#   Copyright 2020 <NAME>
#
#   Licensed under the Apache License, Version 2.0 (the "License");
#   you may not use this file except in compliance with the License.
#   You may obtain a copy of the License at
#
#       http://www.apache.org/licenses/LICENSE-2.0
#
#   Unless required by applicable law or agreed to in writing, software
#   distributed under the License is distributed on an "AS IS" BASIS,
#   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
#   See the License for the specific language governing permissions and
#   limitations under the License.

"""_pcbo.py.

Contains the PCBO class. See ``help(qubovert.PCBO)``.

"""

from . import PUBO
from .utils import QUBOVertWarning, num_bits, approximate_pubo_extrema
from .sat import OR, XOR, BUFFER, NOT, AND


__all__ = 'PCBO', 'boolean_var', 'integer_var'


# special constraint forms

def _special_constraints_eq_zero(pcbo, P, lam):
    """_special_constraints_eq_zero.

    See if the constraint that ``P == 0`` matches any special forms.

    Parameters
    ----------
    pcbo : PCBO object.
        The PCBO to add the constraint to.
    P : PUBO object.
        The PUBO constraint such that ``P == 0``.
        are present.
    lam : float > 0 or sympy.Symbol.
        Langrange multiplier to penalize violations of the constraint.

    Return
    ------
    success : bool.
        True if a special constraint was found and added to ``pcbo``, else
        False.

    """
    keys, values = tuple(P.keys()), tuple(P.values())

    # if P is of the form z == x * y. ie z == AND(x, y)
    if (
            not P.offset and P.num_binary_variables == 3 and
            P.num_terms == 2 and values[0] == -values[1] and
            {len(keys[0]), len(keys[1])} == {1, 2}
    ):
        a, = keys[0] if len(keys[0]) == 1 else keys[1]
        b, c = keys[0] if len(keys[0]) == 2 else keys[1]
        pcbo += PCBO().add_constraint_eq_AND(a, b, c, lam=lam)
        return True

    # if P is of the form z == 1 - x * y. ie z == NAND(x, y)

    # if P is of the form z == x * y. ie z == OR(x, y)

    # if P is of the form z == x * y. ie z == NOR(x, y)

    return False


def _special_constraints_le_zero(pcbo, P, lam, log_trick, bounds):
    """_special_constraints_le_zero.

    See if the constraint that ``P <= 0`` matches any special forms.

    Parameters
    ----------
    pcbo : PCBO object.
        The PCBO to add the constraint to.
    P : PUBO object.
        The PUBO constraint such that ``P <= 0``.
        are present.
    lam : float > 0 or sympy.Symbol.
        Langrange multiplier to penalize violations of the constraint.
    log_trick : bool.
        Whether or not to use the log trick to enforce the inequality
        constraint.
    bounds : two element tuple.
        A tuple ``(min, max)``, the minimum and maximum values that the
        PUBO ``P`` can take.

    Return
    ------
    success : bool.
        True if a special constraint was found and added to ``pcbo``, else
        False.

    """
    min_val, max_val = bounds

    # P without offset defined for convenience
    P_wo_offset = P - P.offset

    # if P is of the form sum(x_i) <= 1.
    if P.offset == -1 and all(x == 1 for x in P_wo_offset.values()):
        pcbo += lam * P * P_wo_offset / 2
        return True

    # if P is of the form P_wo_offset <= -P.offset and P_wo_offset is always
    # >= 0, and (important!) log_trick is False!!
    elif (
            not log_trick and not min_val - P.offset and
            P.offset <= 0 and min_val
    ):
        # We have that P_wo_offset <= P.offset and min(P_wo_offset) = 0. So we
        # can do a penalty lam(P_wo_offset - sum(ancillas))**2

