fp_groups.py

"""Finitely Presented Groups and its algorithms. """

from __future__ import print_function, division
from sympy import S
from sympy.combinatorics.free_groups import (FreeGroup, FreeGroupElement,
                                                free_group)
from sympy.combinatorics.rewritingsystem import RewritingSystem
from sympy.combinatorics.coset_table import (CosetTable,
                                             coset_enumeration_r,
                                             coset_enumeration_c)
from sympy.combinatorics import PermutationGroup
from sympy.printing.defaults import DefaultPrinting
from sympy.utilities import public
from sympy.core.compatibility import string_types

from itertools import product


@public
def fp_group(fr_grp, relators=[]):
    _fp_group = FpGroup(fr_grp, relators)
    return (_fp_group,) + tuple(_fp_group._generators)

@public
def xfp_group(fr_grp, relators=[]):
    _fp_group = FpGroup(fr_grp, relators)
    return (_fp_group, _fp_group._generators)

# Does not work. Both symbols and pollute are undefined. Never tested.
@public
def vfp_group(fr_grpm, relators):
    _fp_group = FpGroup(symbols, relators)
    pollute([sym.name for sym in _fp_group.symbols], _fp_group.generators)
    return _fp_group


def _parse_relators(rels):
    """Parse the passed relators."""
    return rels


###############################################################################
#                           FINITELY PRESENTED GROUPS                         #
###############################################################################


class FpGroup(DefaultPrinting):
    """
    The FpGroup would take a FreeGroup and a list/tuple of relators, the
    relators would be specified in such a way that each of them be equal to the
    identity of the provided free group.

    """
    is_group = True
    is_FpGroup = True
    is_PermutationGroup = False

    def __init__(self, fr_grp, relators):
        relators = _parse_relators(relators)
        self.free_group = fr_grp
        self.relators = relators
        self.generators = self._generators()
        self.dtype = type("FpGroupElement", (FpGroupElement,), {"group": self})

        # CosetTable instance on identity subgroup
        self._coset_table = None
        # returns whether coset table on identity subgroup
        # has been standardized
        self._is_standardized = False

        self._order = None
        self._center = None

        self._rewriting_system = RewritingSystem(self)
        self._perm_isomorphism = None
        return

    def _generators(self):
        return self.free_group.generators

    def make_confluent(self):
        '''
        Try to make the group's rewriting system confluent

        '''
        self._rewriting_system.make_confluent()
        return

    def reduce(self, word):
        '''
        Return the reduced form of `word` in `self` according to the group's
        rewriting system. If it's confluent, the reduced form is the unique normal
        form of the word in the group.

        '''
        return self._rewriting_system.reduce(word)

    def equals(self, word1, word2):
        '''
        Compare `word1` and `word2` for equality in the group
        using the group's rewriting system. If the system is
        confluent, the returned answer is necessarily correct.
        (If it isn't, `False` could be returned in some cases
        where in fact `word1 == word2`)

        '''
        if self.reduce(word1*word2**-1) == self.identity:
            return True
        elif self._rewriting_system.is_confluent:
            return False
        return None

    @property
    def identity(self):
        return self.free_group.identity

    def __contains__(self, g):
        return g in self.free_group

    def subgroup(self, gens, C=None, homomorphism=False):
        '''
        Return the subgroup generated by `gens` using the
        Reidemeister-Schreier algorithm
        homomorphism -- When set to True, return a dictionary containing the images
                     of the presentation generators in the original group.

        Examples
        ========

        >>> from sympy.combinatorics.fp_groups import (FpGroup, FpSubgroup)
        >>> from sympy.combinatorics.free_groups import free_group
        >>> F, x, y = free_group("x, y")
        >>> f = FpGroup(F, [x**3, y**5, (x*y)**2])
        >>> H = [x*y, x**-1*y**-1*x*y*x]
        >>> K, T = f.subgroup(H, homomorphism=True)
        >>> T(K.generators)
        [x*y, x**-1*y**2*x**-1]

        '''

        if not all([isinstance(g, FreeGroupElement) for g in gens]):
            raise ValueError("Generators must be `FreeGroupElement`s")
        if not all([g.group == self.free_group for g in gens]):
                raise ValueError("Given generators are not members of the group")
        if homomorphism:
            g, rels, _gens = reidemeister_presentation(self, gens, C=C, homomorphism=True)
        else:
            g, rels = reidemeister_presentation(self, gens, C=C)
        if g:
            g = FpGroup(g[0].group, rels)
        else:
            g = FpGroup(free_group('')[0], [])
        if homomorphism:
            from sympy.combinatorics.homomorphisms import homomorphism
            return g, homomorphism(g, self, g.generators, _gens, check=False)
        return g

    def coset_enumeration(self, H, strategy="relator_based", max_cosets=None,
                                                        draft=None, incomplete=False):
        """
        Return an instance of ``coset table``, when Todd-Coxeter algorithm is
        run over the ``self`` with ``H`` as subgroup, using ``strategy``
        argument as strategy. The returned coset table is compressed but not
        standardized.

        An instance of `CosetTable` for `fp_grp` can be passed as the keyword
        argument `draft` in which case the coset enumeration will start with
        that instance and attempt to complete it.

        When `incomplete` is `True` and the function is unable to complete for
        some reason, the partially complete table will be returned.

