venv/lib/python3.11/site-packages/sympy/combinatorics/coset_table.py

from sympy.combinatorics.free_groups import free_group
from sympy.printing.defaults import DefaultPrinting

from itertools import chain, product
from bisect import bisect_left


###############################################################################
#                           COSET TABLE                                       #
###############################################################################

class CosetTable(DefaultPrinting):
    # coset_table: Mathematically a coset table
    #               represented using a list of lists
    # alpha: Mathematically a coset (precisely, a live coset)
    #       represented by an integer between i with 1 <= i <= n
    #       alpha in c
    # x: Mathematically an element of "A" (set of generators and
    #   their inverses), represented using "FpGroupElement"
    # fp_grp: Finitely Presented Group with < X|R > as presentation.
    # H: subgroup of fp_grp.
    # NOTE: We start with H as being only a list of words in generators
    #       of "fp_grp". Since `.subgroup` method has not been implemented.

    r"""

    Properties
    ==========

    [1] `0 \in \Omega` and `\tau(1) = \epsilon`
    [2] `\alpha^x = \beta \Leftrightarrow \beta^{x^{-1}} = \alpha`
    [3] If `\alpha^x = \beta`, then `H \tau(\alpha)x = H \tau(\beta)`
    [4] `\forall \alpha \in \Omega, 1^{\tau(\alpha)} = \alpha`

    References
    ==========

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

    .. [2] John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
           Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490.
           "Implementation and Analysis of the Todd-Coxeter Algorithm"

    """
    # default limit for the number of cosets allowed in a
    # coset enumeration.
    coset_table_max_limit = 4096000
    # limit for the current instance
    coset_table_limit = None
    # maximum size of deduction stack above or equal to
    # which it is emptied
    max_stack_size = 100

    def __init__(self, fp_grp, subgroup, max_cosets=None):
        if not max_cosets:
            max_cosets = CosetTable.coset_table_max_limit
        self.fp_group = fp_grp
        self.subgroup = subgroup
        self.coset_table_limit = max_cosets
        # "p" is setup independent of Omega and n
        self.p = [0]
        # a list of the form `[gen_1, gen_1^{-1}, ... , gen_k, gen_k^{-1}]`
        self.A = list(chain.from_iterable((gen, gen**-1) \
                for gen in self.fp_group.generators))
        #P[alpha, x] Only defined when alpha^x is defined.
        self.P = [[None]*len(self.A)]
        # the mathematical coset table which is a list of lists
        self.table = [[None]*len(self.A)]
        self.A_dict = {x: self.A.index(x) for x in self.A}
        self.A_dict_inv = {}
        for x, index in self.A_dict.items():
            if index % 2 == 0:
                self.A_dict_inv[x] = self.A_dict[x] + 1
            else:
                self.A_dict_inv[x] = self.A_dict[x] - 1
        # used in the coset-table based method of coset enumeration. Each of
        # the element is called a "deduction" which is the form (alpha, x) whenever
        # a value is assigned to alpha^x during a definition or "deduction process"
        self.deduction_stack = []
        # Attributes for modified methods.
        H = self.subgroup
        self._grp = free_group(', ' .join(["a_%d" % i for i in range(len(H))]))[0]
        self.P = [[None]*len(self.A)]
        self.p_p = {}

    @property
    def omega(self):
        """Set of live cosets. """
        return [coset for coset in range(len(self.p)) if self.p[coset] == coset]

    def copy(self):
        """
        Return a shallow copy of Coset Table instance ``self``.

        """
        self_copy = self.__class__(self.fp_group, self.subgroup)
        self_copy.table = [list(perm_rep) for perm_rep in self.table]
        self_copy.p = list(self.p)
        self_copy.deduction_stack = list(self.deduction_stack)
        return self_copy

    def __str__(self):
        return "Coset Table on %s with %s as subgroup generators" \
                % (self.fp_group, self.subgroup)

    __repr__ = __str__

    @property
    def n(self):
        """The number `n` represents the length of the sublist containing the
        live cosets.

        """
        if not self.table:
            return 0
        return max(self.omega) + 1

    # Pg. 152 [1]
    def is_complete(self):
        r"""
        The coset table is called complete if it has no undefined entries
        on the live cosets; that is, `\alpha^x` is defined for all
        `\alpha \in \Omega` and `x \in A`.

        """
        return not any(None in self.table[coset] for coset in self.omega)

    # Pg. 153 [1]
    def define(self, alpha, x, modified=False):
        r"""
        This routine is used in the relator-based strategy of Todd-Coxeter
        algorithm if some `\alpha^x` is undefined. We check whether there is
        space available for defining a new coset. If there is enough space
        then we remedy this by adjoining a new coset `\beta` to `\Omega`
        (i.e to set of live cosets) and put that equal to `\alpha^x`, then
        make an assignment satisfying Property[1]. If there is not enough space
        then we halt the Coset Table creation. The maximum amount of space that
        can be used by Coset Table can be manipulated using the class variable
        ``CosetTable.coset_table_max_limit``.

