Control.Monad.Omega

A monad for enumerating sets: like the list monad, but impervious to infinite descent.

A depth-first search of a data structure can fail to give a full traversal if it has an infinitely deep path. Likewise, a breadth-first search of a data structure can fall short if it has an infinitely branching node. Omega addresses this problem by using a "diagonal" traversal that gracefully dissolves such data.

So while liftM2 (,) [0..] [0..] gets "stuck" generating tuples whose first element is zero, "runOmega" $ liftM2 (,) ("each" [0..]) ("each" [0..]) generates all pairs of naturals.

More precisely, if x appears at a finite index in xs, and y appears at a finite index in f x, then y will appear at a finite index in each xs >>= f.

This monad gets its name because it is a monad over sets of order type omega.

Warning: Omega is only a monad when the results of runOmega are interpreted as a set; that is, a valid transformation according to the monad laws may change the order of the results. However, the same set of results will always be reachable. If you are using this as a monad, I recommend that you use the newer weighted-search package instead (it's also faster).

Documentation

diagonal :: [[a]] -> [a]

This is the hinge algorithm of the Omega monad, exposed because it can be useful on its own. Joins a list of lists with the property that for every i j there is an n such that xs !! i !! j == diagonal xs !! n. In particular, n <= (i+j)*(i+j+1)/2 + j.

data Omega a

Instances

Monad Omega

Defined in Control.Monad.Omega

Methods

(>>=) :: Omega a -> (a -> Omega b) -> Omega b

(>>) :: Omega a -> Omega b -> Omega b

return :: a -> Omega a

fail :: String -> Omega a

Functor Omega

Defined in Control.Monad.Omega

Methods

fmap :: (a -> b) -> Omega a -> Omega b

(<$) :: a -> Omega b -> Omega a

MonadFail Omega

Defined in Control.Monad.Omega

Methods

fail :: String -> Omega a

Applicative Omega

Defined in Control.Monad.Omega

Methods

pure :: a -> Omega a

(<*>) :: Omega (a -> b) -> Omega a -> Omega b

liftA2 :: (a -> b -> c) -> Omega a -> Omega b -> Omega c

(*>) :: Omega a -> Omega b -> Omega b

(<*) :: Omega a -> Omega b -> Omega a

Foldable Omega

Defined in Control.Monad.Omega

Methods

fold :: Monoid m => Omega m -> m

foldMap :: Monoid m => (a -> m) -> Omega a -> m

foldr :: (a -> b -> b) -> b -> Omega a -> b

foldr' :: (a -> b -> b) -> b -> Omega a -> b

foldl :: (b -> a -> b) -> b -> Omega a -> b

foldl' :: (b -> a -> b) -> b -> Omega a -> b

foldr1 :: (a -> a -> a) -> Omega a -> a

foldl1 :: (a -> a -> a) -> Omega a -> a

toList :: Omega a -> [a]

null :: Omega a -> Bool

length :: Omega a -> Int

elem :: Eq a => a -> Omega a -> Bool

maximum :: Ord a => Omega a -> a

minimum :: Ord a => Omega a -> a

sum :: Num a => Omega a -> a

product :: Num a => Omega a -> a

Traversable Omega

Defined in Control.Monad.Omega

Methods

traverse :: Applicative f => (a -> f b) -> Omega a -> f (Omega b)

sequenceA :: Applicative f => Omega (f a) -> f (Omega a)

mapM :: Monad m => (a -> m b) -> Omega a -> m (Omega b)

sequence :: Monad m => Omega (m a) -> m (Omega a)

Alternative Omega

Defined in Control.Monad.Omega

Methods

empty :: Omega a

(<|>) :: Omega a -> Omega a -> Omega a

some :: Omega a -> Omega [a]

many :: Omega a -> Omega [a]

MonadPlus Omega

Defined in Control.Monad.Omega

Methods

mzero :: Omega a

mplus :: Omega a -> Omega a -> Omega a

runOmega :: Omega a -> [a]

each :: [a] -> Omega a