        """
        if not max_cosets:
            max_cosets = CosetTable.coset_table_max_limit
        if strategy == 'relator_based':
            C = coset_enumeration_r(self, H, max_cosets=max_cosets,
                                                    draft=draft, incomplete=incomplete)
        else:
            C = coset_enumeration_c(self, H, max_cosets=max_cosets,
                                                    draft=draft, incomplete=incomplete)
        if C.is_complete():
            C.compress()
        return C

    def standardize_coset_table(self):
        """
        Standardized the coset table ``self`` and makes the internal variable
        ``_is_standardized`` equal to ``True``.

        """
        self._coset_table.standardize()
        self._is_standardized = True

    def coset_table(self, H, strategy="relator_based", max_cosets=None,
                                                 draft=None, incomplete=False):
        """
        Return the mathematical coset table of ``self`` in ``H``.

        """
        if not H:
            if self._coset_table is not None:
                if not self._is_standardized:
                    self.standardize_coset_table()
            else:
                C = self.coset_enumeration([], strategy, max_cosets=max_cosets,
                                            draft=draft, incomplete=incomplete)
                self._coset_table = C
                self.standardize_coset_table()
            return self._coset_table.table
        else:
            C = self.coset_enumeration(H, strategy, max_cosets=max_cosets,
                                            draft=draft, incomplete=incomplete)
            C.standardize()
            return C.table

    def order(self, strategy="relator_based"):
        """
        Returns the order of the finitely presented group ``self``. It uses
        the coset enumeration with identity group as subgroup, i.e ``H=[]``.

        Examples
        ========

        >>> from sympy.combinatorics.free_groups import free_group
        >>> from sympy.combinatorics.fp_groups import FpGroup
        >>> F, x, y = free_group("x, y")
        >>> f = FpGroup(F, [x, y**2])
        >>> f.order(strategy="coset_table_based")
        2

        """
        from sympy import S, gcd
        if self._order is not None:
            return self._order
        if self._coset_table is not None:
            self._order = len(self._coset_table.table)
        elif len(self.relators) == 0:
            self._order = self.free_group.order()
        elif len(self.generators) == 1:
            self._order = abs(gcd([r.array_form[0][1] for r in self.relators]))
        elif self._is_infinite():
            self._order = S.Infinity
        else:
            gens, C = self._finite_index_subgroup()
            if C:
                ind = len(C.table)
                self._order = ind*self.subgroup(gens, C=C).order()
            else:
                self._order = self.index([])
        return self._order

    def _is_infinite(self):
        '''
        Test if the group is infinite. Return `True` if the test succeeds
        and `None` otherwise

        '''
        used_gens = set()
        for r in self.relators:
            used_gens.update(r.contains_generators())
        if any([g not in used_gens for g in self.generators]):
            return True
        # Abelianisation test: check is the abelianisation is infinite
        abelian_rels = []
        from sympy.polys.solvers import RawMatrix as Matrix
        from sympy.polys.domains import ZZ
        from sympy.matrices.normalforms import invariant_factors
        for rel in self.relators:
            abelian_rels.append([rel.exponent_sum(g) for g in self.generators])
        m = Matrix(abelian_rels)
        setattr(m, "ring", ZZ)
        if 0 in invariant_factors(m):
            return True
        else:
            return None


    def _finite_index_subgroup(self, s=[]):
        '''
        Find the elements of `self` that generate a finite index subgroup
        and, if found, return the list of elements and the coset table of `self` by
        the subgroup, otherwise return `(None, None)`

        '''
        gen = self.most_frequent_generator()
        rels = list(self.generators)
        rels.extend(self.relators)
        if not s:
            if len(self.generators) == 2:
                s = [gen] + [g for g in self.generators if g != gen]
            else:
                rand = self.free_group.identity
                i = 0
                while ((rand in rels or rand**-1 in rels or rand.is_identity)
                        and i<10):
                    rand = self.random()
                    i += 1
                s = [gen, rand] + [g for g in self.generators if g != gen]
        mid = (len(s)+1)//2
        half1 = s[:mid]
        half2 = s[mid:]
        draft1 = None
        draft2 = None
        m = 200
        C = None
        while not C and (m/2 < CosetTable.coset_table_max_limit):
            m = min(m, CosetTable.coset_table_max_limit)
            draft1 = self.coset_enumeration(half1, max_cosets=m,
                                 draft=draft1, incomplete=True)
            if draft1.is_complete():
                C = draft1
                half = half1
            else:
                draft2 = self.coset_enumeration(half2, max_cosets=m,
                                 draft=draft2, incomplete=True)
                if draft2.is_complete():
                    C = draft2
                    half = half2
            if not C:
                m *= 2
        if not C:
            return None, None
        C.compress()
        return half, C

    def most_frequent_generator(self):
        gens = self.generators
        rels = self.relators
        freqs = [sum([r.generator_count(g) for r in rels]) for g in gens]
        return gens[freqs.index(max(freqs))]

    def random(self):
        import random
        r = self.free_group.identity
        for i in range(random.randint(2,3)):
            r = r*random.choice(self.generators)**random.choice([1,-1])
        return r

    def index(self, H, strategy="relator_based"):
        """
        Return the index of subgroup ``H`` in group ``self``.

