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# Source code for fl

    # -*- coding: utf-8 -*-
    """Finite languages and related automata manipulation

    Finite languages manipulation

    .. *Authors:* Rogério Reis & Nelma Moreira

    .. *This is part of FAdo project*   <http://fado.dcc.fc.up.pt

    .. *Version:* 0.9.5

    .. *Copyright*: 1999-2014 Rogério Reis & Nelma Moreira {rvr,nam}@dcc.fc.up.pt

    .. This program is free software; you can redistribute it and/or modify
       it under the terms of the GNU General Public License as published by
       the Free Software Foundation; either version 2 of the License, or
       (at your option) any later version.

       This program is distributed in the hope that it will be useful,
       but WITHOUT ANY WARRANTY; without even the implied warranty of
       MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
       GNU General Public License for more details.

       You should have received a copy of the GNU General Public License
       along with this program; if not, write to the Free Software
       Foundation, Inc., 675 Mass Ave, Cambridge, MA 02139, USA."""

    import fa
    from copy import copy
    from common import *
    import random

    class FL(object):
        """Finite Language Class

        :var Words: the elements of the language
        :var Sigma: the alphabet"""
        def __init__(self, wordsList=None, Sigma=set([])):
            if not wordsList:
                wordsList = []
            self.Words = set(wordsList)
            self.Sigma = Sigma
            for w in self.Words:
                if w != Epsilon:
                    for l in w:
                        self.Sigma.add(l)

        def __str__(self):
            return "(%s,%s)" % (list(self.Words), list(self.Sigma))

        def __repr__(self):
            return "FL%s" % self.__str__()

        def __len__(self):
            return len(self.Words)

        def reunion(self, other):
            """Reunion of FL:   a | b

            :param FL other: right hand operand
            :rtype: FL
            :raises FAdoGeneralError: if both arguments are not FL"""
            return self.__or__(other)

        def __or__(self, other):
            if type(other) != type(self):
                raise FAdoGeneralError("Incompatible objects")
            new = FL()
            new.Sigma = self.Sigma | other.Sigma
            new.Words = self.Words | other.Words
            return new

        def intersection(self, other):
            """Intersection of FL: a & b

            :param FL other: right hand operand
            :raises FAdoGeneralError: if both arguments are not FL"""
            return self.__and__(other)

        def __and__(self, other):
            if type(other) != type(self):
                raise FAdoGeneralError("Incompatible objects")
            new = FL()
            new.Sigma = self.Sigma | other.Sigma
            new.Words = self.Words & other.Words
            return new

        def diff(self, other):
            """Difference of FL: a - b

            :param FL other: right hand operand
            :rtype: FL
            :raises FAdoGeneralError: if both arguments are not FL"""
            return self.__sub__(other)

        def __sub__(self, other):
            if type(other) != type(self):
                raise FAdoGeneralError("Incompatible objects")
            new = FL()
            new.Sigma = self.Sigma | other.Sigma
            new.Words = self.Words - other.Words
            return new

        def setSigma(self, Sigma, Strict=False):
            """Sets the alphabet of a FL

            :param set Sigma: alphabet
            :param bool Strict: behaviour

            .. attention::
               Unless Strict flag is set to True, alphabet can only be enlarged.  The resulting alphabet is  in fact the
               union of the former alphabet with the new one. If flag is set to True, the alphabet is simply replaced."""
            if Strict:
                self.Sigma = Sigma
            else:
                self.Sigma = self.Sigma.union(Sigma)

        def addWords(self, wList):
            """Adds a list of words to a FL

            :param list wList: words to add"""
            self.Words = self.Words | set(wList)
            for w in wList:
                for c in w:
                    self.Sigma.add(c)

        def suffixClosedP(self):
            """Tests if a language is suffix closed

            :rtype: bool"""
            wrds = list(copy(self.Words))
            if Epsilon not in wrds:
                return False
            else:
                wrds.remove(Epsilon)
            wrds.sort(lambda x, y: len(x) - len(y))
            while wrds:
                w = wrds.pop()
                for w1 in suffixes(w):
                    if w1 not in self.Words:
                        return False
                    else:
                        if w1 in wrds:
                            wrds.remove(w1)
            return True

        def filter(self, automata):
            """Separates a language in two other using a DFA of NFA as a filter