        See Also
        ========

        define_c

        """
        A = self.A
        table = self.table
        len_table = len(table)
        if len_table >= self.coset_table_limit:
            # abort the further generation of cosets
            raise ValueError("the coset enumeration has defined more than "
                    "%s cosets. Try with a greater value max number of cosets "
                    % self.coset_table_limit)
        table.append([None]*len(A))
        self.P.append([None]*len(self.A))
        # beta is the new coset generated
        beta = len_table
        self.p.append(beta)
        table[alpha][self.A_dict[x]] = beta
        table[beta][self.A_dict_inv[x]] = alpha
        # P[alpha][x] = epsilon, P[beta][x**-1] = epsilon
        if modified:
            self.P[alpha][self.A_dict[x]] = self._grp.identity
            self.P[beta][self.A_dict_inv[x]] = self._grp.identity
            self.p_p[beta] = self._grp.identity

    def define_c(self, alpha, x):
        r"""
        A variation of ``define`` routine, described on Pg. 165 [1], used in
        the coset table-based strategy of Todd-Coxeter algorithm. It differs
        from ``define`` routine in that for each definition it also adds the
        tuple `(\alpha, x)` to the deduction stack.

        See Also
        ========

        define

        """
        A = self.A
        table = self.table
        len_table = len(table)
        if len_table >= self.coset_table_limit:
            # abort the further generation of cosets
            raise ValueError("the coset enumeration has defined more than "
                    "%s cosets. Try with a greater value max number of cosets "
                    % self.coset_table_limit)
        table.append([None]*len(A))
        # beta is the new coset generated
        beta = len_table
        self.p.append(beta)
        table[alpha][self.A_dict[x]] = beta
        table[beta][self.A_dict_inv[x]] = alpha
        # append to deduction stack
        self.deduction_stack.append((alpha, x))

    def scan_c(self, alpha, word):
        """
        A variation of ``scan`` routine, described on pg. 165 of [1], which
        puts at tuple, whenever a deduction occurs, to deduction stack.

        See Also
        ========

        scan, scan_check, scan_and_fill, scan_and_fill_c

        """
        # alpha is an integer representing a "coset"
        # since scanning can be in two cases
        # 1. for alpha=0 and w in Y (i.e generating set of H)
        # 2. alpha in Omega (set of live cosets), w in R (relators)
        A_dict = self.A_dict
        A_dict_inv = self.A_dict_inv
        table = self.table
        f = alpha
        i = 0
        r = len(word)
        b = alpha
        j = r - 1
        # list of union of generators and their inverses
        while i <= j and table[f][A_dict[word[i]]] is not None:
            f = table[f][A_dict[word[i]]]
            i += 1
        if i > j:
            if f != b:
                self.coincidence_c(f, b)
            return
        while j >= i and table[b][A_dict_inv[word[j]]] is not None:
            b = table[b][A_dict_inv[word[j]]]
            j -= 1
        if j < i:
            # we have an incorrect completed scan with coincidence f ~ b
            # run the "coincidence" routine
            self.coincidence_c(f, b)
        elif j == i:
            # deduction process
            table[f][A_dict[word[i]]] = b
            table[b][A_dict_inv[word[i]]] = f
            self.deduction_stack.append((f, word[i]))
        # otherwise scan is incomplete and yields no information

    # alpha, beta coincide, i.e. alpha, beta represent the pair of cosets where
    # coincidence occurs
    def coincidence_c(self, alpha, beta):
        """
        A variation of ``coincidence`` routine used in the coset-table based
        method of coset enumeration. The only difference being on addition of
        a new coset in coset table(i.e new coset introduction), then it is
        appended to ``deduction_stack``.

        See Also
        ========

        coincidence

        """
        A_dict = self.A_dict
        A_dict_inv = self.A_dict_inv
        table = self.table
        # behaves as a queue
        q = []
        self.merge(alpha, beta, q)
        while len(q) > 0:
            gamma = q.pop(0)
            for x in A_dict:
                delta = table[gamma][A_dict[x]]
                if delta is not None:
                    table[delta][A_dict_inv[x]] = None
                    # only line of difference from ``coincidence`` routine
                    self.deduction_stack.append((delta, x**-1))
                    mu = self.rep(gamma)
                    nu = self.rep(delta)
                    if table[mu][A_dict[x]] is not None:
                        self.merge(nu, table[mu][A_dict[x]], q)
                    elif table[nu][A_dict_inv[x]] is not None:
                        self.merge(mu, table[nu][A_dict_inv[x]], q)
                    else:
                        table[mu][A_dict[x]] = nu
                        table[nu][A_dict_inv[x]] = mu

    def scan(self, alpha, word, y=None, fill=False, modified=False):
        r"""
        ``scan`` performs a scanning process on the input ``word``.
        It first locates the largest prefix ``s`` of ``word`` for which
        `\alpha^s` is defined (i.e is not ``None``), ``s`` may be empty. Let
        ``word=sv``, let ``t`` be the longest suffix of ``v`` for which
        `\alpha^{t^{-1}}` is defined, and let ``v=ut``. Then three
        possibilities are there:

        1. If ``t=v``, then we say that the scan completes, and if, in addition
        `\alpha^s = \alpha^{t^{-1}}`, then we say that the scan completes
        correctly.

