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|
----------------------------------------------------------------------
-- |
-- Module : FiniteState
-- Maintainer : BB
-- Stability : (stable)
-- Portability : (portable)
--
-- > CVS $Date: 2005/09/22 17:08:48 $
-- > CVS $Author: bringert $
-- > CVS $Revision: 1.13 $
--
-- A simple finite state network module.
-----------------------------------------------------------------------------
module GF.Speech.FiniteState (FA, State, NFA, DFA,
startState, finalStates,
states, transitions,
newFA,
addFinalState,
newState, newTransition,
mapStates, mapTransitions,
moveLabelsToNodes, minimize,
prFAGraphviz) where
import Data.List
import Data.Maybe (catMaybes,fromJust)
import GF.Data.Utilities
import qualified GF.Visualization.Graphviz as Dot
type State = Int
data FA n a b = FA (Graph n a b) n [n]
type NFA a = FA State () (Maybe a)
type DFA a = FA [State] () a
startState :: FA n a b -> n
startState (FA _ s _) = s
finalStates :: FA n a b -> [n]
finalStates (FA _ _ ss) = ss
states :: FA n a b -> [(n,a)]
states (FA g _ _) = nodes g
transitions :: FA n a b -> [(n,n,b)]
transitions (FA g _ _) = edges g
newFA :: Enum n => a -- ^ Start node label
-> FA n a b
newFA l = FA g s []
where (g,s) = newNode l (newGraph [toEnum 0..])
addFinalState :: n -> FA n a b -> FA n a b
addFinalState f (FA g s ss) = FA g s (f:ss)
newState :: a -> FA n a b -> (FA n a b, n)
newState x (FA g s ss) = (FA g' s ss, n)
where (g',n) = newNode x g
newTransition :: n -> n -> b -> FA n a b -> FA n a b
newTransition f t l = onGraph (newEdge (f,t,l))
mapStates :: (a -> c) -> FA n a b -> FA n c b
mapStates f = onGraph (nmap f)
mapTransitions :: (b -> c) -> FA n a b -> FA n a c
mapTransitions f = onGraph (emap f)
minimize :: Eq a => NFA a -> NFA a
minimize = dfa2nfa . determinize . reverseNFA . dfa2nfa . determinize . reverseNFA
onGraph :: (Graph n a b -> Graph n c d) -> FA n a b -> FA n c d
onGraph f (FA g s ss) = FA (f g) s ss
-- | Transform a standard finite automaton with labelled edges
-- to one where the labels are on the nodes instead. This can add
-- up to one extra node per edge.
moveLabelsToNodes :: (Ord n,Eq a) => FA n () (Maybe a) -> FA n (Maybe a) ()
moveLabelsToNodes = removeTrivialEmptyNodes . onGraph f
where f gr@(Graph c _ _) = Graph c' ns (concat ess)
where is = incoming gr
(c',is') = mapAccumL fixIncoming c is
(ns,ess) = unzip (concat is')
-- | Remove nodes which are not start or final, and have
-- exactly one incoming or exactly one outgoing edge.
removeTrivialEmptyNodes :: FA n (Maybe a) () -> FA n (Maybe a) ()
removeTrivialEmptyNodes = id -- FIXME: implement
fixIncoming :: (Eq n, Eq a) => [n] -> (Node n (),[Edge n (Maybe a)]) -> ([n],[(Node n (Maybe a),[Edge n ()])])
fixIncoming cs c@((n,()),es) = (cs'', ((n,Nothing),es'):newContexts)
where ls = nub $ map getLabel es
(cs',cs'') = splitAt (length ls) cs
newNodes = zip cs' ls
es' = [ (x,n,()) | x <- map fst newNodes ]
-- separate cyclic and non-cyclic edges
(cyc,ncyc) = partition (\ (f,_,_) -> f == n) es
-- keep all incoming non-cyclic edges with the right label
to (x,l) = [ (f,x,()) | (f,_,l') <- ncyc, l == l']
-- for each cyclic edge with the right label,
-- add an edge from each of the new nodes (including this one)
++ [ (y,x,()) | (f,_,l') <- cyc, l == l', (y,_) <- newNodes]
newContexts = [ (v, to v) | v <- newNodes ]
alphabet :: Eq b => Graph n a (Maybe b) -> [b]
alphabet = nub . catMaybes . map getLabel . edges
determinize :: Eq a => NFA a -> DFA a
determinize (FA g s f) = let (ns,es) = h [start] [] []
in FA (Graph (freshDFANodes g) [(n,()) | n <- ns] es) start (filter isDFAFinal ns)
where sigma = alphabet g
out = outgoing g
start = closure out [s]
isDFAFinal n = not (null (f `intersect` n))
freshDFANodes (Graph ns _ _) = map (:[]) ns
-- Get the new DFA states and edges produced by a set of DFA states.