            :param DFA|NFA automata: the automata to be used as a filter
            :returns: the accepted/unaccepted pair of languages
            :rtype: tuple of FL"""
            a, b = (FL(), FL())
            a.setSigma(self.Sigma)
            b.setSigma(self.Sigma)
            for w in self.Words:
                if automata.evalWord(w):
                    a.addWords([w])
                else:
                    b.addWords([w])
            return a, b

        def trieFA(self):
            """Generates the trie automaton that recognises this language

            :returns: the trie automaton
            :rtype: ADFA"""
            new = ADFA()
            new.setSigma(copy(self.Sigma))
            i = new.addState()
            new.setInitial(i)
            for w in self.Words:
                if w == Epsilon:
                    new.addFinal(i)
                else:
                    s = i
                    for c in w:
                        if c not in new.delta.get(s, []):
                            sn = new.addState()
                            new.addTransition(s, c, sn)
                            s = sn
                        else:
                            s = new.delta[s][c]
                    new.addFinal(s)
            return new

        def toDFA(self):
            """ Generates a DFA recognizing the language

            :rtype: ADFA

            .. versionadded:: 1.2"""
            return self.trieFA()

        def toNFA(self):
            """Generates a NFA recognizing the language

            :rtype: ANFA

            .. versionadded:: 1.2"""
            return self.toDFA().toANFA()

        # noinspection PyUnboundLocalVariable
        def multiLineAutomaton(self):
            """Generates the trivial linear ANFA equivalent to this language

            :rtype: ANFA"""
            new = ANFA()
            new.setSigma(copy(self.Sigma))
            for w in self.Words:
                s = new.addState()
                new.addInitial(s)
                for c in w:
                    s1 = new.addState()
                    new.addTransition(s, c, s1)
                    s = s1
                new.addFinal(s1)
            return new

    class DCFA(fa.DFA):
        """Deterministic Cover Automata class

        .. inheritance-diagram:: DCFA"""

        def __init__(self):
            super(DCFA, self).__init__()
            self.length = None

        @property
        def length(self):
            """
            :return: size of the longest word
            :rtype: int"""
            return self.length

        @length.setter
        def length(self, value):
            """Setter
            :param int value: size"""
            self.length = value

        @length.deleter
        def length(self):
            """Length deleter"""
            self.length = None

    class AFA(object):
        """Base class for Acyclic Finite Automata

        .. inheritance-diagram:: AFA

        .. note::
           This is just a container for some common methods. **Not to be used directly!!**"""
        def __init__(self):
            self.Dead = None
            self.delta = dict()
            self.Initial = None
            self.States = []
            self.Final = set()

        @abstractmethod
        def addState(self):
            """
            :rtype: int"""
            pass

        def setDeadState(self, sti):
            """Identifies the dead state

            :param int sti: index of the dead state

            .. attention::
               nothing is done to ensure that the state given is legitimate

            .. note::
               without dead state identified, most of the methods for acyclic automata can not be applied"""
            self.Dead = sti

        def ensureDead(self):
            """Ensures that a state is defined as dead"""
            try:
                _ = self.Dead
            except AttributeError:
                x = self.addState()
                self.setDeadState(x)

        def ordered(self):
            """Orders states names in its topological order

            :returns: ordered list of state indexes
            :rtype: list of int

            .. note::
               one could use the FA.toposort() method, but special care must be taken with the dead state for the
               algorithms related with cover automata."""