        # create ancillas
        ancillas = PUBO()
        for i in range(num_bits(-P.offset, log_trick)):
            ancillas[(pcbo._next_ancilla,)] += 1

        diff = P_wo_offset - ancillas
        pcbo += lam * diff * diff

        return True

    # if P is of the form 1 <= x + y, then it is the same as OR(x, y)
    elif (P.offset == 1 and len(P_wo_offset) == 2 and
          set(P_wo_offset.values()) == {-1}):
        variables = tuple(P_wo_offset.keys())
        x, y = AND(*variables[0]), AND(*variables[1])
        pcbo += PCBO().add_constraint_OR(x, y, lam=lam)
        return True

    # if P is of the form x <= y
    elif not P.offset and len(P) == 2 and set(P.values()) == {1, -1}:
        coef = {v: k for k, v in P.items()}
        x, y = AND(*coef[1]), AND(*coef[-1])
        pcbo += lam * x * (1 - y)
        return True

    return False


# helpers

def _get_bounds(P, bounds):
    """_get_bounds.

    Compute the missing bounds of ``P``.

    Parameters
    ----------
    P : PUBO object.
    bounds : two element tuple or None.
        A tuple ``(min, max)``, the minimum and maximum values that the
        PUBO ``P`` can take. If ``bounds`` is None, then they will be
        calculated (approximately), or if either of the elements of
        ``bounds`` is None, then that element will be calculated
        (approximately).

    Return
    ------
    res : two element tuple.
        The bounds of ``P``.

    """
    if bounds is None or bounds == (None, None):
        bounds = approximate_pubo_extrema(P)
    elif bounds[0] is None:
        bounds = approximate_pubo_extrema(P)[0], bounds[1]
    elif bounds[1] is None:
        bounds = bounds[0], approximate_pubo_extrema(P)[1]

    return bounds


def boolean_var(name):
    """boolean_var.

    Create a PCBO (see ``qubovert.PCBO``) from a single boolean variable.

    Parameters
    ----------
    name : any hashable object.
        Name of the boolean variable.

    Return
    ------
    pcbo : qubovert.PCBO object.
        The model representing the boolean variable.

    Examples
    --------
    >>> from qubovert import boolean_var, PCBO
    >>>
    >>> x0 = boolean_var("x0")
    >>> print(x0)
    {('x0',): 1}
    >>> print(isinstance(x0, PCBO))
    True
    >>> print(x0.name)
    x0

    >>> x = [boolean_var('x{}'.format(i)) for i in range(5)]
    >>> pcbo = sum(x)
    >>> print(pcbo)
    {('x0',): 1, ('x1',): 1, ('x2',): 1, ('x3',): 1, ('x4',): 1}
    >>> pcbo **= 2
    >>> print(pcbo)
    {('x0',): 1, ('x0', 'x1'): 2, ('x2', 'x0'): 2, ('x3', 'x0'): 2,
     ('x4', 'x0'): 2, ('x1',): 1, ('x2', 'x1'): 2, ('x3', 'x1'): 2,
     ('x4', 'x1'): 2, ('x2',): 1, ('x2', 'x3'): 2, ('x2', 'x4'): 2, ('x3',): 1,
     ('x4', 'x3'): 2, ('x4',): 1}
    >>> pcbo *= -1
    >>> print(pcbo.solve_bruteforce())
    {'x0': 1, 'x1': 1, 'x2': 1, 'x3': 1, 'x4': 1}
    >>> pcbo.add_constraint_eq_zero(x[0] + x[1])
    >>> print(pcbo.solve_bruteforce())
    {'x0': 0, 'x1': 0, 'x2': 1, 'x3': 1, 'x4': 1}

    Notes
    -----
    ``qubovert.boolean_var(name)`` is equivalent to
    ``qubovert.PCBO.create_var(name)``.

    """
    return PCBO.create_var(name)


def integer_var(prefix, num_bits, log_trick=True):
    """integer_var.

    Return a PCBO object representing an integer variable with `num_bits`
    bits.

    Parameters
    ----------
    prefix : str.
        The prefix for the boolean variable names.
    num_bits : int.
        Number of bits to represent the integer variable with.
    log_trick : bool (optional, defaults to True).
        Whether or not to use a log encoding for the integer.

