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1 files changed, 159 insertions, 220 deletions
diff --git a/src/Lib.hs b/src/Lib.hs
index 0e093e3..7bd27d2 100644
--- a/src/Lib.hs
+++ b/src/Lib.hs
@@ -1,88 +1,85 @@
{-# LANGUAGE TupleSections #-}
-{-# LANGUAGE BangPatterns #-}
module Lib where
-import Data.Char ( chr
- , ord
- )
-import qualified Data.List as L
-import qualified Data.Map as M
-import Data.Maybe ( fromJust )
-import qualified Data.Set as S
-import Debug.Trace ( traceShowId
- , traceShow
- )
-import Parser
-import qualified Text.Parsec as Parsec
+import qualified Data.List as L
+import qualified Data.Map as M
+import Data.Maybe (fromJust)
+import qualified Data.Set as S
+import Parser
+ ( Addend (..),
+ Matrix (..),
+ QN (Neg, Qua),
+ QNMFormula (..),
+ Quantifier (Exists, ForAll),
+ )
mget :: (Ord a) => a -> M.Map a b -> b
mget a m = fromJust $ M.lookup a m
automatizem :: M.Map Addend Int -> Matrix -> DFA [Int] Int
automatizem dir (MEq as bs) =
- eq (fromIntegral $ length dir) (map (`mget` dir) as) (map (`mget` dir) bs)
-automatizem dir (MLe a b) = le (mget a dir) (mget b dir)
-automatizem dir (MLt a b) = lt (mget a dir) (mget b dir)
+ eq (length dir) (map (`mget` dir) as) (map (`mget` dir) bs)
+automatizem dir (MLe a b) = le (mget a dir) (mget b dir)
+automatizem dir (MLt a b) = lt (mget a dir) (mget b dir)
automatizem dir (MAnd x y) = conj (automatizem dir x) (automatizem dir y)
-automatizem dir (MOr x y) = disj (automatizem dir x) (automatizem dir y)
-automatizem dir (MNot x ) = compl (automatizem dir x)
+automatizem dir (MOr x y) = disj (automatizem dir x) (automatizem dir y)
+automatizem dir (MNot x) = compl (automatizem dir x)
automatize :: M.Map Addend Int -> QNMFormula -> NFA [Int] Int
automatize dir (QNMFormula [] m) =
- let dfa@( DFA s _ _ _) = automatizem dir m
- mdfa@(DFA s' _ _ _) = rmdist dfa (fromIntegral (length dir))
- !t = (traceShowId $ (length s, length s'))
- in minimize (fromIntegral $ length dir) $ nondeterminize $ mdfa
+ let dfa = automatizem dir m
+ mdfa = minDistinguishability dfa (length dir)
+ in minReachability (length dir) $ nondeterminize mdfa
automatize dir (QNMFormula (Neg : quas) m) =
let nfa = automatize dir (QNMFormula quas m)
- in minimize (fromIntegral (length dir)) (ncompl nfa)
- where
- rmmin d = rmdist d (fromIntegral (length dir))
- ncompl = nondeterminize . compl . rmmin . determinize
+ in minReachability (length dir) (ncompl nfa)
+ where
+ md dfa = minDistinguishability dfa (length dir)
+ ncompl = nondeterminize . compl . md . determinize
automatize dir (QNMFormula (Qua (Exists, v) : quas) m) =
let nfa = automatize dir (QNMFormula quas m)
- in case M.lookup (Var v) dir of
- Nothing -> nfa
- (Just i) -> minimize (fromIntegral $ length dir) $ existentialize i nfa
+ in case M.lookup (Var v) dir of
+ Nothing -> nfa
+ (Just i) -> minReachability (length dir) $ existentialize i nfa
+automatize _ (QNMFormula (Qua (ForAll, _) : _) _) =
+ error "should be universal-free"
assign :: S.Set Addend -> M.Map Addend Int
assign xs = M.fromList (zip (S.toList xs) [0 ..])