        2. It can also happen that scan does not complete, but `|u|=1`; that
        is, the word ``u`` consists of a single generator `x \in A`. In that
        case, if `\alpha^s = \beta` and `\alpha^{t^{-1}} = \gamma`, then we can
        set `\beta^x = \gamma` and `\gamma^{x^{-1}} = \beta`. These assignments
        are known as deductions and enable the scan to complete correctly.

        3. See ``coicidence`` routine for explanation of third condition.

        Notes
        =====

        The code for the procedure of scanning `\alpha \in \Omega`
        under `w \in A*` is defined on pg. 155 [1]

        See Also
        ========

        scan_c, scan_check, scan_and_fill, scan_and_fill_c

        Scan and Fill
        =============

        Performed when the default argument fill=True.

        Modified Scan
        =============

        Performed when the default argument modified=True

        """
        # alpha is an integer representing a "coset"
        # since scanning can be in two cases
        # 1. for alpha=0 and w in Y (i.e generating set of H)
        # 2. alpha in Omega (set of live cosets), w in R (relators)
        A_dict = self.A_dict
        A_dict_inv = self.A_dict_inv
        table = self.table
        f = alpha
        i = 0
        r = len(word)
        b = alpha
        j = r - 1
        b_p = y
        if modified:
            f_p = self._grp.identity
        flag = 0
        while fill or flag == 0:
            flag = 1
            while i <= j and table[f][A_dict[word[i]]] is not None:
                if modified:
                    f_p = f_p*self.P[f][A_dict[word[i]]]
                f = table[f][A_dict[word[i]]]
                i += 1
            if i > j:
                if f != b:
                    if modified:
                        self.modified_coincidence(f, b, f_p**-1*y)
                    else:
                        self.coincidence(f, b)
                return
            while j >= i and table[b][A_dict_inv[word[j]]] is not None:
                if modified:
                    b_p = b_p*self.P[b][self.A_dict_inv[word[j]]]
                b = table[b][A_dict_inv[word[j]]]
                j -= 1
            if j < i:
                # we have an incorrect completed scan with coincidence f ~ b
                # run the "coincidence" routine
                if modified:
                    self.modified_coincidence(f, b, f_p**-1*b_p)
                else:
                    self.coincidence(f, b)
            elif j == i:
                # deduction process
                table[f][A_dict[word[i]]] = b
                table[b][A_dict_inv[word[i]]] = f
                if modified:
                    self.P[f][self.A_dict[word[i]]] = f_p**-1*b_p
                    self.P[b][self.A_dict_inv[word[i]]] = b_p**-1*f_p
                return
            elif fill:
                self.define(f, word[i], modified=modified)
            # otherwise scan is incomplete and yields no information

    # used in the low-index subgroups algorithm
    def scan_check(self, alpha, word):
        r"""
        Another version of ``scan`` routine, described on, it checks whether
        `\alpha` scans correctly under `word`, it is a straightforward
        modification of ``scan``. ``scan_check`` returns ``False`` (rather than
        calling ``coincidence``) if the scan completes incorrectly; otherwise
        it returns ``True``.

        See Also
        ========

        scan, scan_c, scan_and_fill, scan_and_fill_c

        """
        # alpha is an integer representing a "coset"
        # since scanning can be in two cases
        # 1. for alpha=0 and w in Y (i.e generating set of H)
        # 2. alpha in Omega (set of live cosets), w in R (relators)
        A_dict = self.A_dict
        A_dict_inv = self.A_dict_inv
        table = self.table
        f = alpha
        i = 0
        r = len(word)
        b = alpha
        j = r - 1
        while i <= j and table[f][A_dict[word[i]]] is not None:
            f = table[f][A_dict[word[i]]]
            i += 1
        if i > j:
            return f == b
        while j >= i and table[b][A_dict_inv[word[j]]] is not None:
            b = table[b][A_dict_inv[word[j]]]
            j -= 1
        if j < i:
            # we have an incorrect completed scan with coincidence f ~ b
            # return False, instead of calling coincidence routine
            return False
        elif j == i:
            # deduction process
            table[f][A_dict[word[i]]] = b
            table[b][A_dict_inv[word[i]]] = f
        return True

    def merge(self, k, lamda, q, w=None, modified=False):
        """
        Merge two classes with representatives ``k`` and ``lamda``, described
        on Pg. 157 [1] (for pseudocode), start by putting ``p[k] = lamda``.
        It is more efficient to choose the new representative from the larger
        of the two classes being merged, i.e larger among ``k`` and ``lamda``.
        procedure ``merge`` performs the merging operation, adds the deleted
        class representative to the queue ``q``.

        Parameters
        ==========

        'k', 'lamda' being the two class representatives to be merged.

        Notes
        =====

        Pg. 86-87 [1] contains a description of this method.

