-- Copyright 2009 Mikael Vejdemo Johansson
-- 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))