collect :: Matrix -> S.Set Addend
-collect (MEq as bs) = S.fromList (as ++ bs)
-collect (MLe a b ) = S.fromList [a, b]
-collect (MLt a b ) = S.fromList [a, b]
-collect (MAnd x y ) = S.union (collect x) (collect y)
-collect (MOr x y ) = S.union (collect x) (collect y)
-collect (MNot x ) = collect x
+collect (MEq as bs) = S.fromList (as ++ bs)
+collect (MLe a b) = S.fromList [a, b]
+collect (MLt a b) = S.fromList [a, b]
+collect (MAnd x y) = S.union (collect x) (collect y)
+collect (MOr x y) = S.union (collect x) (collect y)
+collect (MNot x) = collect x
literals :: M.Map Addend Int -> [[Int]]
literals m =
- let
- addends = L.sortOn snd (M.toList m)
- reversedLiterals = map (\(a, i) -> addendReverseBinary a) addends
- max = L.maximum (map length reversedLiterals)
- paddedReversed =
- map (\x -> x ++ replicate (max - length x) 0) reversedLiterals
- padded = map reverse paddedReversed
- in
- L.transpose padded
+ let addends = L.sortOn snd (M.toList m)
+ reversedLiterals = map (\(a, _) -> addendReverseBinary a) addends
+ padTo = L.maximum (map length reversedLiterals)
+ paddedReversed =
+ map (\x -> x ++ replicate (padTo - length x) 0) reversedLiterals
+ padded = map reverse paddedReversed
+ in L.transpose padded
addendReverseBinary :: Addend -> [Int]
-addendReverseBinary (Var x) = []
+addendReverseBinary (Var _) = []
addendReverseBinary (Con n) = reverseBinary n
-reverseBinary :: Integer -> [Int]
+reverseBinary :: Int -> [Int]
reverseBinary 0 = []
-reverseBinary n = fromIntegral (mod n 2) : reverseBinary (div n 2)
+reverseBinary n = mod n 2 : reverseBinary (div n 2)
eval :: QNMFormula -> Bool
-eval f@(QNMFormula q m) =
- let dir = (assign $ collect m)
- nfa = automatize dir f
+eval f@(QNMFormula _ m) =
+ let dir = (assign $ collect m)
+ nfa = automatize dir f
input = literals (assign $ collect m)
- in runNFA nfa input
+ in runNFA nfa input
data State a = S a | Double (State a, State a) | Multi [State a] deriving (Eq, Ord, Show)
@@ -93,137 +90,76 @@ data NFA c a = NFA [State a] [State a] [State a] (State a -> c -> [State a])
runDFA :: (Ord a) => DFA c a -> [c] -> Bool
runDFA (DFA _ start accepts f) cs = foldl f start cs `elem` accepts
-minimize :: (Ord a) => Integer -> NFA [Int] a -> NFA [Int] a
-minimize n nfa@(NFA _ starts accepts f) = NFA states' starts' accepts' f
- where
- states' = closure nfa (chars n) starts
- starts' = starts `L.intersect` states'
- accepts' = accepts `L.intersect` states'
-
-
-rmdist :: (Ord a, Show a) => DFA [Int] a -> Integer -> DFA [Int] a
-rmdist dfa@(DFA states start accepts f) n =
- let
- statecandidates = minimize2 dfa n
- states' = map head statecandidates
- start' = head $ head $ filter (\c -> L.elem start c) statecandidates
- accepts' = L.intersect accepts states'
- f' s c =
- let s' = f s c
- in head $ head $ filter (\c -> L.elem s' c) statecandidates
- in
- DFA states' start' accepts' f'
-
-
-minimize2 :: (Ord a, Show a) => DFA [Int] a -> Integer -> [[State a]]
-minimize2 dfa@(DFA states start accepts f) n =
- let p0 = [states L.\\ accepts, accepts] in minrec dfa n p0
-
-minrec
- :: (Ord a, Show a) => DFA [Int] a -> Integer -> [[State a]] -> [[State a]]