        See Also
        ========

        coincidence, rep

        """
        p = self.p
        rep = self.rep
        phi = rep(k, modified=modified)
        psi = rep(lamda, modified=modified)
        if phi != psi:
            mu = min(phi, psi)
            v = max(phi, psi)
            p[v] = mu
            if modified:
                if v == phi:
                    self.p_p[phi] = self.p_p[k]**-1*w*self.p_p[lamda]
                else:
                    self.p_p[psi] = self.p_p[lamda]**-1*w**-1*self.p_p[k]
            q.append(v)

    def rep(self, k, modified=False):
        r"""
        Parameters
        ==========

        `k \in [0 \ldots n-1]`, as for ``self`` only array ``p`` is used

        Returns
        =======

        Representative of the class containing ``k``.

        Returns the representative of `\sim` class containing ``k``, it also
        makes some modification to array ``p`` of ``self`` to ease further
        computations, described on Pg. 157 [1].

        The information on classes under `\sim` is stored in array `p` of
        ``self`` argument, which will always satisfy the property:

        `p[\alpha] \sim \alpha` and `p[\alpha]=\alpha \iff \alpha=rep(\alpha)`
        `\forall \in [0 \ldots n-1]`.

        So, for `\alpha \in [0 \ldots n-1]`, we find `rep(self, \alpha)` by
        continually replacing `\alpha` by `p[\alpha]` until it becomes
        constant (i.e satisfies `p[\alpha] = \alpha`):w

        To increase the efficiency of later ``rep`` calculations, whenever we
        find `rep(self, \alpha)=\beta`, we set
        `p[\gamma] = \beta \forall \gamma \in p-chain` from `\alpha` to `\beta`

        Notes
        =====

        ``rep`` routine is also described on Pg. 85-87 [1] in Atkinson's
        algorithm, this results from the fact that ``coincidence`` routine
        introduces functionality similar to that introduced by the
        ``minimal_block`` routine on Pg. 85-87 [1].

        See Also
        ========

        coincidence, merge

        """
        p = self.p
        lamda = k
        rho = p[lamda]
        if modified:
            s = p[:]
        while rho != lamda:
            if modified:
                s[rho] = lamda
            lamda = rho
            rho = p[lamda]
        if modified:
            rho = s[lamda]
            while rho != k:
                mu = rho
                rho = s[mu]
                p[rho] = lamda
                self.p_p[rho] = self.p_p[rho]*self.p_p[mu]
        else:
            mu = k
            rho = p[mu]
            while rho != lamda:
                p[mu] = lamda
                mu = rho
                rho = p[mu]
        return lamda

    # alpha, beta coincide, i.e. alpha, beta represent the pair of cosets
    # where coincidence occurs
    def coincidence(self, alpha, beta, w=None, modified=False):
        r"""
        The third situation described in ``scan`` routine is handled by this
        routine, described on Pg. 156-161 [1].

        The unfortunate situation when the scan completes but not correctly,
        then ``coincidence`` routine is run. i.e when for some `i` with
        `1 \le i \le r+1`, we have `w=st` with `s = x_1 x_2 \dots x_{i-1}`,
        `t = x_i x_{i+1} \dots x_r`, and `\beta = \alpha^s` and
        `\gamma = \alpha^{t-1}` are defined but unequal. This means that
        `\beta` and `\gamma` represent the same coset of `H` in `G`. Described
        on Pg. 156 [1]. ``rep``

        See Also
        ========

        scan

        """
        A_dict = self.A_dict
        A_dict_inv = self.A_dict_inv
        table = self.table
        # behaves as a queue
        q = []
        if modified:
            self.modified_merge(alpha, beta, w, q)
        else:
            self.merge(alpha, beta, q)
        while len(q) > 0:
            gamma = q.pop(0)
            for x in A_dict:
                delta = table[gamma][A_dict[x]]
                if delta is not None:
                    table[delta][A_dict_inv[x]] = None
                    mu = self.rep(gamma, modified=modified)
                    nu = self.rep(delta, modified=modified)
                    if table[mu][A_dict[x]] is not None:
                        if modified:
                            v = self.p_p[delta]**-1*self.P[gamma][self.A_dict[x]]**-1
                            v = v*self.p_p[gamma]*self.P[mu][self.A_dict[x]]
                            self.modified_merge(nu, table[mu][self.A_dict[x]], v, q)
                        else:
                            self.merge(nu, table[mu][A_dict[x]], q)
                    elif table[nu][A_dict_inv[x]] is not None:
                        if modified:
                            v = self.p_p[gamma]**-1*self.P[gamma][self.A_dict[x]]
                            v = v*self.p_p[delta]*self.P[mu][self.A_dict_inv[x]]
                            self.modified_merge(mu, table[nu][self.A_dict_inv[x]], v, q)
                        else:
                            self.merge(mu, table[nu][A_dict_inv[x]], q)
                    else:
                        table[mu][A_dict[x]] = nu
                        table[nu][A_dict_inv[x]] = mu
                        if modified:
                            v = self.p_p[gamma]**-1*self.P[gamma][self.A_dict[x]]*self.p_p[delta]
                            self.P[mu][self.A_dict[x]] = v
                            self.P[nu][self.A_dict_inv[x]] = v**-1

    # method used in the HLT strategy
    def scan_and_fill(self, alpha, word):
        """
        A modified version of ``scan`` routine used in the relator-based
        method of coset enumeration, described on pg. 162-163 [1], which
        follows the idea that whenever the procedure is called and the scan
        is incomplete then it makes new definitions to enable the scan to
        complete; i.e it fills in the gaps in the scan of the relator or
        subgroup generator.

