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-rw-r--r--src/Lib.hs332
1 files changed, 218 insertions, 114 deletions
diff --git a/src/Lib.hs b/src/Lib.hs
index e8bc698..0e093e3 100644
--- a/src/Lib.hs
+++ b/src/Lib.hs
@@ -1,60 +1,73 @@
-{-# LANGUAGE GADTs #-}
-{-# LANGUAGE LambdaCase #-}
-{-# LANGUAGE RankNTypes #-}
{-# 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 ()
-import Parser
-import qualified Text.Parsec as Parsec
+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
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)
+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)
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) = minimize (fromIntegral $ length dir) $ nondeterminize $ automatizem dir m
+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
automatize dir (QNMFormula (Neg : quas) m) =
let nfa = automatize dir (QNMFormula quas m)
- in minimize (fromIntegral (length dir)) (ncompl nfa)
+ in minimize (fromIntegral (length dir)) (ncompl nfa)
+ where
+ rmmin d = rmdist d (fromIntegral (length dir))
+ ncompl = nondeterminize . compl . rmmin . 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
+ in case M.lookup (Var v) dir of
+ Nothing -> nfa
(Just i) -> minimize (fromIntegral $ length dir) $ existentialize i nfa
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, 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
addendReverseBinary :: Addend -> [Int]
addendReverseBinary (Var x) = []
@@ -66,12 +79,12 @@ reverseBinary n = fromIntegral (mod n 2) : reverseBinary (div n 2)
eval :: QNMFormula -> Bool
eval f@(QNMFormula q m) =
- let dir = (assign $ collect m)
- nfa = automatize dir f
+ 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 = Single a | Double (State a, State a) | Multi [State a] deriving (Eq, Ord, Show)
+data State a = S a | Double (State a, State a) | Multi [State a] deriving (Eq, Ord, Show)
data DFA c a = DFA [State a] (State a) [State a] (State a -> c -> State a)
@@ -80,35 +93,137 @@ 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)
+
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 /= []
+ 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
+ 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 = Single <$> [- (length js - 1) .. length is - 1 + 1]
- start = Single 0
- accepts = [Single 0]
- rejector = last states
- f :: State Int -> [Int] -> State Int
- f carrystate@(Single carry) c =
- if carrystate == rejector
- then rejector
- else
- let si = sum (map (c !!) is)
- sj = sum (map (c !!) js)
- parityok = mod (carry + si) 2 == mod sj 2
- newcarry = div (carry + si - sj) 2
- in if parityok
- then Single newcarry
- else rejector
- dfa = DFA states start accepts f
+ where
+ states = S <$> [-(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
@@ -119,36 +234,37 @@ lt = less LessThan
data LessType = LessEqual | LessThan deriving (Eq, Ord, Show)
less :: LessType -> Int -> Int -> DFA [Int] Int
-less lt i j = DFA [Single 0, Single 1, Single 2] (Single 0) accepts f
- where
- accepts = if lt == LessEqual then [Single 0, Single 2] else [Single 2]
- f s c = case (s, (c !! i, c !! j)) of
- (Single 0, (1, 1)) -> Single 0
- (Single 0, (0, 0)) -> Single 0
- (Single 0, (1, 0)) -> Single 1
- (Single 0, _) -> Single 2
- (Single 1, _) -> Single 1
- (_, _) -> Single 2
+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
prod :: [a] -> [b] -> [(a, b)]
-prod xs [] = []
-prod [] ys = []
-prod (x : xs) ys = fmap (x,) ys ++ prod xs ys
+prod xs [] = []
+prod [] ys = []
+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
@@ -157,14 +273,11 @@ disj :: (Ord a) => DFA c a -> DFA c a -> DFA c a
disj = junction Disj
compl :: (Ord a) => DFA c a -> DFA c a
-compl (DFA states start accepts f) =
- DFA states start (states L.\\ accepts) f
+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]
+nondeterminize (DFA states start accepts f) = NFA states [start] accepts f'
+ where f' s c = [f s c]
change :: [a] -> Int -> a -> [a]
change xs idx b = take idx xs ++ [b] ++ drop (idx + 1) xs
@@ -172,45 +285,36 @@ 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 =
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
+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
-ncompl :: (Ord a) => NFA c a -> NFA c a
-ncompl = nondeterminize . compl . determinize
chars :: Integer -> [[Int]]
chars 0 = [[]]
-chars n =
- let r = chars (n -1)
- in map (1 :) r ++ map (0 :) r
-
-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'
+chars n = let r = chars (n - 1) in map (1 :) r ++ map (0 :) r