            def _dealS(st1):
                if st1 not in torder:
                    torder.append(st1)
                    if st1 in self.delta.keys():
                        for k in self.delta[st1]:
                            for dest in forceIterable(self.delta[st1][k]):
                                if dest not in torder and dest != self.Dead:
                                    queue.append(dest)

            try:
                dead = self.Dead
            except AttributeError:
                raise FAdoGeneralError("ADFA has not dead state identified")
            torder, queue = [], []
            _dealS(self.Initial)
            while queue:
                st = queue.pop()
                _dealS(st)
            torder.append(dead)
            return torder

        def _getRdelta(self):
            """
            :returns: pair, map of number of sons map, of reverse conectivity
            :rtype: dict"""
            done = set()
            deltaC, rdelta = {}, {}
            notDone = set(forceIterable(self.Initial))
            while notDone:
                sts = uSet(notDone)
                done.add(sts)
                l = set()
                for k in self.delta.get(sts, []):
                    for std in forceIterable(self.delta[sts][k]):
                        l.add(std)
                        rdelta.setdefault(std, set([])).add(sts)
                        if std not in done:
                            notDone.add(std)
                deltaC[sts] = len(l)
                notDone.remove(sts)
            for s in forceIterable(self.Initial):
                if s not in rdelta:
                    rdelta[s] = set()
            return deltaC, rdelta

        def directRank(self):
            """Compute rank function

            :return: ranf map
            :rtype: dict"""
            r, _ = self.evalRank()
            n = {}
            for x in r:
                for i in r[x]:
                    n[i] = x
            return n

        def evalRank(self):
            """Evaluates the rank map of a automaton

            :return: pair of sets of states by rank map, reverse delta accessability map
            :rtype: tuple"""
            (deltaC, rdelta) = self._getRdelta()
            rank, deltai = {}, {}
            for s in xrange(len(self.States)):
                deltai.setdefault(deltaC[s], set([])).add(s)
            i = -1
            notDone = copy(range(len(self.States)))
            deltaC[self.Dead] = 0
            deltai[1].remove(self.Dead)
            deltai[0] = {self.Dead}
            rdelta[self.Dead].remove(self.Dead)
            while notDone:
                rank[i] = deepcopy(deltai[0])
                deltai[0] = set()
                for s in rank[i]:
                    for s1 in rdelta[s]:
                        l = deltaC[s1]
                        deltaC[s1] = l - 1
                        deltai[l].remove(s1)
                        deltai.setdefault(l - 1, set()).add(s1)
                    notDone.remove(s)
                i += 1
            return rank, rdelta

        def getLeaves(self):
            """The set of leaves, i.e. final states for last symbols of language words

            :return: set of leaves
            :rtype: set"""

            # noinspection PyUnresolvedReferences
            def _last(s1):
                queue, done = {s1}, set()
                while queue:
                    q = queue.pop()
                    done.add(q)
                    for k in self.delta.get(q, {}):
                        for s1 in forceIterable(self.delta[q][k]):
                            if self.finalP(s1):
                                return False
                            elif s1 not in done:
                                queue.add(s1)
                return True

            leaves = set()
            for s in self.Final:
                if _last(s):
                    leaves.add(self.States[s])
            return leaves

    class ADFA(fa.DFA, AFA):
        """Acyclic Deterministic Finite Automata class

        .. inheritance-diagram:: ADFA"""
        def __init__(self):
            fa.DFA.__init__(self)
            AFA.__init__(self)

        def __repr__(self):
            return 'ADFA({0:s})'.format(self.__str__())

        def complete(self, dead=None):
            """Make the ADFA complete

            :param int dead: a state to be identified as dead state if one was not identified yet
            :rtype: ADFA

            .. attention::
               The object is modified in place"""
            if dead is not None:
                self.Dead = dead
            else:
                try:
                    _ = self.Dead
                except AttributeError:
                    foo = self.addState()
                    self.Dead = foo
            for st in range(len(self.States)):
                for k in self.Sigma:
                    if k not in self.delta.get(st, {}).keys():
                        self.addTransition(st, k, self.Dead)
            return self

        def dup(self):
            """Duplicate the basic structure into a new ADFA. Basically a copy.deep.