        """
        self.scan(alpha, word, fill=True)

    def scan_and_fill_c(self, alpha, word):
        """
        A modified version of ``scan`` routine, described on Pg. 165 second
        para. [1], with modification similar to that of ``scan_anf_fill`` the
        only difference being it calls the coincidence procedure used in the
        coset-table based method i.e. the routine ``coincidence_c`` is used.

        See Also
        ========

        scan, scan_and_fill

        """
        A_dict = self.A_dict
        A_dict_inv = self.A_dict_inv
        table = self.table
        r = len(word)
        f = alpha
        i = 0
        b = alpha
        j = r - 1
        # loop until it has filled the alpha row in the table.
        while True:
            # do the forward scanning
            while i <= j and table[f][A_dict[word[i]]] is not None:
                f = table[f][A_dict[word[i]]]
                i += 1
            if i > j:
                if f != b:
                    self.coincidence_c(f, b)
                return
            # forward scan was incomplete, scan backwards
            while j >= i and table[b][A_dict_inv[word[j]]] is not None:
                b = table[b][A_dict_inv[word[j]]]
                j -= 1
            if j < i:
                self.coincidence_c(f, b)
            elif j == i:
                table[f][A_dict[word[i]]] = b
                table[b][A_dict_inv[word[i]]] = f
                self.deduction_stack.append((f, word[i]))
            else:
                self.define_c(f, word[i])

    # method used in the HLT strategy
    def look_ahead(self):
        """
        When combined with the HLT method this is known as HLT+Lookahead
        method of coset enumeration, described on pg. 164 [1]. Whenever
        ``define`` aborts due to lack of space available this procedure is
        executed. This routine helps in recovering space resulting from
        "coincidence" of cosets.

        """
        R = self.fp_group.relators
        p = self.p
        # complete scan all relators under all cosets(obviously live)
        # without making new definitions
        for beta in self.omega:
            for w in R:
                self.scan(beta, w)
                if p[beta] < beta:
                    break

    # Pg. 166
    def process_deductions(self, R_c_x, R_c_x_inv):
        """
        Processes the deductions that have been pushed onto ``deduction_stack``,
        described on Pg. 166 [1] and is used in coset-table based enumeration.

        See Also
        ========

        deduction_stack

        """
        p = self.p
        table = self.table
        while len(self.deduction_stack) > 0:
            if len(self.deduction_stack) >= CosetTable.max_stack_size:
                self.look_ahead()
                del self.deduction_stack[:]
                continue
            else:
                alpha, x = self.deduction_stack.pop()
                if p[alpha] == alpha:
                    for w in R_c_x:
                        self.scan_c(alpha, w)
                        if p[alpha] < alpha:
                            break
            beta = table[alpha][self.A_dict[x]]
            if beta is not None and p[beta] == beta:
                for w in R_c_x_inv:
                    self.scan_c(beta, w)
                    if p[beta] < beta:
                        break

    def process_deductions_check(self, R_c_x, R_c_x_inv):
        """
        A variation of ``process_deductions``, this calls ``scan_check``
        wherever ``process_deductions`` calls ``scan``, described on Pg. [1].

        See Also
        ========

        process_deductions

        """
        table = self.table
        while len(self.deduction_stack) > 0:
            alpha, x = self.deduction_stack.pop()
            if not all(self.scan_check(alpha, w) for w in R_c_x):
                return False
            beta = table[alpha][self.A_dict[x]]
            if beta is not None:
                if not all(self.scan_check(beta, w) for w in R_c_x_inv):
                    return False
        return True

    def switch(self, beta, gamma):
        r"""Switch the elements `\beta, \gamma \in \Omega` of ``self``, used
        by the ``standardize`` procedure, described on Pg. 167 [1].

        See Also
        ========

        standardize

        """
        A = self.A
        A_dict = self.A_dict
        table = self.table
        for x in A:
            z = table[gamma][A_dict[x]]
            table[gamma][A_dict[x]] = table[beta][A_dict[x]]
            table[beta][A_dict[x]] = z
            for alpha in range(len(self.p)):
                if self.p[alpha] == alpha:
                    if table[alpha][A_dict[x]] == beta:
                        table[alpha][A_dict[x]] = gamma
                    elif table[alpha][A_dict[x]] == gamma:
                        table[alpha][A_dict[x]] = beta

    def standardize(self):
        r"""
        A coset table is standardized if when running through the cosets and
        within each coset through the generator images (ignoring generator
        inverses), the cosets appear in order of the integers
        `0, 1, \dots, n`. "Standardize" reorders the elements of `\Omega`
        such that, if we scan the coset table first by elements of `\Omega`
        and then by elements of A, then the cosets occur in ascending order.
        ``standardize()`` is used at the end of an enumeration to permute the
        cosets so that they occur in some sort of standard order.