    Return
    ------
    i : qubovert.PCBO object.

    Example
    -------
    >>> from qubovert import integer_var
    >>> var = integer_var('a', 4)
    >>> print(var)
    {('a0',): 1, ('a1',): 2, ('a2',): 4, ('a3',): 8}
    >>> print(var.name)
    a

    >>> from qubovert import integer_var
    >>> var = integer_var('a', 4, log_trick=False)
    >>> print(var)
    {('a0',): 1, ('a1',): 1, ('a2',): 1, ('a3',): 1}

    """
    var = PCBO()
    var.name = prefix
    for i in range(num_bits):
        var[(str(prefix) + str(i),)] = pow(2, i) if log_trick else 1
    return var


# main class

class PCBO(PUBO):
    """PCBO.

    This class deals with Polynomial Constrained Boolean Optimization problems.
    PCBO inherits some methods and attributes from the ``PUBO`` class. See
    ``help(qubovert.PUBO)``.

    ``PCBO`` has all the same methods as ``PUBO``, but adds some constraint
    methods; namely

    - ``add_constraint_eq_zero(P, lam=1, ...)`` enforces that ``P == 0`` by
      penalizing with ``lam``,
    - ``add_constraint_ne_zero(P, lam=1, ...)`` enforces that ``P != 0`` by
      penalizing with ``lam``,
    - ``add_constraint_lt_zero(P, lam=1, ...)`` enforces that ``P < 0`` by
      penalizing with ``lam``,
    - ``add_constraint_le_zero(P, lam=1, ...)`` enforces that ``P <= 0`` by
      penalizing with ``lam``,
    - ``add_constraint_gt_zero(P, lam=1, ...)`` enforces that ``P > 0`` by
      penalizing with ``lam``, and
    - ``add_constraint_ge_zero(P, lam=1, ...)`` enforces that ``P >= 0`` by
      penalizing with ``lam``.

    Each of these takes in a PUBO ``P`` and a lagrange multiplier ``lam`` that
    defaults to 1. See each of their docstrings for important details on their
    implementation.

    We then implement logical operations:

    - ``add_constraint_AND``, ``add_constraint_NAND``,
    - ``add_constraint_eq_AND``, ``add_constraint_eq_NAND``,
    - ``add_constraint_OR``, ``add_constraint_NOR```,
    - ``add_constraint_eq_OR``, ``add_constraint_eq_NOR``,
    - ``add_constraint_XOR``, ``add_constraint_XNOR``,
    - ``add_constraint_eq_XOR``, ``add_constraint_eq_XNOR``,
    - ``add_constraint_BUFFER``, ``add_constraint_NOT``,
    - ``add_constraint_eq_BUFFER``, ``add_constraint_eq_NOT``.

    See each of their docstrings for important details on their implementation.

    Notes
    -----
    - Variables names that begin with ``"__a"`` should not be used since they
      are used internally to deal with some ancilla variables to enforce
      constraints.
    - The ``self.solve_bruteforce`` method will solve the PCBO ensuring that
      all the inputted constraints are satisfied. Whereas
      ``qubovert.utils.solve_pubo_bruteforce(self)`` or
      ``qubovert.utils.solve_pubo_bruteforce(self.to_pubo())`` will solve the
      PUBO created from the PCBO. If the inputted constraints are not enforced
      strong enough (ie too small lagrange multipliers) then these may not give
      the correct result, whereas ``self.solve_bruteforce()`` will always give
      the correct result (ie one that satisfies all the constraints).

    Examples
    --------
    See ``qubovert.PUBO`` for more examples of using PCBO without constraints.