-minrec dfa n pklast =
- let next = concatMap (\part -> minimize3 dfa n pklast [] part) pklast
- in if normpartition next == normpartition pklast
- then next
- else minrec dfa n next
-
-normpartition :: Ord a => [[a]] -> S.Set (S.Set a)
-normpartition xs = S.fromList (map S.fromList xs)
-
-
-minimize3
- :: (Ord a, Show a)
- => DFA [Int] a
- -> Integer
- -> [[State a]]
- -> [[State a]]
- -> [State a]
- -> [[State a]]
-minimize3 dfa@(DFA states starts accepts f) n pklast pk [] = pk
-minimize3 dfa@(DFA states starts accepts f) n pklast [] (x : xs) =
- minimize3 dfa n pklast [[x]] xs
-minimize3 dfa@(DFA states starts accepts f) n pklast pk (x : xs) =
- let ys = filter (\s -> equivall dfa pklast (chars n) (head s) x) pk
- in if ys == []
- then minimize3 dfa n pklast ([x] : pk) xs
- else
- let coll = (head ys)
- trunc = L.delete coll pk
- new = x : coll
- pk' = new : trunc
- in minimize3 dfa n pklast pk' xs
-
-equivall
- :: (Eq a)
- => DFA [Int] a
- -> [[State a]]
- -> [[Int]]
- -> (State a)
- -> (State a)
- -> Bool
-equivall dfa partition cs x y = all (\c -> not $ dist dfa partition c x y) cs
-
-states = S <$> [0, 1, 2, 3, 4]
-start = S 0
-accepts = [S 4]
-f (S 0) [0] = S 1
-f (S 0) [1] = S 2
-f (S 1) [0] = S 1
-f (S 1) [1] = S 3
-f (S 2) [0] = S 1
-f (S 2) [1] = S 2
-f (S 3) [0] = S 1
-f (S 3) [1] = S 4
-f (S 4) [0] = S 1
-f (S 4) [1] = S 2
-dfa = DFA states start accepts f
-p0 = [states L.\\ accepts, accepts]
-p1 = concatMap (minimize3 dfa 1 p0 []) p0
-p2 = concatMap (minimize3 dfa 1 p1 []) p1
-minp = minimize2 dfa 1
-
-
-dist
- :: (Eq a)
- => DFA [Int] a
- -> [[State a]]
- -> [Int]
- -> (State a)
- -> (State a)
- -> Bool
-dist (DFA states starts accepts f) partition c x y =
- (findset partition (f x c)) /= (findset partition (f y c))
-
-findset :: (Eq a) => [[State a]] -> State a -> [State a]
-findset xs x = head (filter (\s -> L.elem x s) xs)
+minReachability :: (Ord a) => Int -> NFA [Int] a -> NFA [Int] a
+minReachability n nfa@(NFA _ starts accepts f) = NFA states' starts' accepts' f
+ where
+ states' = closure nfa (chars n) starts
+ starts' = starts `L.intersect` states'
+ accepts' = accepts `L.intersect` states'
+
+minDistinguishability :: (Ord a, Show a) => DFA [Int] a -> Int -> DFA [Int] a
+minDistinguishability dfa@(DFA _ start accepts f) n =
+ let partition = hopcroftPartitionRefinement dfa n
+ states' = map head partition
+ start' = head $ head $ filter (L.elem start) partition
+ accepts' = L.intersect accepts states'
+ f' s c = let s' = f s c in head $ head $ filter (L.elem s') partition
+ in DFA states' start' accepts' f'
+
+hopcroftPartitionRefinement ::
+ (Ord a, Show a) => DFA [Int] a -> Int -> [[State a]]
+hopcroftPartitionRefinement (DFA states _ accepts f) n =
+ let p0 = [states L.\\ accepts, accepts] in recurse p0
+ where
+ sigma = chars n
+ normalizePartition partition = S.fromList (map S.fromList partition)
+ findInPartition xs x = fromJust $ L.find (L.elem x) xs
+ equalUnder partition x y =
+ findInPartition partition x == findInPartition partition y
+ indistinguishable partition x y =
+ all (\c -> equalUnder partition (f x c) (f y c)) sigma
+ partialRefine _ pk [] = pk