            :rtype: ADFA"""
            return deepcopy(self)

        def minimalP(self, method=None):
            """Tests if the DFA is minimal

            :param method: minimization algorithm (here void)
            :rtype: bool"""
            foo = self.minimal()
            if self.completeP():
                foo.complete()
            return len(foo) == len(self)

        def forceToDCFA(self):
            """ Conversion to DCFA

            :rtype: DFA"""
            new = fa.DFA()
            new.States = deepcopy(self.States)
            new.Sigma = deepcopy(self.Sigma)
            new.Initial = self.Initial
            new.Final = copy(self.Final)
            for s in self.delta:
                for c in self.delta[s]:
                    new.addTransition(s, c, self.delta[s][c])
            return new

        def wordGenerator(self):
            """Creates a random word generator

            :return: the random word generator
            :rtype: RndWGen

            .. versionadded:: 1.2"""
            return RndWGen(self)

        def minimal(self):
            """Finds the minimal equivalent ADFA

            .. seealso:: [TCS 92 pp 181-189] Minimisation of acyclic deterministic automata in linear time, Dominique Revuz

            :returns: the minimal equivalent ADFA
            :rtype: ADFA"""

            def _getListDelta(ss):
                """returns [([sons,final?],s) for s in ss].sort"""
                l = []
                for s in ss:
                    dl = [new.delta[s][k] for k in new.Sigma]
                    dl.append(s in new.Final)
                    l.append((dl, s))
                l.sort()
                return l

            def _collapse(r1, r2):
                """redirects all transitions going to r2 to r1 and adds r2 to toBeDeleted"""
                for s in rdelta[r2]:
                    for k in new.delta[s]:
                        if new.delta[s][k] == r2:
                            new.delta[s][k] = r1
                toBeDeleted.append(r2)

            if len(self.States) == 1:
                return self
            new = deepcopy(self)
            new.trim()
            new.complete()
            if new.Dead is None:
                deadName = None
            else:
                deadName = new.States[new.Dead]
            (rank, rdelta) = new.evalRank()
            toBeDeleted = []
            maxr = len(rank) - 2
            for r in xrange(maxr + 1):
                ls = _getListDelta(rank[r])
                (d0, s0) = ls[0]
                j = 1
                while j < len(ls):
                    (d1, s1) = ls[j]
                    if d0 == d1:
                        _collapse(s0, s1)
                    else:
                        (d0, s0) = (d1, s1)
                    j += 1
            new.deleteStates(toBeDeleted)
            if deadName is not None:
                new.Dead = new.stateIndex(deadName)
            return new

        def level(self):
            """Computes the level  for each state

            :returns: levels of states
            :rtype: dict

            .. versionadded:: 0.9.8"""
            lvl = {}
            done, alvl = set(), [self.Initial]
            l = 0
            while alvl:
                nlvl = set()
                for i in alvl:
                    lvl[i] = l
                    done.add(i)
                    for c in self.delta[i]:
                        j = self.delta[i][c]
                        if j not in done and j not in alvl:
                            nlvl.add(j)
                l += 1
                alvl = copy(nlvl)
            return lvl

        def _gap(self, l, lvl):
            """Computes the gap value for each pair of states.

            The automata is supposed to have its states named numerically in such way that the initial is zero

            :param int l: length of the longest word
            :param dict lvl: level of each state
            :returns: gap function
            :rtype: dict"""
            def _range(r, s):
                return l - max(lvl[r], lvl[s])
            gp = {}
            n = len(self.States) - 1
            for i in range(n):
                gp[(self.stateIndex(i), self.stateIndex(n))] = l
            if lvl[self.stateIndex(n)] <= l:
                for i in self.Final:
                    gp[(i, self.stateIndex(n))] = 0
            for i in range(n):
                for j in range(i + 1, n):
                    if not self.same_nullability(self.stateIndex(i), self.stateIndex(j)):
                        gp[(self.stateIndex(i), self.stateIndex(j))] = 0
                    else:
                        gp[(self.stateIndex(i), self.stateIndex(j))] = l
            for i in range(n - 2, -1, -1):
                for j in range(n, i, -1):
                    for c in self.Sigma:
                        i1, j1 = self.delta[self.stateIndex(i)][c], self.delta[self.stateIndex(j)][c]
                        if i1 != j1:
                            if int(self.States[i1]) < int(self.States[j1]):
                                g = gp[(i1, j1)]
                            else:
                                g = gp[(j1, i1)]
                            if g + 1 <= _range(self.stateIndex(i), self.stateIndex(j)):
                                gp[(self.stateIndex(i), self.stateIndex(j))] = min(gp[(self.stateIndex(i),
                                                                                       self.stateIndex(j))], g + 1)
            return gp

        def minDFCA(self):
            """Generates a minimal deterministic cover automata from a DFA

