Subject: traversal transformations Newsgroups: gmane.comp.lang.haskell.cafe Date: Sunday 14th June 2009 23:29:04 UTC (over 8 years ago) Hi, While playing with Church Encodings of data structures, I realized there are generalisations in the same way Data.Foldable and Data.Traversable are generalisations of lists. The normal Church Encoding of lists is like this: > newtype List a = L { unL :: forall b. (a -> b -> b) -> b -> b } It represents a list by a right fold: > foldr f z l = unL l f z List can be constructed with cons and nil: > nil = L $ \f -> id > cons a l = L $ \f -> f a . unL l f Oleg has written about this: http://okmij.org/ftp/Haskell/zip-folds.lhs Now function of type (b -> b) are endomorphisms which have a Data.Monoid instance, so the type can be generalized: > newtype FM a = FM { unFM :: forall b. Monoid b => (a -> b) -> b } > fmnil = FM $ \f -> mempty > fmcons a l = FM $ \f -> f a `mappend` unFM l f Now lists are represented by (almost) their foldMap function: > instance Foldable FM where > foldMap = flip unFM But notice that there is now nothing list specific in the FM type, nothing prevents us to add other constructor functions. > fmsnoc l a = FM $ \f -> unFM l f `mappend` f a > fmlist = fmcons 2 $ fmcons 3 $ fmnil `fmsnoc` 4 `fmsnoc` 5 *Main> getProduct $ foldMap Product fmlist 120 Now that we have a container type represented by foldMap, there's nothing stopping us to do a container type represented by traverse from Data.Traversable: {-# LANGUAGE RankNTypes #-} import Data.Monoid import Data.Foldable import Data.Traversable import Control.Monad import Control.Applicative newtype Container a = C { travC :: forall f b . Applicative f => (a -> f b) -> f (Container b) } czero :: Container a cpure :: a -> Container a ccons :: a -> Container a -> Container a csnoc :: Container a -> a -> Container a cpair :: Container a -> Container a -> Container a cnode :: Container a -> a -> Container a -> Container a ctree :: a -> Container (Container a) -> Container a cflat :: Container (Container a) -> Container a czero = C $ \f -> pure czero cpure x = C $ \f -> cpure <$> f x ccons x l = C $ \f -> ccons <$> f x <*> travC l f csnoc l x = C $ \f -> csnoc <$> travC l f <*> f x cpair l r = C $ \f -> cpair <$> travC l f <*> travC r f cnode l x r = C $ \f -> cnode <$> travC l f <*> f x <*> travC r f ctree x l = C $ \f -> ctree <$> f x <*> travC l (traverse f) cflat l = C $ \f -> cflat <$> travC l (traverse f) instance Functor Container where fmap g c = C $ \f -> travC c (f . g) instance Foldable Container where foldMap = foldMapDefault instance Traversable Container where traverse = flip travC instance Monad Container where return = cpure m >>= f = cflat $ fmap f m instance Monoid (Container a) where mempty = czero mappend = cpair Note that there are all kinds of "constructors", and they can all be combined. Writing their definitions is similar to how you would write Traversable instances. So I'm not sure what we have here, as I just ran into it, I wasn't looking for a solution to a problem. It is also all quite abstract, and I'm not sure I understand what is going on everywhere. Is this useful? Has this been done before? Are there better implementations of foldMap and (>>=) for Container? Finally, a little example. A Show instance (for debugging purposes) which shows the nesting structure. newtype ShowContainer a = ShowContainer { doShowContainer :: String } instance Functor ShowContainer where fmap _ (ShowContainer x) = ShowContainer $ "(" ++ x ++ ")" instance Applicative ShowContainer where pure _ = ShowContainer "()" ShowContainer l <*> ShowContainer r = ShowContainer $ init l ++ "," ++ r ++ ")" instance Show a => Show (Container a) where show = doShowContainer . traverse (ShowContainer . show) greetings, -- Sjoerd Visscher [email protected] |
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