+ partialRefine pkPrev [] (x : xs) = partialRefine pkPrev [[x]] xs
+ partialRefine pkPrev pk (x : xs) =
+ case filter (\s -> indistinguishable pkPrev (head s) x) pk of
+ [] -> partialRefine pkPrev ([x] : pk) xs
+ [match] -> partialRefine pkPrev ((x : match) : L.delete match pk) xs
+ _ -> error "should be unreachable"
+ recurse pkPrev =
+ let pk = concatMap (partialRefine pkPrev []) pkPrev
+ in if normalizePartition pk == normalizePartition pkPrev
+ then pk
+ else recurse pk
runNFA :: (Ord a) => NFA c a -> [c] -> Bool
runNFA (NFA _ starts accepts f) cs =
foldl (\xs c -> L.nub $ concatMap (`f` c) xs) starts cs
`L.intersect` accepts
- /= []
+ /= []
reversal :: (Ord a) => NFA c a -> NFA c a
reversal (NFA states starts accepts f) = NFA states accepts starts f'
- where f' s c = filter (\state -> s `elem` f state c) states
-
-eq :: Integer -> [Int] -> [Int] -> DFA [Int] Int
-eq n is js = determinize $ minimize n $ reversal $ nondeterminize dfa
- where
- states = S <$> [-(length js - 1) .. length is - 1 + 1]
- start = S 0
- accepts = [S 0]
- rejector = last states
- f :: State Int -> [Int] -> State Int
- f carrystate@(S carry) c = if carrystate == rejector
- then rejector
- else
- let si = sum (map ((c) !!) (is))
- sj = sum (map (c !!) (js))
- parityok = mod (carry + si) 2 == mod sj 2
- newcarry = div (carry + si - sj) 2
- in if parityok then S newcarry else rejector
- dfa = DFA states start accepts f
+ where
+ f' s c = filter (\state -> s `elem` f state c) states
+
+eq :: Int -> [Int] -> [Int] -> DFA [Int] Int
+eq n is js = determinize $ minReachability n $ reversal $ nondeterminize dfa
+ where
+ states = S <$> [- (length js - 1) .. length is - 1 + 1]
+ start = S 0
+ accepts = [S 0]
+ rejector = last states
+ f :: State Int -> [Int] -> State Int
+ f carrystate@(S carry) c =
+ if carrystate == rejector
+ then rejector
+ else
+ let si = sum (map (c !!) is)
+ sj = sum (map (c !!) js)
+ parityok = mod (carry + si) 2 == mod sj 2
+ newcarry = div (carry + si - sj) 2
+ in if parityok then S newcarry else rejector
+ dfa = DFA states start accepts f
le :: Int -> Int -> DFA [Int] Int
le = less LessEqual
@@ -234,37 +170,38 @@ lt = less LessThan
data LessType = LessEqual | LessThan deriving (Eq, Ord, Show)
less :: LessType -> Int -> Int -> DFA [Int] Int
-less lt i j = DFA [S 0, S 1, S 2] (S 0) accepts f
- where
- accepts = if lt == LessEqual then [S 0, S 2] else [S 2]
- f s c = case (s, (c !! i, c !! j)) of
- (S 0, (1, 1)) -> S 0
- (S 0, (0, 0)) -> S 0
- (S 0, (1, 0)) -> S 1
- (S 0, _ ) -> S 2
- (S 1, _ ) -> S 1
- (_ , _ ) -> S 2
+less op i j = DFA [S 0, S 1, S 2] (S 0) accepts f
+ where
+ accepts = if op == LessEqual then [S 0, S 2] else [S 2]
+ f s c = case (s, (c !! i, c !! j)) of
+ (S 0, (1, 1)) -> S 0
+ (S 0, (0, 0)) -> S 0
+ (S 0, (1, 0)) -> S 1
+ (S 0, _) -> S 2
+ (S 1, _) -> S 1
+ (_, _) -> S 2
prod :: [a] -> [b] -> [(a, b)]
-prod xs [] = []
-prod [] ys = []
-prod (x : xs) ys = fmap (x, ) ys ++ prod xs ys
+prod _ [] = []
+prod [] _ = []
+prod (x : xs) ys = fmap (x,) ys ++ prod xs ys
data JunctionType = Conj | Disj deriving (Eq, Ord, Show)
junction :: (Ord a) => JunctionType -> DFA c a -> DFA c a -> DFA c a