        Examples
        ========

        >>> from sympy.combinatorics.free_groups import free_group
        >>> from sympy.combinatorics.fp_groups import FpGroup
        >>> F, x, y = free_group("x, y")
        >>> f = FpGroup(F, [x**5, y**4, y*x*y**3*x**3])
        >>> f.index([x])
        4

        """
        # TODO: use |G:H| = |G|/|H| (currently H can't be made into a group)
        # when we know |G| and |H|

        if H == []:
            return self.order()
        else:
            C = self.coset_enumeration(H, strategy)
            return len(C.table)

    def __str__(self):
        if self.free_group.rank > 30:
            str_form = "<fp group with %s generators>" % self.free_group.rank
        else:
            str_form = "<fp group on the generators %s>" % str(self.generators)
        return str_form

    __repr__ = __str__

#==============================================================================
#                       PERMUTATION GROUP METHODS
#==============================================================================

    def _to_perm_group(self):
        '''
        Return an isomorphic permutation group and the isomorphism.
        The implementation is dependent on coset enumeration so
        will only terminate for finite groups.

        '''
        from sympy.combinatorics import Permutation, PermutationGroup
        from sympy.combinatorics.homomorphisms import homomorphism
        if self.order() == S.Infinity:
            raise NotImplementedError("Permutation presentation of infinite "
                                                  "groups is not implemented")
        if self._perm_isomorphism:
            T = self._perm_isomorphism
            P = T.image()
        else:
            C = self.coset_table([])
            gens = self.generators
            images = [[C[i][2*gens.index(g)] for i in range(len(C))] for g in gens]
            images = [Permutation(i) for i in images]
            P = PermutationGroup(images)
            T = homomorphism(self, P, gens, images, check=False)
            self._perm_isomorphism = T
        return P, T

    def _perm_group_list(self, method_name, *args):
        '''
        Given the name of a `PermutationGroup` method (returning a subgroup
        or a list of subgroups) and (optionally) additional arguments it takes,
        return a list or a list of lists containing the generators of this (or
        these) subgroups in terms of the generators of `self`.

        '''
        P, T = self._to_perm_group()
        perm_result = getattr(P, method_name)(*args)
        single = False
        if isinstance(perm_result, PermutationGroup):
            perm_result, single = [perm_result], True
        result = []
        for group in perm_result:
            gens = group.generators
            result.append(T.invert(gens))
        return result[0] if single else result

    def derived_series(self):
        '''
        Return the list of lists containing the generators
        of the subgroups in the derived series of `self`.

        '''
        return self._perm_group_list('derived_series')

    def lower_central_series(self):
        '''
        Return the list of lists containing the generators
        of the subgroups in the lower central series of `self`.

        '''
        return self._perm_group_list('lower_central_series')

    def center(self):
        '''
        Return the list of generators of the center of `self`.

        '''
        return self._perm_group_list('center')


    def derived_subgroup(self):
        '''
        Return the list of generators of the derived subgroup of `self`.

        '''
        return self._perm_group_list('derived_subgroup')


    def centralizer(self, other):
        '''
        Return the list of generators of the centralizer of `other`
        (a list of elements of `self`) in `self`.

        '''
        T = self._to_perm_group()[1]
        other = T(other)
        return self._perm_group_list('centralizer', other)

    def normal_closure(self, other):
        '''
        Return the list of generators of the normal closure of `other`
        (a list of elements of `self`) in `self`.

        '''
        T = self._to_perm_group()[1]
        other = T(other)
        return self._perm_group_list('normal_closure', other)

    def _perm_property(self, attr):
        '''
        Given an attribute of a `PermutationGroup`, return
        its value for a permutation group isomorphic to `self`.

        '''
        P = self._to_perm_group()[0]
        return getattr(P, attr)

    @property
    def is_abelian(self):
        '''
        Check if `self` is abelian.

        '''
        return self._perm_property("is_abelian")

    @property
    def is_nilpotent(self):
        '''
        Check if `self` is nilpotent.

        '''
        return self._perm_property("is_nilpotent")

    @property
    def is_solvable(self):
        '''
        Check if `self` is solvable.

        '''
        return self._perm_property("is_solvable")

    @property
    def elements(self):
        '''
        List the elements of `self`.

        '''
        P, T = self._to_perm_group()
        return T.invert(P._elements)

    @property
    def is_cyclic(self):
        """
        Return ``True`` if group is Cyclic.

        """
        if len(self.generators) <= 1:
            return True
        try:
            P, T = self._to_perm_group()
        except NotImplementedError:
            raise NotImplementedError("Check for infinite Cyclic group "
                                      "is not implemented")
        return P.is_cyclic

    def abelian_invariants(self):
        """
        Returns Abelian Invariants of a group.
        """
        try:
            P, T = self._to_perm_group()
        except NotImplementedError:
            raise NotImplementedError("abelian invariants is not implemented"
                                      "for infinite group")
        return P.abelian_invariants()


class FpSubgroup(DefaultPrinting):
    '''
    The class implementing a subgroup of an FpGroup or a FreeGroup
    (only finite index subgroups are supported at this point). This
    is to be used if one wishes to check if an element of the original
    group belongs to the subgroup

    '''
    def __init__(self, G, gens, normal=False):
        super(FpSubgroup,self).__init__()
        self.parent = G
        self.generators = list(set([g for g in gens if g != G.identity]))
        self._min_words = None #for use in __contains__
        self.C = None
        self.normal = normal

    def __contains__(self, g):

        if isinstance(self.parent, FreeGroup):
            if self._min_words is None:
                # make _min_words - a list of subwords such that
                # g is in the subgroup if and only if it can be
                # partitioned into these subwords. Infinite families of
                # subwords are presented by tuples, e.g. (r, w)
                # stands for the family of subwords r*w**n*r**-1

                def _process(w):
                    # this is to be used before adding new words
                    # into _min_words; if the word w is not cyclically
                    # reduced, it will generate an infinite family of
                    # subwords so should be written as a tuple;
                    # if it is, w**-1 should be added to the list
                    # as well
                    p, r = w.cyclic_reduction(removed=True)
                    if not r.is_identity:
                        return [(r, p)]
                    else:
                        return [w, w**-1]

