Bool #
(Functor f, Ord1 f) => Ord (Nu f) Source #
Instance details Defined in Data.Fix Methods compare :: Nu f -> Nu f -> Ordering # (<) :: Nu f -> Nu f -> Bool # (<=) :: Nu f -> Nu f -> Bool # (>) :: Nu f -> Nu f -> Bool # (>=) :: Nu f -> Nu f -> Bool # max :: Nu f -> Nu f -> Nu f # min :: Nu f -> Nu f -> Nu f #

hoistNu :: (forall a. f a -> g a) -> Nu f -> Nu g Source #

Change base functor in Nu.

foldNu :: Functor f => (f a -> a) -> Nu f -> a Source #

Fold Nu.

>>> let nu = unfoldNu (\i -> if i < 4 then Cons i (i + 1) else Nil) (0 :: Int)
>>> foldNu (elimListF 0 (+)) nu
6

unfoldNu :: (a -> f a) -> a -> Nu f Source #

Unfold Nu.

>>> unfoldNu (\i -> if i < 4 then Cons i (i + 1) else Nil) (0 :: Int)
unfoldNu unFix (Fix (Cons 0 (Fix (Cons 1 (Fix (Cons 2 (Fix (Cons 3 (Fix Nil)))))))))

wrapNu :: Functor f => f (Nu f) -> Nu f Source #

Wrap Nu.

>>> let x = unfoldNu (\i -> if i < 3 then Cons i (i + 1) else Nil) (0 :: Int)
>>> wrapNu (Cons 10 x)
unfoldNu unFix (Fix (Cons 10 (Fix (Cons 0 (Fix (Cons 1 (Fix (Cons 2 (Fix Nil)))))))))

Since: 0.3.2

unwrapNu :: Functor f => Nu f -> f (Nu f) Source #

Unwrap Nu.

>>> let x = unfoldNu (\i -> if i < 3 then Cons i (i + 1) else Nil) (0 :: Int)
>>> unwrapNu x
Cons 0 (unfoldNu unFix (Fix (Cons 1 (Fix (Cons 2 (Fix Nil))))))

Since: 0.3.2

Refolding

refold :: Functor f => (f b -> b) -> (a -> f a) -> a -> b Source #

Refold one recursive type into another, one layer at the time.

Monadic variants

foldFixM :: (Monad m, Traversable t) => (t a -> m a) -> Fix t -> m a Source #

Monadic foldFix.

unfoldFixM :: (Monad m, Traversable t) => (a -> m (t a)) -> a -> m (Fix t) Source #

Monadic anamorphism.

refoldM :: (Monad m, Traversable t) => (t b -> m b) -> (a -> m (t a)) -> a -> m b Source #

Monadic hylomorphism.

Deprecated aliases

cata :: Functor f => (f a -> a) -> Fix f -> a Source #

Deprecated: Use foldFix

Catamorphism or generic function fold.

ana :: Functor f => (a -> f a) -> a -> Fix f Source #

Deprecated: Use unfoldFix

Anamorphism or generic function unfold.

hylo :: Functor f => (f b -> b) -> (a -> f a) -> a -> b Source #

Deprecated: Use refold

Hylomorphism is anamorphism followed by catamorphism.

cataM :: (Monad m, Traversable t) => (t a -> m a) -> Fix t -> m a Source #

Deprecated: Use foldFixM

Monadic catamorphism.

anaM :: (Monad m, Traversable t) => (a -> m (t a)) -> a -> m (Fix t) Source #

Deprecated: Use unfoldFixM

Monadic anamorphism.

hyloM :: (Monad m, Traversable t) => (t b -> m b) -> (a -> m (t a)) -> a -> m b Source #

Deprecated: Use refoldM

Monadic hylomorphism.