            :rtype: DCFA

            .. versionadded:: 0.9.8

            .. seealso::
                Cezar Campeanu, Andrei Päun, and Sheng Yu, An efficient algorithm for constructing minimal cover
                automata for finite languages, IJFCS"""
            new = self.dup()
            if not self.completeP():
                new.complete()
            rank = new.directRank()
            irank = dict((v, [k for (k, xx) in filter(lambda (key, value): value == v, rank.items())])
                         for v in set(rank.values()))
            l = rank[new.Initial]
            lvl = new.level()
            lnames = []
            foo = [x for x in irank]
            foo.sort(reverse=True)
            for i in foo:
                for j in irank[i]:
                    lnames.append(j)
            new.renameStates(lnames)
            g = new._gap(l, lvl)
            P = [False for _ in new.States]
            toMerge = []
            for i in range(len(new.States) - 1):
                if not P[i]:
                    for j in range(i + 1, len(new.States)):
                        if not P[j] and g[(new.stateIndex(i), new.stateIndex(j))] == l:
                            toMerge.append((new.stateIndex(i), new.stateIndex(j)))
                            P[j] = True
            for (a, b) in toMerge:
                new.mergeStates(b, a)
            new.trim()
            new = new.forceToDCFA()
            new.Length = l
            return new

        def trim(self):
            """Remove states that do not lead to a final state, or, inclusively, that can't be reached from the initial
            state. Only useful states remain.

            .. attention:: in place transformation"""
            fa.OFA.trim(self)
            try:
                del self.Dead
            except AttributeError:
                pass
            return self

        def toANFA(self):
            """Converts the ADFA in a equivalent ANFA

            :rtype: ANFA"""
            new = ANFA()
            new.setSigma(copy(self.Sigma))
            new.States = copy(self.States)
            for s in xrange(len(self.States)):
                for k in self.delta.get(s, {}):
                    new.addTransition(s, k, self.delta[s][k])
            new.addInitial(self.Initial)
            for s in self.Final:
                new.addFinal(s)
            return new

        def toNFA(self):
            """Converts the ADFA in a equivalent NFA

            :rtype: ANFA

            .. versionadded:: 1.2"""
            return self.toANFA()

    class RndWGen(object):
        """Word random generator class

        .. versionadded:: 1.2"""
        def __init__(self, aut):
            """
            :param aut: automata recognizing the language
            :type aut: ADFA """
            self.Sigma = list(aut.Sigma)
            self.table = dict()
            self.aut = aut.minimal()
            rank, _ = self.aut.evalRank()
            deltai = self.aut.deltaR()
            mrank = max(rank)
            for i in range(0, mrank + 1):
                for s in rank[i]:
                    self.table.setdefault(s, {})
                    if self.aut.finalP(s):
                        final = 1
                    else:
                        final = 0
                    self.table[s][None] = sum([self.table[s].get(c, 0) for c in self.Sigma])
                    for c in self.Sigma:
                        rs = deltai[s].get(c, [])
                        for r in rs:
                            self.table.setdefault(r, {})
                            self.table[r][c] = self.table[s][None] + final

        @staticmethod
        def _rndChoose(l):
            sm = sum(l)
            r = random.randint(1,sm)
            for i, j in enumerate(l):
                if r <= j:
                    return i
                else:
                    r -= j

        def next(self):
            """Next word

            :return: a new random word"""
            word = Word()
            s = self.aut.Initial
            while True:
                if self.aut.finalP(s) and random.randint(1, self.table[s][None] + 1) == 1:
                    return word
                i = self._rndChoose([self.table[s].get(c, 0) for c in self.Sigma])
                word.append(self.Sigma[i])
                s = self.aut.delta[s][self.Sigma[i]]