        Notes
        =====

        procedure is described on pg. 167-168 [1], it also makes use of the
        ``switch`` routine to replace by smaller integer value.

        Examples
        ========

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

        # Example 5.3 from [1]
        >>> f = FpGroup(F, [x**2*y**2, x**3*y**5])
        >>> C = coset_enumeration_r(f, [])
        >>> C.compress()
        >>> C.table
        [[1, 3, 1, 3], [2, 0, 2, 0], [3, 1, 3, 1], [0, 2, 0, 2]]
        >>> C.standardize()
        >>> C.table
        [[1, 2, 1, 2], [3, 0, 3, 0], [0, 3, 0, 3], [2, 1, 2, 1]]

        """
        A = self.A
        A_dict = self.A_dict
        gamma = 1
        for alpha, x in product(range(self.n), A):
            beta = self.table[alpha][A_dict[x]]
            if beta >= gamma:
                if beta > gamma:
                    self.switch(gamma, beta)
                gamma += 1
                if gamma == self.n:
                    return

    # Compression of a Coset Table
    def compress(self):
        """Removes the non-live cosets from the coset table, described on
        pg. 167 [1].

        """
        gamma = -1
        A = self.A
        A_dict = self.A_dict
        A_dict_inv = self.A_dict_inv
        table = self.table
        chi = tuple([i for i in range(len(self.p)) if self.p[i] != i])
        for alpha in self.omega:
            gamma += 1
            if gamma != alpha:
                # replace alpha by gamma in coset table
                for x in A:
                    beta = table[alpha][A_dict[x]]
                    table[gamma][A_dict[x]] = beta
                    # XXX: The line below uses == rather than = which means
                    # that it has no effect. It is not clear though if it is
                    # correct simply to delete the line or to change it to
                    # use =. Changing it causes some tests to fail.
                    #
                    # https://github.com/sympy/sympy/issues/27633
                    table[beta][A_dict_inv[x]] == gamma # noqa: B015
        # all the cosets in the table are live cosets
        self.p = list(range(gamma + 1))
        # delete the useless columns
        del table[len(self.p):]
        # re-define values
        for row in table:
            for j in range(len(self.A)):
                row[j] -= bisect_left(chi, row[j])

    def conjugates(self, R):
        R_c = list(chain.from_iterable((rel.cyclic_conjugates(), \
                (rel**-1).cyclic_conjugates()) for rel in R))
        R_set = set()
        for conjugate in R_c:
            R_set = R_set.union(conjugate)
        R_c_list = []
        for x in self.A:
            r = {word for word in R_set if word[0] == x}
            R_c_list.append(r)
            R_set.difference_update(r)
        return R_c_list

    def coset_representative(self, coset):
        '''
        Compute the coset representative of a given coset.

        Examples
        ========

        >>> from sympy.combinatorics import free_group
        >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_r
        >>> F, x, y = free_group("x, y")
        >>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
        >>> C = coset_enumeration_r(f, [x])
        >>> C.compress()
        >>> C.table
        [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]]
        >>> C.coset_representative(0)
        <identity>
        >>> C.coset_representative(1)
        y
        >>> C.coset_representative(2)
        y**-1

        '''
        for x in self.A:
            gamma = self.table[coset][self.A_dict[x]]
            if coset == 0:
                return self.fp_group.identity
            if gamma < coset:
                return self.coset_representative(gamma)*x**-1

    ##############################
    #      Modified Methods      #
    ##############################

    def modified_define(self, alpha, x):
        r"""
        Define a function p_p from from [1..n] to A* as
        an additional component of the modified coset table.

        Parameters
        ==========

        \alpha \in \Omega
        x \in A*

        See Also
        ========

        define

        """
        self.define(alpha, x, modified=True)

    def modified_scan(self, alpha, w, y, fill=False):
        r"""
        Parameters
        ==========
        \alpha \in \Omega
        w \in A*
        y \in (YUY^-1)
        fill -- `modified_scan_and_fill` when set to True.

        See Also
        ========

        scan
        """
        self.scan(alpha, w, y=y, fill=fill, modified=True)

    def modified_scan_and_fill(self, alpha, w, y):
        self.modified_scan(alpha, w, y, fill=True)

    def modified_merge(self, k, lamda, w, q):
        r"""
        Parameters
        ==========

        'k', 'lamda' -- the two class representatives to be merged.
        q -- queue of length l of elements to be deleted from `\Omega` *.
        w -- Word in (YUY^-1)

        See Also
        ========

        merge
        """
        self.merge(k, lamda, q, w=w, modified=True)

    def modified_rep(self, k):
        r"""
        Parameters
        ==========

        `k \in [0 \ldots n-1]`

        See Also
        ========

        rep
        """
        self.rep(k, modified=True)

    def modified_coincidence(self, alpha, beta, w):
        r"""
        Parameters
        ==========

        A coincident pair `\alpha, \beta \in \Omega, w \in Y \cup Y^{-1}`

        See Also
        ========

        coincidence

        """
        self.coincidence(alpha, beta, w=w, modified=True)

###############################################################################
#                           COSET ENUMERATION                                 #
###############################################################################

# relator-based method
def coset_enumeration_r(fp_grp, Y, max_cosets=None, draft=None,
                                    incomplete=False, modified=False):
    """
    This is easier of the two implemented methods of coset enumeration.
    and is often called the HLT method, after Hazelgrove, Leech, Trotter
    The idea is that we make use of ``scan_and_fill`` makes new definitions
    whenever the scan is incomplete to enable the scan to complete; this way
    we fill in the gaps in the scan of the relator or subgroup generator,
    that's why the name relator-based method.

