-- Copyright 2009 Mikael Vejdemo Johansson <mik@stanford.edu> -- Released under a BSD license -- | Implements decorated and ordered trees, and most operations acting only on these. -- The decorated tree is the most fundamental part of the implementation, and the internal -- structure of the tree is to the largest extent what determines the actual identity -- of a given element of the free operad. module Math.Operad.OrderedTree where import Prelude hiding (mapM) import Data.Foldable (Foldable, foldMap) import Data.Traversable import Data.List (sort, sortBy, intersperse, nub, findIndices) import Control.Applicative import Data.Ord import Control.Monad.State hiding (mapM) import Data.Monoid import Math.Operad.PPrint -- * Decorated and ordered trees -- | The fundamental tree data type used. Leaves carry labels - most often integral - -- and these are expected to control, e.g., composition points in shuffle operad compositions. -- The vertices carry labels, used for the ordering on trees and to distinguish different -- basis corollas of the same arity. data (Ord a, Show a) => PreDecoratedTree a b = DTLeaf !b | DTVertex { vertexType :: !a, subTrees :: ![PreDecoratedTree a b]} deriving (Eq, Ord, Read, Show) -- | The arity of a corolla vertexArity :: (Ord a, Show a) => PreDecoratedTree a b -> Int vertexArity t = length (subTrees t) instance (Ord a, Show a) => Functor (PreDecoratedTree a) where fmap f (DTLeaf b) = DTLeaf (f b) fmap f (DTVertex t ts) = DTVertex t (map (fmap f) ts) instance (Ord a, Show a) => Foldable (PreDecoratedTree a) where foldMap f (DTLeaf b) = f b foldMap f (DTVertex _ ts) = mconcat (map (foldMap f) ts) instance (Ord a, Show a) => Traversable (PreDecoratedTree a) where traverse f (DTLeaf b) = DTLeaf <$> f b traverse f (DTVertex t ts) = DTVertex t <$> traverse (traverse f) ts instance (Ord a, Show a, Show b) => PPrint (PreDecoratedTree a b) where pp (DTLeaf x) = show x pp (DTVertex t ts) = "m" ++ show t ++ "(" ++ concat (intersperse "," (map pp ts)) ++ ")" -- | Apply a function @f@ to all the internal vertex labels of a PreDecoratedTree. vertexMap :: (Ord a, Show a, Ord b, Show b) => (a -> b) -> PreDecoratedTree a c -> PreDecoratedTree b c vertexMap _ (DTLeaf i) = DTLeaf i vertexMap f (DTVertex t ts) = DTVertex (f t) (map (vertexMap f) ts) -- | If a tree has trees as labels for its leaves, we can replace the leaves with the roots of -- those label trees. Thus we may glue together trees, as required by the compositions. glueTrees :: (Ord a, Show a) => PreDecoratedTree a (PreDecoratedTree a b) -> PreDecoratedTree a b glueTrees (DTLeaf newRoot) = newRoot glueTrees (DTVertex t ts) = DTVertex t (map glueTrees ts) instance (Ord a, Show a) => Monad (PreDecoratedTree a) where return = DTLeaf; a >>= f = glueTrees (fmap f a) -- | This is the fundamental datatype of the whole project. Monomials in a free operad -- are decorated trees, and we build a type for decorated trees here. We require our -- trees to have Int labels, limiting us to at most 2 147 483 647 leaf labels. type DecoratedTree a = PreDecoratedTree a Int -- | Monomial orderings on the free operad requires us to choose an ordering for the -- trees. These are parametrized by types implementing the type class 'TreeOrdering', -- and this is a data type for a tree carrying its comparison type. We call these -- /ordered trees/. data (Ord a, Show a, TreeOrdering t) => OrderedTree a t = OT (DecoratedTree a) t deriving (Eq, Show, Read) instance (Ord a, Show a, TreeOrdering t) => PPrint (OrderedTree a t) where pp (OT dect _) = pp dect -- | Monomial ordering for trees. We require this to be a total well-ordering, compatible -- with the operadic compositions. instance (Ord a, Show a, TreeOrdering t) => Ord (OrderedTree a t) where compare (OT s o1) (OT t _) = treeCompare o1 s t -- | Building an ordered tree with 'PathLex' ordering from a decorated tree. ot :: (Ord a, Show a, TreeOrdering t) => DecoratedTree a -> OrderedTree a t ot t = OT t ordering -- | Extracting the underlying tree from an ordered tree. dt :: (Ord a, Show a, TreeOrdering t) => OrderedTree a t -> DecoratedTree a dt (OT t _) = t -- ** Monomial orderings on the free operad -- | The type class that parametrizes types implementing tree orderings. class (Eq t, Show t) => TreeOrdering t where treeCompare :: (Ord a, Show a) => t -> DecoratedTree a -> DecoratedTree a -> Ordering treeCompare o t1 t2 = comparePathSequence o t1 (orderedPathSequence t1) t2 (orderedPathSequence t2) comparePathSequence :: (Ord a, Show a) => t -> DecoratedTree