    # noinspection PyUnresolvedReferences
    class ANFA(fa.NFA, AFA):
        """Acyclic Nondeterministic Finite Automata class

        .. inheritance-diagram:: ANFA"""

        def moveFinal(self, st, stf):
            """Unsets a set as final transfering transition to another final
            :param int st: the state to be 'moved'
            :param int stf: the destination final state

            .. note::
               stf must be a 'last' final state, i.e., must have no out transitions to anywhere but to a possible dead
               state

            .. attention:: the object is modified in place"""
            (rdelta, _) = self._getRdelta()
            for s in rdelta[st]:
                l = []
                for k in self.delta[s]:
                    if st in self.delta[s][k]:
                        l.append(k)
                for k in l:
                    self.addTransition(s, k, stf)
                self.delFinal(s)

        def mergeStates(self, s1, s2):
            """Merge state s2 into state s1

            :param int s1: state
            :param int s2: state

            .. note::
               no attempt is made to check if the merging preserves the language of teh automaton

            .. attention:: the object is modified in place"""
            (_, rdelta) = self._getRdelta()
            for s in rdelta[s2]:
                l = []
                for k in self.delta[s]:
                    if s2 in self.delta[s][k]:
                        l.append(k)
                for k in l:
                    self.delta[s][k].remove(s2)
                    self.addTransition(s, k, s1)
            for k in self.delta.get(s2, {}):
                for ss in self.delta[s2][k]:
                    self.delta.setdefault(s1, {}).setdefault(k, set()).add(ss)
            self.deleteState(s2)

        def mergeLeaves(self):
            """Merge leaves

            .. attention:: object is modified in place"""
            l = self.getLeaves()
            if len(l):
                s0n = l.pop()
                while l:
                    s0 = self.stateIndex(s0n)
                    s = self.stateIndex(l.pop())
                    self.mergeStates(s0, s)

        def mergeInitial(self):
            """Merge initial states

            .. attention:: object is modified in place"""
            l = copy(self.Initial)
            s0 = self.StateName(l.pop())
            while l:
                s = self.stateIndex(l.pop())
                self.mergeStates(s0, s)

    def sigmaInitialSegment(Sigma, l, exact=False):
        """Generates the ADFA recognizing Sigma^i for i<=l
        :param set Sigma: the alphabet
        :param int l: length
        :param bool exact: only the words with exactly that length?
        :returns: the automaton
        :rtype: ADFA"""
        new = ADFA()
        new.setSigma(Sigma)
        s = new.addState()
        if not exact:
            new.addFinal(s)
        new.setInitial(s)
        for i in range(l):
            s1 = new.addState()
            if not exact or i == l - 1:
                new.addFinal(s1)
            for k in Sigma:
                new.addTransition(s, k, s1)
            s = s1
        return new

    # noinspection PyUnboundLocalVariable
    def genRndTrieBalanced(maxL, Sigma, safe=True):
        """Generates a random trie automaton for a binary language of balanced words of a given leght for max word
        :param int maxL: length of the max word
        :param set Sigma: alphabet to be used
        :param bool safe: should a word of size maxl be present in every language?
        :return: the generated trie automaton
        :rtype: ADFA"""

        def _genEnsurance(m, alphabet):
            l = len(alphabet)
            fair = m / l
            if m % l == 0:
                odd = 0
            else:
                odd = 1
            pool = copy(alphabet)
            c = {}
            sl = []
            while len(sl) < m:
                s1 = random.choice(pool)
                c[s1] = c.get(s1, 0) + 1
                if c[s1] == fair + odd:
                    pool.remove(s1)
                sl.append(s1)
            return sl

        def _legal(cont):
            l = [cont[k1] for k1 in cont]
            return max(l) - min(l) <= 1