    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.

    # TODO: complete the docstring

    See Also
    ========

    scan_and_fill,

    Examples
    ========

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

    # Example 5.1 from [1]
    >>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
    >>> C = coset_enumeration_r(f, [x])
    >>> for i in range(len(C.p)):
    ...     if C.p[i] == i:
    ...         print(C.table[i])
    [0, 0, 1, 2]
    [1, 1, 2, 0]
    [2, 2, 0, 1]
    >>> C.p
    [0, 1, 2, 1, 1]

    # Example from exercises Q2 [1]
    >>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
    >>> C = coset_enumeration_r(f, [])
    >>> C.compress(); C.standardize()
    >>> C.table
    [[1, 2, 3, 4],
    [5, 0, 6, 7],
    [0, 5, 7, 6],
    [7, 6, 5, 0],
    [6, 7, 0, 5],
    [2, 1, 4, 3],
    [3, 4, 2, 1],
    [4, 3, 1, 2]]

    # Example 5.2
    >>> f = FpGroup(F, [x**2, y**3, (x*y)**3])
    >>> Y = [x*y]
    >>> C = coset_enumeration_r(f, Y)
    >>> for i in range(len(C.p)):
    ...     if C.p[i] == i:
    ...         print(C.table[i])
    [1, 1, 2, 1]
    [0, 0, 0, 2]
    [3, 3, 1, 0]
    [2, 2, 3, 3]

    # Example 5.3
    >>> f = FpGroup(F, [x**2*y**2, x**3*y**5])
    >>> Y = []
    >>> C = coset_enumeration_r(f, Y)
    >>> for i in range(len(C.p)):
    ...     if C.p[i] == i:
    ...         print(C.table[i])
    [1, 3, 1, 3]
    [2, 0, 2, 0]
    [3, 1, 3, 1]
    [0, 2, 0, 2]

    # Example 5.4
    >>> F, a, b, c, d, e = free_group("a, b, c, d, e")
    >>> f = FpGroup(F, [a*b*c**-1, b*c*d**-1, c*d*e**-1, d*e*a**-1, e*a*b**-1])
    >>> Y = [a]
    >>> C = coset_enumeration_r(f, Y)
    >>> for i in range(len(C.p)):
    ...     if C.p[i] == i:
    ...         print(C.table[i])
    [0, 0, 0, 0, 0, 0, 0, 0, 0, 0]

    # example of "compress" method
    >>> C.compress()
    >>> C.table
    [[0, 0, 0, 0, 0, 0, 0, 0, 0, 0]]

    # Exercises Pg. 161, Q2.
    >>> F, x, y = free_group("x, y")
    >>> f = FpGroup(F, [x**2*y**2, y**-1*x*y*x**-3])
    >>> Y = []
    >>> C = coset_enumeration_r(f, Y)
    >>> C.compress()
    >>> C.standardize()
    >>> C.table
    [[1, 2, 3, 4],
    [5, 0, 6, 7],
    [0, 5, 7, 6],
    [7, 6, 5, 0],
    [6, 7, 0, 5],
    [2, 1, 4, 3],
    [3, 4, 2, 1],
    [4, 3, 1, 2]]

    # John J. Cannon; Lucien A. Dimino; George Havas; Jane M. Watson
    # Mathematics of Computation, Vol. 27, No. 123. (Jul., 1973), pp. 463-490
    # from 1973chwd.pdf
    # Table 1. Ex. 1
    >>> F, r, s, t = free_group("r, s, t")
    >>> E1 = FpGroup(F, [t**-1*r*t*r**-2, r**-1*s*r*s**-2, s**-1*t*s*t**-2])
    >>> C = coset_enumeration_r(E1, [r])
    >>> for i in range(len(C.p)):
    ...     if C.p[i] == i:
    ...         print(C.table[i])
    [0, 0, 0, 0, 0, 0]

    Ex. 2
    >>> F, a, b = free_group("a, b")
    >>> Cox = FpGroup(F, [a**6, b**6, (a*b)**2, (a**2*b**2)**2, (a**3*b**3)**5])
    >>> C = coset_enumeration_r(Cox, [a])
    >>> index = 0
    >>> for i in range(len(C.p)):
    ...     if C.p[i] == i:
    ...         index += 1
    >>> index
    500

    # Ex. 3
    >>> F, a, b = free_group("a, b")
    >>> B_2_4 = FpGroup(F, [a**4, b**4, (a*b)**4, (a**-1*b)**4, (a**2*b)**4, \
            (a*b**2)**4, (a**2*b**2)**4, (a**-1*b*a*b)**4, (a*b**-1*a*b)**4])
    >>> C = coset_enumeration_r(B_2_4, [a])
    >>> index = 0
    >>> for i in range(len(C.p)):
    ...     if C.p[i] == i:
    ...         index += 1
    >>> index
    1024

