Intro
Fixed points are such arguments to a function that it would return unchanged: 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 iteration from some starting point. For example, (\x -> x^2) 0.5 would converge to 0, thus 0 is an attractive fixed point of this function.
Attractive fixed points can be, with luck, approached (and, in some cases, even reached in that many steps) from a suitable non-fixed point by iterating the function from that point. Other times, the iteration will diverge, so there should first be a proof in place that a fixed point will attract the iterating process. For some functions, the proof is common knowledge.
The code
I have tidied up some prior art that accomplishes the task neatly. I then set out to extend the same idea to monadic functions, but to no luck. This is the code I have by now:
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 may be loaded in repl. There are examples to be run in the comments, for illustration.)
The problem
There is an _ in the definition of fixpM above. It is a function of type [m a] -> [m a] that should do, in principle, the same as the function converge above, but kinda lifted. I have come to suspect it can't be written.
I do have composed another, specialized code for 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)
(An example run is, again, found in a comment.)
-- But it is a whole different algorithm, not an extension / generalization of the pure function we started with. In particular, we do not pass the stage where a list of inits up to the first repetition is made available.
Can we not extend the pure algorithm to work on monadic functions?
And why so?
I would admire a hint towards a piece of theory that explains how to either prove impossibility or construct a solution in a routine fashion, but perhaps this is just a triviality I'm missing while busy typing idle questions, in which case a straightforward counterexample would defeat me.
P.S. I understand this is a somewhat trivial exercise. Still, I want to have become done with it once and forever.
P.S. 2 A better approximation to the pure variant, as suggested by @n-m (retaining iterate), would look like this:
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
Through the use of iterate, its behaviour with regard to the monad is different in that the effects are retained between consecutive approximations. Performance-wise, these functions are of the same complexity.
P.S. 3 A more complete rendition of the ideas offered by @n-m encodes the algorithm, as far as I can see, one to one with the pure variant:
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
Unfortunately, it happens to be rather lengthy. More substantially, both these solutions have the same somewhat undesirable behaviour with regard to the effects of the monad: all the effects are retained between consecutive approximations.