                # make the initial list
                gens = []
                for w in self.generators:
                    if self.normal:
                        w = w.cyclic_reduction()
                    gens.extend(_process(w))

                for w1 in gens:
                    for w2 in gens:
                        # if w1 and w2 are equal or are inverses, continue
                        if w1 == w2 or (not isinstance(w1, tuple)
                                                        and w1**-1 == w2):
                            continue

                        # if the start of one word is the inverse of the
                        # end of the other, their multiple should be added
                        # to _min_words because of cancellation
                        if isinstance(w1, tuple):
                            # start, end
                            s1, s2 = w1[0][0], w1[0][0]**-1
                        else:
                            s1, s2 = w1[0], w1[len(w1)-1]

                        if isinstance(w2, tuple):
                            # start, end
                            r1, r2 = w2[0][0], w2[0][0]**-1
                        else:
                            r1, r2 = w2[0], w2[len(w1)-1]

                        # p1 and p2 are w1 and w2 or, in case when
                        # w1 or w2 is an infinite family, a representative
                        p1, p2 = w1, w2
                        if isinstance(w1, tuple):
                            p1 = w1[0]*w1[1]*w1[0]**-1
                        if isinstance(w2, tuple):
                            p2 = w2[0]*w2[1]*w2[0]**-1

                        # add the product of the words to the list is necessary
                        if r1**-1 == s2 and not (p1*p2).is_identity:
                            new = _process(p1*p2)
                            if not new in gens:
                                gens.extend(new)

                        if r2**-1 == s1 and not (p2*p1).is_identity:
                            new = _process(p2*p1)
                            if not new in gens:
                                gens.extend(new)

                self._min_words = gens

            min_words = self._min_words

            def _is_subword(w):
                # check if w is a word in _min_words or one of
                # the infinite families in it
                w, r = w.cyclic_reduction(removed=True)
                if r.is_identity or self.normal:
                    return w in min_words
                else:
                    t = [s[1] for s in min_words if isinstance(s, tuple)
                                                                and s[0] == r]
                    return [s for s in t if w.power_of(s)] != []

            # store the solution of words for which the result of
            # _word_break (below) is known
            known = {}

            def _word_break(w):
                # check if w can be written as a product of words
                # in min_words
                if len(w) == 0:
                    return True
                i = 0
                while i < len(w):
                    i += 1
                    prefix = w.subword(0, i)
                    if not _is_subword(prefix):
                        continue
                    rest = w.subword(i, len(w))
                    if rest not in known:
                        known[rest] = _word_break(rest)
                    if known[rest]:
                        return True
                return False

            if self.normal:
                g = g.cyclic_reduction()
            return _word_break(g)
        else:
            if self.C is None:
                C = self.parent.coset_enumeration(self.generators)
                self.C = C
            i = 0
            C = self.C
            for j in range(len(g)):
                i = C.table[i][C.A_dict[g[j]]]
            return i == 0

    def order(self):
        from sympy import S
        if not self.generators:
            return 1
        if isinstance(self.parent, FreeGroup):
            return S.Infinity
        if self.C is None:
            C = self.parent.coset_enumeration(self.generators)
            self.C = C
        # This is valid because `len(self.C.table)` (the index of the subgroup)
        # will always be finite - otherwise coset enumeration doesn't terminate
        return self.parent.order()/len(self.C.table)

    def to_FpGroup(self):
        if isinstance(self.parent, FreeGroup):
            gen_syms = [('x_%d'%i) for i in range(len(self.generators))]
            return free_group(', '.join(gen_syms))[0]
        return self.parent.subgroup(C=self.C)

    def __str__(self):
        if len(self.generators) > 30:
            str_form = "<fp subgroup with %s generators>" % len(self.generators)
        else:
            str_form = "<fp subgroup on the generators %s>" % str(self.generators)
        return str_form

    __repr__ = __str__


###############################################################################
#                           LOW INDEX SUBGROUPS                               #
###############################################################################

def low_index_subgroups(G, N, Y=[]):
    """
    Implements the Low Index Subgroups algorithm, i.e find all subgroups of
    ``G`` upto a given index ``N``. This implements the method described in
    [Sim94]. This procedure involves a backtrack search over incomplete Coset
    Tables, rather than over forced coincidences.

    Parameters
    ==========

    G: An FpGroup < X|R >
    N: positive integer, representing the maximum index value for subgroups
    Y: (an optional argument) specifying a list of subgroup generators, such
    that each of the resulting subgroup contains the subgroup generated by Y.