    >>> H = PCBO()
    >>> H.add_constraint_eq_zero({('a', 1): 2, (1, 2): -1, (): -1})
    >>> H
    {('a', 1, 2): -4, (1, 2): 3, (): 1}
    >>> H -= 1
    >>> H
    {('a', 1, 2): -4, (1, 2): 3}

    >>> H = PCBO()
    >>>
    >>> # minimize -x_0 - x_1 - x_2
    >>> for i in (0, 1, 2):
    >>>     H[(i,)] -= 1
    >>>
    >>> # subject to constraints
    >>> H.add_constraint_eq_zero(  # enforce that x_0 x_1 - x_2 == 0
    >>>     {(0, 1): 1, (2,): -1}
    >>> ).add_constraint_lt_zero(  # enforce that x_1 x_2 + x_0 < 1
    >>>     {(1, 2): 1, (0,): 1, (): -1}
    >>> )
    >>> print(H)
    >>> # {(1,): -2, (2,): -1, (0, 1): 2, (1, 2): 2, (0, 1, 2): 2}
    >>>
    >>> print(H.solve_bruteforce(all_solutions=True))
    >>> # [{0: 0, 1: 1, 2: 0}]
    >>>
    >>> Q = H.to_qubo()
    >>> solutions = [H.convert_solution(sol)
    >>>              for sol in Q.solve_bruteforce(all_solutions=True)]
    >>> print(solutions)
    >>> # [{0: 0, 1: 1, 2: 0}]  # matches the PCBO solution!
    >>>
    >>> L = H.to_quso()
    >>> solutions = [H.convert_solution(sol)
    >>>              for sol in L.solve_bruteforce(all_solutions=True)]
    >>> print(solutions)
    >>> # [{0: 0, 1: 1, 2: 0}]  # matches the PCBO solution!

    Enforce that c == a b

    >>> H = PCBO().add_constraint_eq_AND('c', 'a', 'b')
    >>> H
    {('c',): 3, ('b', 'a'): 1, ('c', 'a'): -2, ('c', 'b'): -2}

    >>> from any_module import qubo_solver
    >>> # or from qubovert.utils import solve_qubo_bruteforce as qubo_solver
    >>> H = PCBO()
    >>>
    >>> # make it favorable to AND variables a and b, and variables b and c
    >>> H.add_constraint_AND('a', 'b').add_constraint_AND('b', 'c')
    >>>
    >>> # make it favorable to OR variables b and c
    >>> H.add_constraint_OR('b', 'c')
    >>>
    >>> # make it favorable to (a AND b) OR (c AND d)
    >>> H.add_constraint_OR(['a', 'b'], ['c', 'd'])
    >>>
    >>> H
    {('b', 'a'): -2, (): 4, ('b',): -1, ('c',): -1,
     ('c', 'd'): -1, ('c', 'd', 'b', 'a'): 1}
    >>> Q = H.to_qubo()
    >>> Q
    {(): 4, (0,): -1, (2,): -1, (2, 3): 1, (4,): 6, (0, 4): -4,
     (1, 4): -4, (5,): 6, (2, 5): -4, (3, 5): -4, (4, 5): 1}
    >>> obj_value, sol = qubo_solver(Q)
    >>> sol
    {0: 1, 1: 1, 2: 1, 3: 0, 4: 1, 5: 0}
    >>> solution = H.convert_solution(sol)
    >>> solution
    {'b': 1, 'a': 1, 'c': 1, 'd': 0}

    """

    def __init__(self, *args, **kwargs):
        """__init__.

        This class deals with polynomial constrained boolean optimization.
        Note that it is generally more efficient to initialize an empty PCBO
        object and then build the PCBO, rather than initialize a PCBO object
        with an already built dict.

        Parameters
        ----------
        arguments : define a dictionary with ``dict(*args, **kwargs)``.
            The dictionary will be initialized to follow all the convensions of
            the class. Alternatively, ``args[0]`` can be a PCBO object.

