Introduction
Fixed points are the arguments of the function that it will not change: f x == x. An example would be (\x -> x^2) 1 == 1- here the fixed point is 1.
Attractive fixed points are those fixed points that can be found by iterating from some starting point. For example, it (\x -> x^2) 0.5will converge to 0, so 0 is an attractive fixed point of this function.
Attractive fixed points can, with luck, come up (and, in some cases, even achieved on this set of steps) from a suitable non-fixed point, iterating the function from this point. In other cases, the iteration will diverge, so there must first be evidence that a fixed point will attract the iteration process. For some functions, the proof is well known.
The code
I embellished some prior art that performs the task neatly. Then I decided to extend the same idea to monadic functions, but no luck. This is the code I have:
module Fix where
-- | Take elements from a list until met two equal adjacent elements. Of those,
-- take only the first one, then be done with it.
--
-- This function is intended to operate on infinite lists, but it will still
-- work on finite ones.
converge :: Eq a => [a] -> [a]
converge = convergeBy (==)
-- \ r a = \x -> (x + a / x) / 2
-- \ -- ^ A method of computing square roots due to Isaac Newton.
-- \ take 8 $ iterate (r 2) 1
-- [1.0,1.5,1.4166666666666665,1.4142156862745097,1.4142135623746899,
-- 1.414213562373095,1.414213562373095,1.414213562373095]
-- \ converge $ iterate (r 2) 1
-- [1.0,1.5,1.4166666666666665,1.4142156862745097,1.4142135623746899,1.414213562373095]
-- | Find a fixed point of a function. May present a non-terminating function
-- if applied carelessly!
fixp :: Eq a => (a -> a) -> a -> a
fixp f = last . converge . iterate f
-- \ fixp (r 2) 1
-- 1.414213562373095
-- | Non-overloaded counterpart to `converge`.
convergeBy :: (a -> a -> Bool) -> [a] -> [a]
convergeBy _ [ ] = [ ]
convergeBy _ [x] = [x]
convergeBy eq (x: xs @(y: _))
| x `eq` y = [x]
| otherwise = x : convergeBy eq xs
-- \ convergeBy (\x y -> abs (x - y) < 0.001) $ iterate (r 2) 1
-- [1.0,1.5,1.4166666666666665,1.4142156862745097]
-- | Non-overloaded counterpart to `fixp`.
fixpBy :: (a -> a -> Bool) -> (a -> a) -> a -> a
fixpBy eq f = last . convergeBy eq . iterate f
-- \ fixpBy (\x y -> abs (x - y) < 0.001) (r 2) 1
-- 1.4142156862745097
-- | Find a fixed point of a monadic function. May present a non-terminating
-- function if applied carelessly!
-- TODO
fixpM :: (Eq a, Monad m) => (m a -> m a) -> m a -> m a
fixpM f = last . _ . iterate f
(It can be downloaded at repl. Examples are provided to illustrate.)
Problem
fixpM _. [m a] -> [m a], , converge , . , .
fixpM:
fixpM :: (Eq a, Monad m) => (a -> m a) -> a -> m a
fixpM f x = do
y <- f x
if x == y
then return x
else fixpM f y
-- \ fixpM (\x -> (".", x^2)) 0.5
-- ("............",0.0)
( .)
- , / , . , , inits .
?
?
, , , , , , , , , .
P.S. , . , .
P.S. 2 , @n-m ( iterate), :
fixpM :: (Eq a, Monad m) => (m a -> m a) -> m a -> m a
fixpM f = collapse . iterate f
where
collapse (mx: mxs @(my: _)) = do
x <- mx
y <- my
if x == y
then return x
else collapse mxs
iterate, , . , .
P.S. 3 , @n-m, , , :
fixpM :: (Eq a, Monad m) => (m a -> m a) -> m a -> m a
fixpM f = lastM . convergeM . iterate (f >>= \x -> return x )
convergeM :: (Monad m, Eq a) => [m a] -> m [a]
convergeM = convergeByM (==)
convergeByM :: (Monad m, Eq a) => (a -> a -> Bool) -> [m a] -> m [a]
convergeByM _ [ ] = return [ ]
convergeByM _ [mx] = mx >>= \x -> return [x]
convergeByM eq xs = do
case xs of
[ ] -> return [ ]
[mx] -> mx >>= \x -> return [x]
(mx: mxs @(my: _)) -> do
x <- mx
y <- my
if x `eq` y
then return [x]
else do
xs <- convergeM mxs
return (x:xs)
lastM :: Monad m => m [a] -> m a
lastM mxs = mxs >>= \xs -> case xs of
[] -> error "Fix.lastM: No last element!"
xs -> return . head . reverse $ xs
, . , : .