junction jt (DFA states1 start1 accepts1 f1) (DFA states2 start2 accepts2 f2) =
DFA states' start' accepts' f'
- where
- newStates = prod states1 states2
- states' = Double <$> newStates
- start' = Double (start1, start2)
- accepts' = if jt == Conj
- then Double <$> prod accepts1 accepts2
- else
- Double
- <$> filter (\(s, t) -> s `elem` accepts1 || t `elem` accepts2) newStates
- f' (Double (s, t)) c = Double (f1 s c, f2 t c)
+ where
+ newStates = prod states1 states2
+ states' = Double <$> newStates
+ start' = Double (start1, start2)
+ accepts' =
+ if jt == Conj
+ then Double <$> prod accepts1 accepts2
+ else
+ Double
+ <$> filter (\(s, t) -> s `elem` accepts1 || t `elem` accepts2) newStates
+ f' (Double (s, t)) c = Double (f1 s c, f2 t c)
conj :: (Ord a) => DFA c a -> DFA c a -> DFA c a
conj = junction Conj
@@ -277,44 +214,46 @@ compl (DFA states start accepts f) = DFA states start (states L.\\ accepts) f
nondeterminize :: (Ord a) => DFA c a -> NFA c a
nondeterminize (DFA states start accepts f) = NFA states [start] accepts f'
- where f' s c = [f s c]
+ where
+ f' s c = [f s c]
change :: [a] -> Int -> a -> [a]
change xs idx b = take idx xs ++ [b] ++ drop (idx + 1) xs
closure :: (Ord a) => NFA c a -> [c] -> [State a] -> [State a]
-closure nfa@(NFA states starts accepts f) cs initstates =
+closure nfa@(NFA _ _ _ f) cs initstates =
let new = concatMap (\state -> concatMap (f state) cs) initstates
- in if L.nub new L.\\ L.nub initstates /= []
+ in if L.nub new L.\\ L.nub initstates /= []
then closure nfa cs (L.nub $ new ++ initstates)
else L.nub initstates
existentialize :: (Ord a) => Int -> NFA [Int] a -> NFA [Int] a
-existentialize idx nfa@(NFA states starts accepts f) = NFA states
- starts'
- accepts
- f'
- where
- zeroer = replicate 50 0
- oneer = change zeroer idx 1
- starts' = closure nfa [zeroer, oneer] starts
- f' s c = f s (change c idx 0) ++ f s (change c idx 1)
+existentialize idx nfa@(NFA states starts accepts f) =
+ NFA
+ states
+ starts'
+ accepts
+ f'
+ where
+ zeroer = replicate 50 0
+ oneer = change zeroer idx 1
+ starts' = closure nfa [zeroer, oneer] starts
+ f' s c = f s (change c idx 0) ++ f s (change c idx 1)
powerset :: [a] -> [[a]]
-powerset [] = [[]]
+powerset [] = [[]]
powerset (x : xs) = let rest = powerset xs in map (x :) rest ++ rest
determinize :: (Ord a) => NFA c a -> DFA c a
determinize (NFA states start accepts f) = DFA states' start' accepts' f'
- where
- newStates = map L.sort $ powerset states
- states' = Multi <$> newStates
- start' = Multi $ L.sort start
- accepts' =
- Multi <$> filter (\state' -> state' `L.intersect` accepts /= []) newStates
- f' (Multi s) c = Multi $ L.nub $ L.sort $ concatMap (`f` c) s
-
-
-chars :: Integer -> [[Int]]
+ where
+ newStates = map L.sort $ powerset states
+ states' = Multi <$> newStates
+ start' = Multi $ L.sort start
+ accepts' =
+ Multi <$> filter (\state' -> state' `L.intersect` accepts /= []) newStates
+ f' (Multi s) c = Multi $ L.nub $ L.sort $ concatMap (`f` c) s
+
+chars :: Int -> [[Int]]
chars 0 = [[]]
chars n = let r = chars (n - 1) in map (1 :) r ++ map (0 :) r
generated by cgit v1.2.3 (git 2.25.1) at 2024-10-16 23:17:31 +0000