        Examples
        --------
        >>> pcbo = PCBO()
        >>> pcbo[('a',)] += 5
        >>> pcbo[(0, 'a')] -= 2
        >>> pcbo -= 1.5
        >>> pcbo
        {('a',): 5, ('a', 0): -2, (): -1.5}
        >>> pcbo.add_constraint_eq_zero({('a',): 1}, lam=5)
        >>> pcbo
        {('a',): 10, ('a', 0): -2, (): -1.5}

        >>> pcbo = PCBO({('a',): 5, (0, 'a', 1): -2, (): -1.5})
        >>> pcbo
        {('a',): 5, ('a', 0, 1): -2, (): -1.5}

        """
        # use self.__class__ here because PCSO uses this code as well.
        super(self.__class__, self).__init__(*args, **kwargs)
        if len(args) == 1 and isinstance(args[0], self.__class__):
            self._constraints = args[0].constraints
            self._ancilla = args[0].num_ancillas
        else:
            self._ancilla, self._constraints = 0, {}

    def update(self, *args, **kwargs):
        """update.

        Update the PCBO but following all the conventions of this class.

        Parameters
        ----------
        arguments : defines a dictionary or PCBO.
            Ie ``d = dict(*args, **kwargs)``.
            Each element in d will be added in place to this instance following
            all the required convensions.

        """
        # use self.__class__ here because PCSO uses this code as well.
        super(self.__class__, self).update(*args, **kwargs)
        if len(args) == 1 and isinstance(args[0], self.__class__):
            for k, v in args[0]._constraints.items():
                self._constraints.setdefault(k, []).extend(v)

    @property
    def constraints(self):
        """constraints.

        Return the constraints of the PCBO.

        Return
        ------
        res : dict.
            The keys of ``res`` are some or all of
            ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, and ``'ge'``.
            The values are lists of ``qubovert.PUBO`` objects. For a
            given key, value pair ``k, v``, the ``v[i]`` element represents
            the PUBO ``v[i]`` being == 0 if ``k == 'eq'``,
            != 0 if ``k == 'ne'``,
            < 0 if ``k == 'lt'``, <= 0 if ``k == 'le'``,
            > 0 if ``k == 'gt'``, >= 0 if ``k == 'ge'``.

        """
        return {k: [x.copy() for x in v] for k, v in self._constraints.items()}

    def _append_constraint(self, key, constraint):
        """_append_constraint.

        Internal method to add a constraint to the constraints dictionary.

        Parameters
        ----------
        key : str.
            One of ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, or
            ``'ge'``.
        constraint : qubovert.PUBO object.

        """
        self._constraints.setdefault(key, []).append(constraint)

    def _pop_constraint(self, key):
        """_append_constraint.

        Internal method to remove the most recently added constraint.

        Parameters
        ----------
        key : str.
            One of ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, or
            ``'ge'``.

        """
        if self._constraints.get(key, []):
            self._constraints[key].pop()
            if not self._constraints[key]:
                self._constraints.pop(key)

    @property
    def num_ancillas(self):
        """num_ancillas.

        Return the number of ancilla variables introduced to the PCBO in
        order to enforce the inputted constraints.

        Returns
        -------
        num : int.
            Number of ancillas in the PCBO.

        """
        return self._ancilla

    @property
    def _next_ancilla(self):
        """_next_ancilla.

        Get the next available ancilla bit and increment.

        Return
        ------
        a : int.
            Ancilla bit label.

        """
        self._ancilla += 1
        return "__a%d" % (self._ancilla - 1)

    @classmethod
    def remove_ancilla_from_solution(cls, solution):
        """remove_ancilla_from_solution.

        Take a solution to the PCBO and remove all the ancilla variables, (
        represented by `_a` prefixes).

        Parameters
        ----------
        solution : dict.
            Must be the solution in terms of the original variables. Thus if
            ``solution`` is the solution to the ``self.to_pubo``,
            ``self.to_qubo``, ``self.to_puso``, or ``self.to_quso``
            formulations, then you should first call ``self.convert_solution``.
            See ``help(self.convert_solution)``.

        Return
        ------
        res : dict.
            The same as ``solution`` but with all the ancilla bits removed.

        """
        return {k: v for k, v in solution.items() if str(k)[:3] != "__a"}

    def is_solution_valid(self, solution):
        """is_solution_valid.

        Finds whether or not the given solution satisfies the constraints.