    Examples
    ========

    >>> from sympy.combinatorics.free_groups import free_group
    >>> from sympy.combinatorics.fp_groups import FpGroup, low_index_subgroups
    >>> F, x, y = free_group("x, y")
    >>> f = FpGroup(F, [x**2, y**3, (x*y)**4])
    >>> L = low_index_subgroups(f, 4)
    >>> for coset_table in L:
    ...     print(coset_table.table)
    [[0, 0, 0, 0]]
    [[0, 0, 1, 2], [1, 1, 2, 0], [3, 3, 0, 1], [2, 2, 3, 3]]
    [[0, 0, 1, 2], [2, 2, 2, 0], [1, 1, 0, 1]]
    [[1, 1, 0, 0], [0, 0, 1, 1]]

    References
    ==========

    .. [1] Holt, D., Eick, B., O'Brien, E.
           "Handbook of Computational Group Theory"
           Section 5.4

    .. [2] Marston Conder and Peter Dobcsanyi
           "Applications and Adaptions of the Low Index Subgroups Procedure"

    """
    C = CosetTable(G, [])
    R = G.relators
    # length chosen for the length of the short relators
    len_short_rel = 5
    # elements of R2 only checked at the last step for complete
    # coset tables
    R2 = set([rel for rel in R if len(rel) > len_short_rel])
    # elements of R1 are used in inner parts of the process to prune
    # branches of the search tree,
    R1 = set([rel.identity_cyclic_reduction() for rel in set(R) - R2])
    R1_c_list = C.conjugates(R1)
    S = []
    descendant_subgroups(S, C, R1_c_list, C.A[0], R2, N, Y)
    return S


def descendant_subgroups(S, C, R1_c_list, x, R2, N, Y):
    A_dict = C.A_dict
    A_dict_inv = C.A_dict_inv
    if C.is_complete():
        # if C is complete then it only needs to test
        # whether the relators in R2 are satisfied
        for w, alpha in product(R2, C.omega):
            if not C.scan_check(alpha, w):
                return
        # relators in R2 are satisfied, append the table to list
        S.append(C)
    else:
        # find the first undefined entry in Coset Table
        for alpha, x in product(range(len(C.table)), C.A):
            if C.table[alpha][A_dict[x]] is None:
                # this is "x" in pseudo-code (using "y" makes it clear)
                undefined_coset, undefined_gen = alpha, x
                break
        # for filling up the undefine entry we try all possible values
        # of beta in Omega or beta = n where beta^(undefined_gen^-1) is undefined
        reach = C.omega + [C.n]
        for beta in reach:
            if beta < N:
                if beta == C.n or C.table[beta][A_dict_inv[undefined_gen]] is None:
                    try_descendant(S, C, R1_c_list, R2, N, undefined_coset, \
                            undefined_gen, beta, Y)


def try_descendant(S, C, R1_c_list, R2, N, alpha, x, beta, Y):
    r"""
    Solves the problem of trying out each individual possibility
    for `\alpha^x.

    """
    D = C.copy()
    if beta == D.n and beta < N:
        D.table.append([None]*len(D.A))
        D.p.append(beta)
    D.table[alpha][D.A_dict[x]] = beta
    D.table[beta][D.A_dict_inv[x]] = alpha
    D.deduction_stack.append((alpha, x))
    if not D.process_deductions_check(R1_c_list[D.A_dict[x]], \
            R1_c_list[D.A_dict_inv[x]]):
        return
    for w in Y:
        if not D.scan_check(0, w):
            return
    if first_in_class(D, Y):
        descendant_subgroups(S, D, R1_c_list, x, R2, N, Y)


def first_in_class(C, Y=[]):
    """
    Checks whether the subgroup ``H=G1`` corresponding to the Coset Table
    could possibly be the canonical representative of its conjugacy class.

    Parameters
    ==========

    C: CosetTable

    Returns
    =======

    bool: True/False

    If this returns False, then no descendant of C can have that property, and
    so we can abandon C. If it returns True, then we need to process further
    the node of the search tree corresponding to C, and so we call
    ``descendant_subgroups`` recursively on C.

    Examples
    ========

    >>> from sympy.combinatorics.free_groups import free_group
    >>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, first_in_class
    >>> F, x, y = free_group("x, y")
    >>> f = FpGroup(F, [x**2, y**3, (x*y)**4])
    >>> C = CosetTable(f, [])
    >>> C.table = [[0, 0, None, None]]
    >>> first_in_class(C)
    True
    >>> C.table = [[1, 1, 1, None], [0, 0, None, 1]]; C.p = [0, 1]
    >>> first_in_class(C)
    True
    >>> C.table = [[1, 1, 2, 1], [0, 0, 0, None], [None, None, None, 0]]
    >>> C.p = [0, 1, 2]
    >>> first_in_class(C)
    False
    >>> C.table = [[1, 1, 1, 2], [0, 0, 2, 0], [2, None, 0, 1]]
    >>> first_in_class(C)
    False

    # TODO:: Sims points out in [Sim94] that performance can be improved by
    # remembering some of the information computed by ``first_in_class``. If
    # the ``continue alpha`` statement is executed at line 14, then the same thing
    # will happen for that value of alpha in any descendant of the table C, and so
    # the values the values of alpha for which this occurs could profitably be
    # stored and passed through to the descendants of C. Of course this would
    # make the code more complicated.