        def _descend(s1, ens, safe1, m, cont):
            sons = 0
            if not safe1:
                if _legal(cont):
                    final = random.randint(0, 1)
                else:
                    final = 0
            # noinspection PyUnboundLocalVariable
            if safe1:
                trie.addFinal(s1)
                final = 1
            elif final == 1:
                trie.addFinal(s1)
            if m != 0:
                if safe1:
                    ks = ens.pop()
                else:
                    ks = None
                for k1 in trie.Sigma:
                    ss = trie.addState()
                    trie.addTransition(s1, k1, ss)
                    cont[k1] = cont.get(k1, 0) + 1
                    if _descend(ss, ens, k1 == ks, m - 1, cont):
                        sons += 1
                    cont[k1] -= 1
            if sons == 0 and final == 0:
                trie.deleteState(s1)
                return False
            else:
                return True

        if safe:
            ensurance = _genEnsurance(maxL, Sigma)
        else:
            ensurance = None
        trie = ADFA()
        trie.setSigma(Sigma)
        s = trie.addState()
        trie.setInitial(s)
        contab = {}
        for k in Sigma:
            contab[k] = 0
        _descend(s, ensurance, safe, maxL, contab)
        if random.randint(0, 1) == 1:
            trie.delFinal(s)
        return trie

    # noinspection PyUnboundLocalVariable
    def genRndTrieUnbalanced(maxL, Sigma, ratio, safe=True):
        """Generates a random trie automaton for a binary language of balanced words of a given length for max word

        :param int maxL: length of the max word
        :param set Sigma: alphabet to be used
        :param int ratio: the ratio of the unbalance
        :param bool safe: should a word of size maxl be present in every language?
        :return: the generated trie automaton
        :rtype: ADFA"""

        def _genEnsurance(m, alphabet):
            chief = uSet(alphabet)
            fair = m / (ratio + 1)
            pool = list(copy(alphabet))
            c = {}
            sl = []
            while len(sl) < m:
                s1 = random.choice(pool)
                c[s1] = c.get(s1, 0) + 1
                if len(sl) - c.get(chief, 0) == fair:
                    pool = [chief]
                sl.append(s1)
            return sl

        def _legal(cont):
            l = [cont[k1] for k1 in cont]
            return (ratio + 1) * cont[uSet(Sigma)] >= sum(l)

        # noinspection PyUnboundLocalVariable
        def _descend(s1, ens, safe1, m, cont):
            sons = 0
            if not safe1:
                if _legal(cont):
                    final = random.randint(0, 1)
                else:
                    final = 0
            if safe1:
                trie.addFinal(s1)
                final = 1
            elif final == 1:
                trie.addFinal(s1)
            if m:
                if safe1:
                    ks = ens.pop()
                else:
                    ks = None
                for k1 in trie.Sigma:
                    ss = trie.addState()
                    trie.addTransition(s1, k1, ss)
                    cont[k1] = cont.get(k1, 0) + 1
                    if _descend(ss, ens, k1 == ks, m - 1, cont):
                        sons += 1
                    cont[k1] -= 1
            if sons == 0 and final == 0:
                trie.deleteState(s1)
                return False
            else:
                return True

        if safe:
            ensurance = _genEnsurance(maxL, Sigma)
        else:
            ensurance = None
        trie = ADFA()
        trie.setSigma(Sigma)
        s = trie.addState()
        trie.setInitial(s)
        contab = {}
        for k in Sigma:
            contab[k] = 0
        _descend(s, ensurance, safe, maxL, contab)
        if random.randint(0, 1) == 1:
            trie.delFinal(s)
        return trie

    # noinspection PyUnboundLocalVariable
    def genRandomTrie(maxL, Sigma, safe=True):
        """Generates a random trie automaton for a finite language with a given length for max word
        :param int maxL: length of the max word
        :param set Sigma: alphabet to be used
        :param bool safe: should a word of size maxl be present in every language?
        :return: the generated trie automaton
        :rtype: ADFA"""

        def _genEnsurance(m, alphabet):
            l = len(alphabet)
            sl = list(alphabet)
            return [sl[random.randint(0, l - 1)] for _ in xrange(m)]