    References
    ==========

    .. [1] Holt, D., Eick, B., O'Brien, E.
           "Handbook of computational group theory"

    """
    # 1. Initialize a coset table C for < X|R >
    C = CosetTable(fp_grp, Y, max_cosets=max_cosets)
    # Define coset table methods.
    if modified:
        _scan_and_fill = C.modified_scan_and_fill
        _define = C.modified_define
    else:
        _scan_and_fill = C.scan_and_fill
        _define = C.define
    if draft:
        C.table = draft.table[:]
        C.p = draft.p[:]
    R = fp_grp.relators
    A_dict = C.A_dict
    p = C.p
    for i in range(len(Y)):
        if modified:
            _scan_and_fill(0, Y[i], C._grp.generators[i])
        else:
            _scan_and_fill(0, Y[i])
    alpha = 0
    while alpha < C.n:
        if p[alpha] == alpha:
            try:
                for w in R:
                    if modified:
                        _scan_and_fill(alpha, w, C._grp.identity)
                    else:
                        _scan_and_fill(alpha, w)
                    # if alpha was eliminated during the scan then break
                    if p[alpha] < alpha:
                        break
                if p[alpha] == alpha:
                    for x in A_dict:
                        if C.table[alpha][A_dict[x]] is None:
                            _define(alpha, x)
            except ValueError as e:
                if incomplete:
                    return C
                raise e
        alpha += 1
    return C

def modified_coset_enumeration_r(fp_grp, Y, max_cosets=None, draft=None,
                                    incomplete=False):
    r"""
    Introduce a new set of symbols y \in Y that correspond to the
    generators of the subgroup. Store the elements of Y as a
    word P[\alpha, x] and compute the coset table similar to that of
    the regular coset enumeration methods.

    Examples
    ========

    >>> from sympy.combinatorics.free_groups import free_group
    >>> from sympy.combinatorics.fp_groups import FpGroup
    >>> from sympy.combinatorics.coset_table import modified_coset_enumeration_r
    >>> F, x, y = free_group("x, y")
    >>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
    >>> C = modified_coset_enumeration_r(f, [x])
    >>> C.table
    [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1], [None, 1, None, None], [1, 3, None, None]]

    See Also
    ========

    coset_enumertation_r

    References
    ==========

    .. [1] Holt, D., Eick, B., O'Brien, E.,
           "Handbook of Computational Group Theory",
           Section 5.3.2
    """
    return coset_enumeration_r(fp_grp, Y, max_cosets=max_cosets, draft=draft,
                             incomplete=incomplete, modified=True)

# Pg. 166
# coset-table based method
def coset_enumeration_c(fp_grp, Y, max_cosets=None, draft=None,
                                                incomplete=False):
    """
    >>> from sympy.combinatorics.free_groups import free_group
    >>> from sympy.combinatorics.fp_groups import FpGroup, coset_enumeration_c
    >>> F, x, y = free_group("x, y")
    >>> f = FpGroup(F, [x**3, y**3, x**-1*y**-1*x*y])
    >>> C = coset_enumeration_c(f, [x])
    >>> C.table
    [[0, 0, 1, 2], [1, 1, 2, 0], [2, 2, 0, 1]]

    """
    # Initialize a coset table C for < X|R >
    X = fp_grp.generators
    R = fp_grp.relators
    C = CosetTable(fp_grp, Y, max_cosets=max_cosets)
    if draft:
        C.table = draft.table[:]
        C.p = draft.p[:]
        C.deduction_stack = draft.deduction_stack
        for alpha, x in product(range(len(C.table)), X):
            if C.table[alpha][C.A_dict[x]] is not None:
                C.deduction_stack.append((alpha, x))
    A = C.A
    # replace all the elements by cyclic reductions
    R_cyc_red = [rel.identity_cyclic_reduction() for rel in R]
    R_c = list(chain.from_iterable((rel.cyclic_conjugates(), (rel**-1).cyclic_conjugates()) \
            for rel in R_cyc_red))
    R_set = set()
    for conjugate in R_c:
        R_set = R_set.union(conjugate)
    # a list of subsets of R_c whose words start with "x".
    R_c_list = []
    for x in C.A:
        r = {word for word in R_set if word[0] == x}
        R_c_list.append(r)
        R_set.difference_update(r)
    for w in Y:
        C.scan_and_fill_c(0, w)
    for x in A:
        C.process_deductions(R_c_list[C.A_dict[x]], R_c_list[C.A_dict_inv[x]])
    alpha = 0
    while alpha < len(C.table):
        if C.p[alpha] == alpha:
            try:
                for x in C.A:
                    if C.p[alpha] != alpha:
                        break
                    if C.table[alpha][C.A_dict[x]] is None:
                        C.define_c(alpha, x)
                        C.process_deductions(R_c_list[C.A_dict[x]], R_c_list[C.A_dict_inv[x]])
            except ValueError as e:
                if incomplete:
                    return C
                raise e
        alpha += 1
    return C