a -> ([[a]],Shuffle) -> DecoratedTree a -> ([[a]],Shuffle) -> Ordering ordering :: t -- | Finding the path sequences. cf. Dotsenko-Khoroshkin. pathSequence :: (Ord a, Show a) => DecoratedTree a -> ([[a]],Shuffle) pathSequence (DTLeaf j) = ([[]],[j]) pathSequence (DTVertex l ts) = let pathSequences = map pathSequence $! ts paths = concatMap fst $! pathSequences leaves = concatMap snd $! pathSequences in (map (l:) $! paths, leaves) -- | Reordering the path sequences to mirror the actual leaf ordering. orderedPathSequence :: (Ord a, Show a) => DecoratedTree a -> ([[a]],Shuffle) orderedPathSequence t = (map fst . sortBy (comparing snd) $ zip ps1 ps2, ps2) where (ps1, ps2) = pathSequence t -- | Changes direction of an ordering. reverseOrder :: Ordering -> Ordering reverseOrder LT = GT reverseOrder GT = LT reverseOrder EQ = EQ -- | Using the path sequence, the leaf orders and order reversal, we can get 8 different orderings -- from one paradigm. These are given by 'PathPerm', 'RPathPerm', 'PathRPerm', 'RPathRPerm' for the -- variations giving (possibly reversed) path sequence comparison precedence over (possibly reversed) -- leaf permutations; additionally, there are 'PermPath', 'RPermPath', 'PermRPath' and 'RPermRPath' -- for the variations with the opposite precedence. data PathPerm = PathPerm deriving (Eq, Ord, Show, Read) instance TreeOrdering PathPerm where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cs = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else comp1) clS coS in if any (/= EQ) cs then head (filter (/=EQ) cs) else compare perms permt ordering = PathPerm data RPathPerm = RPathPerm deriving (Eq, Ord, Show, Read) instance TreeOrdering RPathPerm where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cS = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else reverseOrder comp1) clS coS in if any (/= EQ) cS then head (filter (/=EQ) cS) else compare perms permt ordering = RPathPerm data PathRPerm = PathRPerm deriving (Eq, Ord, Show, Read) instance TreeOrdering PathRPerm where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cs = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else comp1) clS coS in if any (/= EQ) cs then head (filter (/=EQ) cs) else reverseOrder $ compare perms permt ordering = PathRPerm data RPathRPerm = RPathRPerm deriving (Eq, Ord, Show, Read) instance TreeOrdering RPathRPerm where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cS = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else reverseOrder comp1) clS coS in if any (/= EQ) cS then head (filter (/=EQ) cS) else reverseOrder $ compare perms permt ordering = RPathRPerm data PermPath = PermPath deriving (Eq, Ord, Show, Read) instance TreeOrdering PermPath where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cs = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else comp1) clS coS test1 = compare perms permt in if test1 /= EQ then test1 else if any (/= EQ) cs then head (filter (/=EQ) cs) else EQ ordering = PermPath data PermRPath = PermRPath deriving (Eq, Ord, Show, Read) instance TreeOrdering PermRPath where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cS = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else reverseOrder comp1) clS coS test1 = compare perms permt in if test1 /= EQ then test1 else if any (/= EQ) cS then head (filter (/=EQ) cS) else EQ ordering = PermRPath data RPermPath = RPermPath deriving (Eq, Ord, Show, Read) instance TreeOrdering RPermPath where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cs = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else comp1) clS coS test1 = reverseOrder $ compare perms permt in if test1 /= EQ then test1 else if any (/= EQ) cs then head (filter (/=EQ) cs) else EQ ordering = RPermPath data RPermRPath = RPermRPath deriving (Eq, Ord, Show, Read) instance TreeOrdering RPermRPath where treeCompare o s t = if (nLeaves s) /= (nLeaves t) then comparing nLeaves s t else if s == t then EQ else comparePathSequence o s (orderedPathSequence s) t (orderedPathSequence t) comparePathSequence _ _ (paths,perms) _ (patht,permt) = let clS = zipWith (comparing length) paths patht coS = zipWith compare paths patht cS = zipWith (\comp1 comp2 -> if comp1 == EQ then comp2 else reverseOrder comp1) clS coS test1 = reverseOrder $ compare perms permt in if test1 /= EQ then test1 else if any (/= EQ) cS then head (filter (/=EQ) cS) else EQ ordering = RPermRPath -- ** Utility functions on trees -- -- Trees are represented rooted, and all operations act on a specific root, and