        Parameters
        ----------
        solution : dict.
            Must be the solution in terms of the original variables. Thus if
            ``solution`` is the solution to the ``self.to_pubo``,
            ``self.to_qubo``, ``self.to_puso``, or ``self.to_quso``
            formulations, then you should first call ``self.convert_solution``.
            See ``help(self.convert_solution)``.

        Return
        ------
        valid : bool.
            Whether or not the given solution satisfies the constraints.

        """
        if any(v.value(solution) != 0
               for v in self._constraints.get('eq', [])):
            return False

        if any(v.value(solution) == 0
               for v in self._constraints.get('ne', [])):
            return False

        if any(v.value(solution) >= 0
               for v in self._constraints.get("lt", [])):
            return False

        if any(v.value(solution) > 0
               for v in self._constraints.get("le", [])):
            return False

        if any(v.value(solution) <= 0
               for v in self._constraints.get("gt", [])):
            return False

        if any(v.value(solution) < 0
               for v in self._constraints.get("ge", [])):
            return False

        return True

    # override
    def __round__(self, ndigits=None):
        """round.

        Round values of the PCBO object.

        Parameters
        ----------
        ndigits : int.
            Number of decimal digits to round to.

        Returns
        -------
        res : PCBO object.
            Copy of self but with each value rounded to ``ndigits`` decimal
            digits. Each value has a type according to the docstring
            specifications of ``round``, see ``help(round)``.

        """
        # use self.__class__ here because PCSO uses this code as well.
        d = super(self.__class__, self).__round__(ndigits)
        d._constraints = self.constraints
        return d

    # override
    def subs(self, *args, **kwargs):
        """subs.

        Replace any ``sympy`` symbols that are used in the dict with values.
        Please see ``help(sympy.Symbol.subs)`` for more info.

        Parameters
        ----------
        arguments : substitutions.
            Same parameters as are inputted into ``sympy.Symbol.subs``.

        Returns
        -------
        res : PCBO object.
            Same as ``self`` but with all the symbols replaced with values.

        """
        # use self.__class__ here because PCSO uses this code as well.
        d = super(self.__class__, self).subs(*args, **kwargs)
        d._constraints = {
            k: [P.subs(*args, **kwargs) for P in v]
            for k, v in self._constraints.items()
        }
        return d

    def add_constraint_eq_zero(self,
                               P, lam=1,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_eq_zero.

        Enforce that ``P == 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P == 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they may be
            calculated (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        The following enforces that :math:`\prod_{i=0}^{3} x_i == 0`.

        >>> H = PCBO()
        >>> H.add_constraint_eq_zero({(0, 1, 2, 3): 1})
        >>> H
        {(0, 1, 2, 3): 1}

        The following enforces that :math:`\sum_{i=1}^{3} i x_i x_{i+1} == 0`.

        >>> H = PCBO()
        >>> H.add_constraint_eq_zero({(1, 2): 1, (2, 3): 2, (3, 4): 3})
        >>> H
        {(1, 2): 1, (1, 2, 3): 4, (1, 2, 3, 4): 6,
         (2, 3): 4, (2, 3, 4): 12, (3, 4): 9}

        Here we show how operations can be strung together.

        >>> H = PCBO()
        >>> H.add_constraint_eq_zero(
                {(0, 1): 1}
            ).add_constraint_eq_zero(
                {(1, 2): 1, (): -1}
            )
        >>> H
        {(0, 1): 1, (1, 2): -1, (): 1}

        """
        P = PUBO(P)
        self._append_constraint("eq", P)
        if not lam:
            return self

        if _special_constraints_eq_zero(self, P, lam):
            return self

        min_val, max_val = _get_bounds(P, bounds)

        if min_val == max_val == 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        elif min_val > 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam * P
        elif max_val < 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self -= lam * P
        elif min_val == 0:
            self += lam * P
        elif max_val == 0:
            self -= lam * P
        else:
            self += lam * P * P

        return self

    def add_constraint_ne_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_ne_zero.