    # The code below is taken directly from the function on page 208 of [Sim94]
    # nu[alpha]

    """
    n = C.n
    # lamda is the largest numbered point in Omega_c_alpha which is currently defined
    lamda = -1
    # for alpha in Omega_c, nu[alpha] is the point in Omega_c_alpha corresponding to alpha
    nu = [None]*n
    # for alpha in Omega_c_alpha, mu[alpha] is the point in Omega_c corresponding to alpha
    mu = [None]*n
    # mutually nu and mu are the mutually-inverse equivalence maps between
    # Omega_c_alpha and Omega_c
    next_alpha = False
    # For each 0!=alpha in [0 .. nc-1], we start by constructing the equivalent
    # standardized coset table C_alpha corresponding to H_alpha
    for alpha in range(1, n):
        # reset nu to "None" after previous value of alpha
        for beta in range(lamda+1):
            nu[mu[beta]] = None
        # we only want to reject our current table in favour of a preceding
        # table in the ordering in which 1 is replaced by alpha, if the subgroup
        # G_alpha corresponding to this preceding table definitely contains the
        # given subgroup
        for w in Y:
            # TODO: this should support input of a list of general words
            # not just the words which are in "A" (i.e gen and gen^-1)
            if C.table[alpha][C.A_dict[w]] != alpha:
                # continue with alpha
                next_alpha = True
                break
        if next_alpha:
            next_alpha = False
            continue
        # try alpha as the new point 0 in Omega_C_alpha
        mu[0] = alpha
        nu[alpha] = 0
        # compare corresponding entries in C and C_alpha
        lamda = 0
        for beta in range(n):
            for x in C.A:
                gamma = C.table[beta][C.A_dict[x]]
                delta = C.table[mu[beta]][C.A_dict[x]]
                # if either of the entries is undefined,
                # we move with next alpha
                if gamma is None or delta is None:
                    # continue with alpha
                    next_alpha = True
                    break
                if nu[delta] is None:
                    # delta becomes the next point in Omega_C_alpha
                    lamda += 1
                    nu[delta] = lamda
                    mu[lamda] = delta
                if nu[delta] < gamma:
                    return False
                if nu[delta] > gamma:
                    # continue with alpha
                    next_alpha = True
                    break
            if next_alpha:
                next_alpha = False
                break
    return True

#========================================================================
#                    Simplifying Presentation
#========================================================================

def simplify_presentation(*args, **kwargs):
    '''
    For an instance of `FpGroup`, return a simplified isomorphic copy of
    the group (e.g. remove redundant generators or relators). Alternatively,
    a list of generators and relators can be passed in which case the
    simplified lists will be returned.

    By default, the generators of the group are unchanged. If you would
    like to remove redundant generators, set the keyword argument
    `change_gens = True`.

    '''
    change_gens = kwargs.get("change_gens", False)

    if len(args) == 1:
        if not isinstance(args[0], FpGroup):
            raise TypeError("The argument must be an instance of FpGroup")
        G = args[0]
        gens, rels = simplify_presentation(G.generators, G.relators,
                                              change_gens=change_gens)
        if gens:
            return FpGroup(gens[0].group, rels)
        return FpGroup(FreeGroup([]), [])
    elif len(args) == 2:
        gens, rels = args[0][:], args[1][:]
        if not gens:
            return gens, rels
        identity = gens[0].group.identity
    else:
        if len(args) == 0:
            m = "Not enough arguments"
        else:
            m = "Too many arguments"
        raise RuntimeError(m)

    prev_gens = []
    prev_rels = []
    while not set(prev_rels) == set(rels):
        prev_rels = rels
        while change_gens and not set(prev_gens) == set(gens):
            prev_gens = gens
            gens, rels = elimination_technique_1(gens, rels, identity)
        rels = _simplify_relators(rels, identity)

    if change_gens:
        syms = [g.array_form[0][0] for g in gens]
        F = free_group(syms)[0]
        identity = F.identity
        gens = F.generators
        subs = dict(zip(syms, gens))
        for j, r in enumerate(rels):
            a = r.array_form
            rel = identity
            for sym, p in a:
                rel = rel*subs[sym]**p
            rels[j] = rel
    return gens, rels

def _simplify_relators(rels, identity):
    """Relies upon ``_simplification_technique_1`` for its functioning. """
    rels = rels[:]

    rels = list(set(_simplification_technique_1(rels)))
    rels.sort()
    rels = [r.identity_cyclic_reduction() for r in rels]
    try:
        rels.remove(identity)
    except ValueError:
        pass
    return rels

# Pg 350, section 2.5.1 from [2]
def elimination_technique_1(gens, rels, identity):
    rels = rels[:]
    # the shorter relators are examined first so that generators selected for
    # elimination will have shorter strings as equivalent
    rels.sort()
    gens = gens[:]
    redundant_gens = {}
    redundant_rels = []
    used_gens = set()
    # examine each relator in relator list for any generator occurring exactly
    # once
    for rel in rels:
        # don't look for a redundant generator in a relator which
        # depends on previously found ones
        contained_gens = rel.contains_generators()
        if any([g in contained_gens for g in redundant_gens]):
            continue
        contained_gens = list(contained_gens)
        contained_gens.sort(reverse = True)
        for gen in contained_gens:
            if rel.generator_count(gen) == 1 and gen not in used_gens:
                k = rel.exponent_sum(gen)
                gen_index = rel.index(gen**k)
                bk = rel.subword(gen_index + 1, len(rel))
                fw = rel.subword(0, gen_index)
                chi = bk*fw
                redundant_gens[gen] = chi**(-1*k)
                used_gens.update(chi.contains_generators())
                redundant_rels.append(rel)
                break
    rels = [r for r in rels if r not in redundant_rels]
    # eliminate the redundant generators from remaining relators
    rels = [r.eliminate_words(redundant_gens, _all = True).identity_cyclic_reduction() for r in rels]
    rels = list(set(rels))
    try:
        rels.remove(identity)
    except ValueError:
        pass
    gens = [g for g in gens if g not in redundant_gens]
    return gens, rels

def _simplification_technique_1(rels):
    """
    All relators are checked to see if they are of the form `gen^n`. If any
    such relators are found then all other relators are processed for strings
    in the `gen` known order.