new ns = unzip [ (s, (n,s,c)) | n <- ns, c <- sigma, let s = sort (reachable out c n), not (null s) ]
h currentStates oldStates oldEdges
| null currentStates = (oldStates,oldEdges)
| otherwise = h newStates' allOldStates (newEdges++oldEdges)
where (newStates,newEdges) = new currentStates
allOldStates = currentStates ++ oldStates
newStates' = nub newStates \\ allOldStates
-- | Get all the nodes reachable from a set of nodes by only empty edges.
closure :: Eq n => Outgoing n a (Maybe b) -> [n] -> [n]
closure out = fix closure_
where closure_ r = r `union` [y | x <- r, (_,y,Nothing) <- getOutgoing out x]
-- | Get all nodes reachable from a set of nodes by one edge with the given
-- label and then any number of empty edges.
reachable :: (Eq n, Eq b) => Outgoing n a (Maybe b) -> b -> [n] -> [n]
reachable out c ns = closure out [y | n <- ns, (_,y,Just c') <- getOutgoing out n, c' == c]
reverseNFA :: NFA a -> NFA a
reverseNFA (FA g s fs) = FA g''' s' [s]
where g' = reverseGraph g
(g'',s') = newNode () g'
g''' = newEdges [(s',f,Nothing) | f <- fs] g''
dfa2nfa :: DFA a -> NFA a
dfa2nfa (FA (Graph _ ns es) s fs) = FA (Graph c ns' es') s' fs'
where newNodes = zip (map fst ns) [0..]
newNode n = lookup' n newNodes
c = [length ns..]
ns' = [ (n,()) | (_,n) <- newNodes ]
es' = [ (newNode f, newNode t,Just l) | (f,t,l) <- es]
s' = newNode s
fs' = map newNode fs
--
-- * Visualization
--
prFAGraphviz :: (Eq n,Show n) => FA n String String -> String
prFAGraphviz = Dot.prGraphviz . toGraphviz
prFAGraphviz_ :: (Eq n,Show n,Show a, Show b) => FA n a b -> String
prFAGraphviz_ = Dot.prGraphviz . toGraphviz . mapStates show . mapTransitions show
toGraphviz :: (Eq n,Show n) => FA n String String -> Dot.Graph
toGraphviz (FA (Graph _ ns es) s f) = Dot.Graph Dot.Directed [] (map mkNode ns) (map mkEdge es)
where mkNode (n,l) = Dot.Node (show n) attrs
where attrs = [("label",l)]
++ if n == s then [("shape","box")] else []
++ if n `elem` f then [("style","bold")] else []
mkEdge (x,y,l) = Dot.Edge (show x) (show y) [("label",l)]
--
-- * Graphs
--
data Graph n a b = Graph [n] [Node n a] [Edge n b]
deriving (Eq,Show)
type Node n a = (n,a)
type Edge n b = (n,n,b)
type Incoming n a b = [(Node n a,[Edge n b])]
type Outgoing n a b = [(Node n a,[Edge n b])]
newGraph :: [n] -> Graph n a b
newGraph ns = Graph ns [] []
nodes :: Graph n a b -> [Node n a]
nodes (Graph _ ns _) = ns
edges :: Graph n a b -> [Edge n b]
edges (Graph _ _ es) = es
nmap :: (a -> c) -> Graph n a b -> Graph n c b
nmap f (Graph c ns es) = Graph c [(n,f l) | (n,l) <- ns] es
emap :: (b -> c) -> Graph n a b -> Graph n a c
emap f (Graph c ns es) = Graph c ns [(x,y,f l) | (x,y,l) <- es]
newNode :: a -> Graph n a b -> (Graph n a b,n)
newNode l (Graph (c:cs) ns es) = (Graph cs ((c,l):ns) es, c)
newEdge :: Edge n b -> Graph n a b -> Graph n a b
newEdge e (Graph c ns es) = Graph c ns (e:es)
newEdges :: [Edge n b] -> Graph n a b -> Graph n a b
newEdges es' (Graph c ns es) = Graph c ns (es'++es)
-- | Get a list of all nodes and their incoming edges.
incoming :: Ord n => Graph n a b -> Incoming n a b
incoming = groupEdgesBy getTo
-- | Get a list of all nodes and their outgoing edges.
outgoing :: Ord n => Graph n a b -> Outgoing n a b
outgoing = groupEdgesBy getFrom
getOutgoing :: Eq n => Outgoing n a b -> n -> [Edge n b]
getOutgoing out x = head [ es | ((y,_),es) <- out, x == y ]
groupEdgesBy :: (Ord n) => (Edge n b -> n) -> Graph n a b -> [(Node n a,[Edge n b])]
groupEdgesBy h (Graph _ ns es) =
snd $ mapAccumL f (sortBy (compareBy h) es) (sortBy (compareBy fst) ns)
where f es' v@(n,_) = let (nes,es'') = span ((==n) . h) es' in (es'',(v,nes))
getFrom :: Edge n b -> n
getFrom (f,_,_) = f
getTo :: Edge n b -> n
getTo (_,t,_) = t
getLabel :: Edge n b -> b
getLabel (_,_,l) = l
reverseGraph :: Graph n a b -> Graph n a b
reverseGraph (Graph c ns es) = Graph c ns [ (t,f,l) | (f,t,l) <- es ]
|