may recurse from there. -- | Build a single corolla in a decorated tree. Takes a list for labels for the leaves, and derives -- the arity of the corolla from those. This, and the composition functions, form the preferred method -- to construct trees. corolla :: (Ord a, Show a) => a -> [Int] -> DecoratedTree a corolla label leaflabels = if null leaflabels then error "The operadic Buchberger, and many other algorithms, require the absence of 0-ary operations." else DTVertex label (map DTLeaf leaflabels) -- | Build a single leaf. leaf :: (Ord a, Show a) => Int -> DecoratedTree a leaf n = DTLeaf n -- | Check whether a given root is a leaf. isLeaf :: (Ord a, Show a) => DecoratedTree a -> Bool isLeaf (DTLeaf _) = True isLeaf _ = False -- | Check whether a given root is a corolla. isCorolla :: (Ord a, Show a) => DecoratedTree a -> Bool isCorolla = not . isLeaf -- | Change the leaves of a tree to take their values from a given list. relabelLeaves :: (Ord a, Show a) => DecoratedTree a -> [b] -> PreDecoratedTree a b relabelLeaves tree newLabels = fst $ runState (mapM (\_ -> do; (x:xs) <- get; put xs; return x) tree) newLabels -- | Find the permutation the leaf labeling ordains for inputs. leafOrder :: (Ord a, Show a) => DecoratedTree a -> [Int] leafOrder = foldMap (:[]) -- | Find the minimal leaf covering any given vertex. minimalLeaf :: (Ord a, Show a, Ord b) => PreDecoratedTree a b -> b minimalLeaf (DTLeaf lbl) = lbl minimalLeaf vertex = minimum $ map minimalLeaf (subTrees vertex) -- | Compute the number of leaves of the entire tree covering a given vertex. nLeaves :: (Ord a, Show a) => DecoratedTree a -> Int nLeaves (DTLeaf _) = 1 nLeaves vertex = sum $ map nLeaves (subTrees vertex) -- | 'arityDegree' is one less than 'nLeaves'. arityDegree :: (Ord a, Show a) => DecoratedTree a -> Int arityDegree t = nLeaves t - 1 -- * Shuffles -- Basic handling functions for building, recognizing and applying shuffle permutations. -- | A shuffle is a special kind of sequence of integers. type Shuffle = [Int] -- | We need to recognize sorted sequences of integers. isSorted :: (Ord a, Show a) => [a] -> Bool isSorted xs = and $ zipWith (<=) xs (tail xs) -- | This tests whether a given sequence of integers really is a shuffle. isShuffle :: Shuffle -> Bool isShuffle as = (sort as) == [1..length as] && any (\k -> isShuffleIJ as k (length as-k)) [1..length as] -- | This tests whether a given sequence of integers is an (i,j)-shuffle isShuffleIJ :: Shuffle -> Int -> Int -> Bool isShuffleIJ as i j = isSorted [a | a <- as, a <= i] && isSorted [a | a <- as, a > i] && length as == i+j -- | This tests whether a given sequence of integers is admissible for a specific composition operation. isShuffleIPQ :: Shuffle -> Int -> Int -> Bool isShuffleIPQ as i p = let initSegment = take i as finalSegment = drop i as upperTree = take p finalSegment latterTree = drop p finalSegment in initSegment == [1 .. length initSegment] && isSorted upperTree && isSorted latterTree -- | This applies the resulting permutation from a shuffle to a set of elements applyPerm :: Show a => Shuffle -> [a] -> [a] applyPerm s is = --trace ("applyPerm " ++ show s ++ " " ++ show (length is) ++ "\n") $ map (is!!) (map (subtract 1) s) -- | Apply the permutation inversely to 'applyPerm'. invApplyPerm :: Shuffle -> [a] -> [a] invApplyPerm sh lst = map snd . sortBy (comparing fst) $ zip sh lst -- | Generate all subsets of length k from a given list. kSubsets :: Int -> [Int] -> [[Int]] kSubsets 0 _ = [[]] kSubsets k ss = if length ss == k then [ss] else if length ss < k then [] else do map sort $ (map ((head ss):) $ kSubsets (k-1) (tail ss)) ++ kSubsets k (tail ss) -- | Applies @f@ only at the @n@th place in a list. applyAt :: (a -> a) -> Int -> [a] -> [a] applyAt f n as = take n as ++ [f (as !! n)] ++ drop (n+1) as -- | Picks out the last nonzero entry in a list. lastNonzero :: (Num a) => [a] -> Int lastNonzero as = let ras = reverse as dwz = dropWhile (==0) ras lnzi = length dwz - 1 in lnzi -- | Generates shuffle permutations by filling buckets. allShPerm :: Int -> [Int] -> [[[Int]]] allShPerm 0 as = [replicate (length as) []] allShPerm n as = do let lastIndex = filter (>=0) [lastNonzero as] indices = nub $ (findIndices (>1) as) ++ lastIndex i <- indices p <- allShPerm (n-1) (applyAt (subtract 1) i as) return (applyAt (++[n]) i p) -- | Generates all shuffles from Sh_i(p,q). allShuffles :: Int -> Int -> Int -> [Shuffle] allShuffles i p q = map concat $ allShPerm (i+p+q) ((replicate (i-1) 1) ++ [p+1] ++ (replicate q 1))