    Examples
    ========

    >>> from sympy.combinatorics.free_groups import free_group
    >>> from sympy.combinatorics.fp_groups import _simplification_technique_1
    >>> F, x, y = free_group("x, y")
    >>> w1 = [x**2*y**4, x**3]
    >>> _simplification_technique_1(w1)
    [x**-1*y**4, x**3]

    >>> w2 = [x**2*y**-4*x**5, x**3, x**2*y**8, y**5]
    >>> _simplification_technique_1(w2)
    [x**-1*y*x**-1, x**3, x**-1*y**-2, y**5]

    >>> w3 = [x**6*y**4, x**4]
    >>> _simplification_technique_1(w3)
    [x**2*y**4, x**4]

    """
    from sympy import gcd

    rels = rels[:]
    # dictionary with "gen: n" where gen^n is one of the relators
    exps = {}
    for i in range(len(rels)):
        rel = rels[i]
        if rel.number_syllables() == 1:
            g = rel[0]
            exp = abs(rel.array_form[0][1])
            if rel.array_form[0][1] < 0:
                rels[i] = rels[i]**-1
                g = g**-1
            if g in exps:
                exp = gcd(exp, exps[g].array_form[0][1])
            exps[g] = g**exp

    one_syllables_words = exps.values()
    # decrease some of the exponents in relators, making use of the single
    # syllable relators
    for i in range(len(rels)):
        rel = rels[i]
        if rel in one_syllables_words:
            continue
        rel = rel.eliminate_words(one_syllables_words, _all = True)
        # if rels[i] contains g**n where abs(n) is greater than half of the power p
        # of g in exps, g**n can be replaced by g**(n-p) (or g**(p-n) if n<0)
        for g in rel.contains_generators():
            if g in exps:
                exp = exps[g].array_form[0][1]
                max_exp = (exp + 1)//2
                rel = rel.eliminate_word(g**(max_exp), g**(max_exp-exp), _all = True)
                rel = rel.eliminate_word(g**(-max_exp), g**(-(max_exp-exp)), _all = True)
        rels[i] = rel
    rels = [r.identity_cyclic_reduction() for r in rels]
    return rels


###############################################################################
#                           SUBGROUP PRESENTATIONS                            #
###############################################################################

# Pg 175 [1]
def define_schreier_generators(C, homomorphism=False):
    '''
    Parameters
    ==========

    C -- Coset table.
    homomorphism -- When set to True, return a dictionary containing the images
                     of the presentation generators in the original group.
    '''
    y = []
    gamma = 1
    f = C.fp_group
    X = f.generators
    if homomorphism:
        # `_gens` stores the elements of the parent group to
        # to which the schreier generators correspond to.
        _gens = {}
        # compute the schreier Traversal
        tau = {}
        tau[0] = f.identity
    C.P = [[None]*len(C.A) for i in range(C.n)]
    for alpha, x in product(C.omega, C.A):
        beta = C.table[alpha][C.A_dict[x]]
        if beta == gamma:
            C.P[alpha][C.A_dict[x]] = "<identity>"
            C.P[beta][C.A_dict_inv[x]] = "<identity>"
            gamma += 1
            if homomorphism:
                tau[beta] = tau[alpha]*x
        elif x in X and C.P[alpha][C.A_dict[x]] is None:
            y_alpha_x = '%s_%s' % (x, alpha)
            y.append(y_alpha_x)
            C.P[alpha][C.A_dict[x]] = y_alpha_x
            if homomorphism:
                _gens[y_alpha_x] = tau[alpha]*x*tau[beta]**-1
    grp_gens = list(free_group(', '.join(y)))
    C._schreier_free_group = grp_gens.pop(0)
    C._schreier_generators = grp_gens
    if homomorphism:
        C._schreier_gen_elem = _gens
    # replace all elements of P by, free group elements
    for i, j in product(range(len(C.P)), range(len(C.A))):
        # if equals "<identity>", replace by identity element
        if C.P[i][j] == "<identity>":
            C.P[i][j] = C._schreier_free_group.identity
        elif isinstance(C.P[i][j], string_types):
            r = C._schreier_generators[y.index(C.P[i][j])]
            C.P[i][j] = r
            beta = C.table[i][j]
            C.P[beta][j + 1] = r**-1

def reidemeister_relators(C):
    R = C.fp_group.relators
    rels = [rewrite(C, coset, word) for word in R for coset in range(C.n)]
    order_1_gens = set([i for i in rels if len(i) == 1])

    # remove all the order 1 generators from relators
    rels = list(filter(lambda rel: rel not in order_1_gens, rels))

    # replace order 1 generators by identity element in reidemeister relators
    for i in range(len(rels)):
        w = rels[i]
        w = w.eliminate_words(order_1_gens, _all=True)
        rels[i] = w

    C._schreier_generators = [i for i in C._schreier_generators
                    if not (i in order_1_gens or i**-1 in order_1_gens)]

    # Tietze transformation 1 i.e TT_1
    # remove cyclic conjugate elements from relators
    i = 0
    while i < len(rels):
        w = rels[i]
        j = i + 1
        while j < len(rels):
            if w.is_cyclic_conjugate(rels[j]):
                del rels[j]
            else:
                j += 1
        i += 1