        # noinspection PyUnboundLocalVariable
        def _descend(s1, ens, safe1, m):
            sons = 0
            if not safe1:
                final = random.randint(0, 1)
            if safe1:
                trie.addFinal(s1)
                final = 1
            elif final == 1:
                trie.addFinal(s1)
            if m:
                if safe1:
                    ks = ens.pop()
                else:
                    ks = None
                for k in trie.Sigma:
                    ss = trie.addState()
                    trie.addTransition(s1, k, ss)
                    if _descend(ss, ens, k == ks, m - 1):
                        sons += 1
            if sons == 0 and final == 0:
                trie.deleteState(s1)
                return False
            else:
                return True

        if safe:
            ensurance = _genEnsurance(maxL, Sigma)
        else:
            ensurance = None
        trie = ADFA()
        trie.setSigma(Sigma)
        s = trie.addState()
        trie.setInitial(s)
        _descend(s, ensurance, safe, maxL)
        if random.randint(0, 1) == 1:
            trie.delFinal(s)
        return trie

    # noinspection PyUnboundLocalVariable
    def genRndTriePrefix(maxL, Sigma, ClosedP=False, safe=True):
        """Generates a random trie automaton for a finite (either prefix free or prefix closed) language with a given
        length for max word
        :param int maxL: length of the max word
        :param set Sigma: alphabet to be used
        :param bool ClosedP: should it be a prefix closed language?
        :param bool safe: should a word of size maxl be present in every language?
        :return: the generated trie automaton
        :rtype: ADFA"""

        def _genEnsurance(m, alphabet):
            l = len(alphabet)
            sl = list(alphabet)
            return [sl[random.randint(0, l - 1)] for _ in xrange(m)]

        def _descend(s1, ens, saf, m):
            sons = ClosedP
            if m is 0:
                final = random.randint(0, 1)
                if saf or final == 1:
                    trie.addFinal(s1)
                    return True
                else:
                    return False
            else:
                if saf is True:
                    ks = ens.pop()
                else:
                    ks = None
                for k in trie.Sigma:
                    ss = trie.addState()
                    trie.addTransition(s1, k, ss)
                    r = _descend(ss, ens, k == ks, m - 1)
                    if not ClosedP:
                        sons |= r
                    else:
                        sons &= 1
                if not ClosedP:
                    if not sons:
                        final = random.randint(0, 1)
                        if final == 1:
                            trie.addFinal(s1)
                            return True
                        else:
                            return False
                    else:
                        return True
                else:
                    if not sons:
                        final = random.randint(0, 1)
                        if final == 1:
                            trie.addFinal(s1)
                            return True
                        else:
                            return False
                    else:
                        trie.addFinal(s1)
                        return True

        if safe:
            ensurance = _genEnsurance(maxL, Sigma)
        trie = ADFA()
        trie.setSigma(Sigma)
        s = trie.addState()
        trie.setInitial(s)
        _descend(s, ensurance, safe, maxL)
        return trie

    def DFAtoADFA(aut):
        """Transforms an acyclic DFA into a ADFA

        :param DFA aut: the automaton to be transformed
        :raises notAcyclic: if the DFA is not acyclic
        :returns: the converted automaton
        :rtype: ADFA"""
        new = deepcopy(aut)
        new.trim()
        if not new.acyclicP(True):
            raise notAcyclic()
        afa = ADFA()
        afa.States = copy(new.States)
        afa.Sigma = copy(new.Sigma)
        afa.Initial = new.Initial
        afa.delta = copy(new.delta)
        afa.Final = copy(new.Final)
        afa.complete()
        return afa

    def stringToADFA(s):
        """Convert a canonical string representation of a ADFA to a ADFA
        :param list s: the string in its canonical order
        :returns: the ADFA
        :rtype: ADFA

        .. seealso::
            Marco Almeida, Nelma Moreira, and Rogério Reis. Exact generation of minimal acyclic deterministic finite
            automata. International Journal of Foundations of Computer Science, 19(4):751-765, August 2008.
        """
        k = len(s[0]) - 1
        new = ADFA()
        new.setSigma([str(c) for c in range(k)])
        for st, sts in enumerate(s):
            new.addState(str(st))
            for c, s1 in enumerate(sts[:-1]):
                new.addTransition(st, str(c), s1)
            if sts[-1]:
                new.addFinal(st)
        new.setInitial(len(s) - 1)
        return new

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