Copyright	(c) Andy Gill 2001 (c) Oregon Graduate Institute of Science and Technology 2002
License	BSD-style (see the file libraries/base/LICENSE)
Maintainer	libraries@haskell.org
Stability	stable
Portability	portable
Safe Haskell	Trustworthy
Language	Haskell2010

Control.Monad.Fix

Description

Monadic fixpoints.

For a detailed discussion, see Levent Erkok's thesis, Value Recursion in Monadic Computations, Oregon Graduate Institute, 2002.

Synopsis

class Monad m => MonadFix (m :: Type -> Type) where
- mfix :: (a -> m a) -> m a
fix :: (a -> a) -> a

Documentation

class Monad m => MonadFix (m :: Type -> Type) where Source #

Monads having fixed points with a 'knot-tying' semantics. Instances of MonadFix should satisfy the following laws:

Purity: mfix (return . h) = return (fix h)
Left shrinking (or Tightening): mfix (\x -> a >>= \y -> f x y) = a >>= \y -> mfix (\x -> f x y)
Sliding: mfix (liftM h . f) = liftM h (mfix (f . h)), for strict h.
Nesting: mfix (\x -> mfix (\y -> f x y)) = mfix (\x -> f x x)

This class is used in the translation of the recursive do notation supported by GHC and Hugs.

Methods

mfix :: (a -> m a) -> m a Source #

The fixed point of a monadic computation. mfix f executes the action f only once, with the eventual output fed back as the input. Hence f should not be strict, for then mfix f would diverge.

Instances

Instances details

MonadFix Complex Source #	Since: base-4.15.0.0
Instance details Defined in Data.Complex Methods mfix :: (a -> Complex a) -> Complex a Source #
MonadFix Identity Source #	Since: base-4.8.0.0
Instance details Defined in Data.Functor.Identity Methods mfix :: (a -> Identity a) -> Identity a Source #
MonadFix First Source #	Since: base-4.8.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> First a) -> First a Source #
MonadFix Last Source #	Since: base-4.8.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Last a) -> Last a Source #
MonadFix Down Source #	Since: base-4.12.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Down a) -> Down a Source #
MonadFix First Source #	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mfix :: (a -> First a) -> First a Source #
MonadFix Last Source #	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mfix :: (a -> Last a) -> Last a Source #
MonadFix Max Source #	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mfix :: (a -> Max a) -> Max a Source #
MonadFix Min Source #	Since: base-4.9.0.0
Instance details Defined in Data.Semigroup Methods mfix :: (a -> Min a) -> Min a Source #
MonadFix Dual Source #	Since: base-4.8.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Dual a) -> Dual a Source #
MonadFix Product Source #	Since: base-4.8.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Product a) -> Product a Source #
MonadFix Sum Source #	Since: base-4.8.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Sum a) -> Sum a Source #
MonadFix NonEmpty Source #	Since: base-4.9.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> NonEmpty a) -> NonEmpty a Source #
MonadFix Par1 Source #	Since: base-4.9.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Par1 a) -> Par1 a Source #
MonadFix IO Source #	Since: base-2.1
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> IO a) -> IO a Source #
MonadFix Maybe Source #	Since: base-2.1
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Maybe a) -> Maybe a Source #
MonadFix Solo Source #	Since: base-4.15
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Solo a) -> Solo a Source #
MonadFix [] Source #	Since: base-2.1
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> [a]) -> [a] Source #
MonadFix (ST s) Source #	Since: base-2.1
Instance details Defined in Control.Monad.ST.Lazy.Imp Methods mfix :: (a -> ST s a) -> ST s a Source #
MonadFix (Either e) Source #	Since: base-4.3.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Either e a) -> Either e a Source #
MonadFix (ST s) Source #	Since: base-2.1
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> ST s a) -> ST s a Source #
MonadFix f => MonadFix (Ap f) Source #	Since: base-4.12.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Ap f a) -> Ap f a Source #
MonadFix f => MonadFix (Alt f) Source #	Since: base-4.8.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Alt f a) -> Alt f a Source #
MonadFix f => MonadFix (Rec1 f) Source #	Since: base-4.9.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> Rec1 f a) -> Rec1 f a Source #
(MonadFix f, MonadFix g) => MonadFix (Product f g) Source #	Since: base-4.9.0.0
Instance details Defined in Data.Functor.Product Methods mfix :: (a -> Product f g a) -> Product f g a Source #
(MonadFix f, MonadFix g) => MonadFix (f :*: g) Source #	Since: base-4.9.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> (f :: g) a) -> (f :: g) a Source #
MonadFix ((->) r) Source #	Since: base-2.1
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> r -> a) -> r -> a Source #
MonadFix f => MonadFix (M1 i c f) Source #	Since: base-4.9.0.0
Instance details Defined in Control.Monad.Fix Methods mfix :: (a -> M1 i c f a) -> M1 i c f a Source #

fix :: (a -> a) -> a Source #

fix f is the least fixed point of the function f, i.e. the least defined x such that f x = x.

When f is strict, this means that because, by the definition of strictness, f ⊥ = ⊥ and such the least defined fixed point of any strict function is ⊥.

Examples

Expand

We can write the factorial function using direct recursion as

>>> let fac n = if n <= 1 then 1 else n * fac (n-1) in fac 5
120

This uses the fact that Haskell’s let introduces recursive bindings. We can rewrite this definition using fix,

Instead of making a recursive call, we introduce a dummy parameter rec; when used within fix, this parameter then refers to fix’s argument, hence the recursion is reintroduced.

>>> fix (\rec n -> if n <= 1 then 1 else n * rec (n-1)) 5
120

Using fix, we can implement versions of repeat as fix . (:) and cycle as fix . (++)

>>> take 10 $ fix (0:)
[0,0,0,0,0,0,0,0,0,0]

>>> map (fix (\rec n -> if n < 2 then n else rec (n - 1) + rec (n - 2))) [1..10]
[1,1,2,3,5,8,13,21,34,55]

Implementation Details

Expand

The current implementation of fix uses structural sharing

fix f = let x = f x in x

A more straightforward but non-sharing version would look like

fix f = f (fix f)