    C._reidemeister_relators = rels


def rewrite(C, alpha, w):
    """
    Parameters
    ==========

    C: CosetTable
    alpha: A live coset
    w: A word in `A*`

    Returns
    =======

    rho(tau(alpha), w)

    Examples
    ========

    >>> from sympy.combinatorics.fp_groups import FpGroup, CosetTable, define_schreier_generators, rewrite
    >>> from sympy.combinatorics.free_groups import free_group
    >>> F, x, y = free_group("x ,y")
    >>> f = FpGroup(F, [x**2, y**3, (x*y)**6])
    >>> C = CosetTable(f, [])
    >>> C.table = [[1, 1, 2, 3], [0, 0, 4, 5], [4, 4, 3, 0], [5, 5, 0, 2], [2, 2, 5, 1], [3, 3, 1, 4]]
    >>> C.p = [0, 1, 2, 3, 4, 5]
    >>> define_schreier_generators(C)
    >>> rewrite(C, 0, (x*y)**6)
    x_4*y_2*x_3*x_1*x_2*y_4*x_5

    """
    v = C._schreier_free_group.identity
    for i in range(len(w)):
        x_i = w[i]
        v = v*C.P[alpha][C.A_dict[x_i]]
        alpha = C.table[alpha][C.A_dict[x_i]]
    return v

# Pg 350, section 2.5.2 from [2]
def elimination_technique_2(C):
    """
    This technique eliminates one generator at a time. Heuristically this
    seems superior in that we may select for elimination the generator with
    shortest equivalent string at each stage.

    >>> from sympy.combinatorics.free_groups import free_group
    >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r, \
            reidemeister_relators, define_schreier_generators, elimination_technique_2
    >>> F, x, y = free_group("x, y")
    >>> f = FpGroup(F, [x**3, y**5, (x*y)**2]); H = [x*y, x**-1*y**-1*x*y*x]
    >>> C = coset_enumeration_r(f, H)
    >>> C.compress(); C.standardize()
    >>> define_schreier_generators(C)
    >>> reidemeister_relators(C)
    >>> elimination_technique_2(C)
    ([y_1, y_2], [y_2**-3, y_2*y_1*y_2*y_1*y_2*y_1, y_1**2])

    """
    rels = C._reidemeister_relators
    rels.sort(reverse=True)
    gens = C._schreier_generators
    for i in range(len(gens) - 1, -1, -1):
        rel = rels[i]
        for j in range(len(gens) - 1, -1, -1):
            gen = gens[j]
            if rel.generator_count(gen) == 1:
                k = rel.exponent_sum(gen)
                gen_index = rel.index(gen**k)
                bk = rel.subword(gen_index + 1, len(rel))
                fw = rel.subword(0, gen_index)
                rep_by = (bk*fw)**(-1*k)
                del rels[i]; del gens[j]
                for l in range(len(rels)):
                    rels[l] = rels[l].eliminate_word(gen, rep_by)
                break
    C._reidemeister_relators = rels
    C._schreier_generators = gens
    return C._schreier_generators, C._reidemeister_relators

def reidemeister_presentation(fp_grp, H, C=None, homomorphism=False):
    """
    Parameters
    ==========

    fp_group: A finitely presented group, an instance of FpGroup
    H: A subgroup whose presentation is to be found, given as a list
    of words in generators of `fp_grp`
    homomorphism: When set to True, return a homomorphism from the subgroup
                    to the parent group

    Examples
    ========

    >>> from sympy.combinatorics.free_groups import free_group
    >>> from sympy.combinatorics.fp_groups import FpGroup, reidemeister_presentation
    >>> F, x, y = free_group("x, y")

    Example 5.6 Pg. 177 from [1]
    >>> f = FpGroup(F, [x**3, y**5, (x*y)**2])
    >>> H = [x*y, x**-1*y**-1*x*y*x]
    >>> reidemeister_presentation(f, H)
    ((y_1, y_2), (y_1**2, y_2**3, y_2*y_1*y_2*y_1*y_2*y_1))

    Example 5.8 Pg. 183 from [1]
    >>> f = FpGroup(F, [x**3, y**3, (x*y)**3])
    >>> H = [x*y, x*y**-1]
    >>> reidemeister_presentation(f, H)
    ((x_0, y_0), (x_0**3, y_0**3, x_0*y_0*x_0*y_0*x_0*y_0))

    Exercises Q2. Pg 187 from [1]
    >>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
    >>> H = [x]
    >>> reidemeister_presentation(f, H)
    ((x_0,), (x_0**4,))

    Example 5.9 Pg. 183 from [1]
    >>> f = FpGroup(F, [x**3*y**-3, (x*y)**3, (x*y**-1)**2])
    >>> H = [x]
    >>> reidemeister_presentation(f, H)
    ((x_0,), (x_0**6,))

    """
    if not C:
        C = coset_enumeration_r(fp_grp, H)
    C.compress(); C.standardize()
    define_schreier_generators(C, homomorphism=homomorphism)
    reidemeister_relators(C)
    gens, rels = C._schreier_generators, C._reidemeister_relators
    gens, rels = simplify_presentation(gens, rels, change_gens=True)

    C.schreier_generators = tuple(gens)
    C.reidemeister_relators = tuple(rels)

    if homomorphism:
        _gens = []
        for gen in gens:
            _gens.append(C._schreier_gen_elem[str(gen)])
        return C.schreier_generators, C.reidemeister_relators, _gens

    return C.schreier_generators, C.reidemeister_relators


FpGroupElement = FreeGroupElement