Control.Monad.Fix

base-4.12.0.0: Basic libraries

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 | experimental
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 where
  - mfix :: (a -> m a) -> m a
- fix :: (a -> a) -> a

# Documentation

class Monad m => MonadFix m where #

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 #

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

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

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

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

For example, 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`,

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

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