        Enforce that ``P != 0`` by penalizing invalid solutions with ``lam``.
        See Notes below for more details.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P != 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... != 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function.
        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\min_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\min_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c != 2`.

        >>> H = PCBO().add_constraint_ne_zero(
                {('a', 'b', 'c'): -1, ('a',): 2,
                ('a', 'b'): -4, ('c',): 3, (): -2}
            )
        >>> print(H)
        {('c', 'a', 'b'): -19, ('__a0', 'c', 'a', 'b'): -4,
         ('__a0', 'c', '__a1', 'a', 'b'): -4, ('c', '__a1', 'a', 'b'): 2,
         ('__a0', 'c', '__a2', 'a', 'b'): -8, ('c', '__a2', 'a', 'b'): 4,
         ('__a0', '__a3', 'c', 'a', 'b'): -16, ('__a3', 'c', 'a', 'b'): 8,
         ('__a0', 'c', '__a4', 'a', 'b'): -32, ('c', '__a4', 'a', 'b'): 16,
         ('a',): -8, ('c', 'a'): 12, ('__a0', 'a'): 8,
         ('__a0', '__a1', 'a'): 8, ('__a1', 'a'): -4,
         ('__a0', '__a2', 'a'): 16, ('__a2', 'a'): -8,
         ('__a0', '__a3', 'a'): 32, ('__a3', 'a'): -16,
         ('__a0', '__a4', 'a'): 64, ('__a4', 'a'): -32, ('a', 'b'): 24,
         ('__a0', 'a', 'b'): -16, ('__a0', '__a1', 'a', 'b'): -16,
         ('__a1', 'a', 'b'): 8, ('__a0', '__a2', 'a', 'b'): -32,
         ('__a2', 'a', 'b'): 16, ('__a0', '__a3', 'a', 'b'): -64,
         ('__a3', 'a', 'b'): 32, ('__a0', '__a4', 'a', 'b'): -128,
         ('__a4', 'a', 'b'): 64, ('__a0', 'c'): 12, ('__a0', 'c', '__a1'): 12,
         ('c', '__a1'): -6, ('__a0', 'c', '__a2'): 24, ('c', '__a2'): -12,
         ('__a0', '__a3', 'c'): 48, ('__a3', 'c'): -24,
         ('__a0', 'c', '__a4'): 96, ('c', '__a4'): -48, ('c',): -9, (): 9,
         ('__a0',): -8, ('__a0', '__a1'): -8, ('__a1',): 7,
         ('__a0', '__a2'): -16, ('__a2',): 16, ('__a3',): 40,
         ('__a0', '__a4'): -64, ('__a4',): 112, ('__a2', '__a1'): 4,
         ('__a3', '__a1'): 8, ('__a4', '__a1'): 16, ('__a3', '__a2'): 16,
         ('__a4', '__a2'): 32, ('__a0', '__a3'): -32, ('__a3', '__a4'): 64}
        >>> print(H.is_solution_valid({'b': 0, 'c': 0, 'a': 1}))
        False
        >>> print(H.is_solution_valid({'b': 0, 'c': 1, 'a': 1}))
        True

        """
        P = PUBO(P)
        self._append_constraint("ne", P)
        if not lam:
            return self

        min_val, max_val = _get_bounds(P, bounds)

        if min_val == max_val == 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam
        elif min_val > 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        elif max_val < 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        elif min_val == 0:
            self.add_constraint_gt_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=suppress_warnings
            )
            self._pop_constraint('gt')
        elif max_val == 0:
            self.add_constraint_lt_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=suppress_warnings
            )
            self._pop_constraint('lt')

        else:
            # don't mutate the P that we put in self._constraints
            P = P.copy()
            sign = 2 * boolean_var(self._next_ancilla) - 1
            P += sign
            max_val += 1
            min_val -= 1
            for i in range(num_bits(max_val - min_val - 1, log_trick)):
                v = pow(2, i) if log_trick else 1
                P += sign * v * boolean_var(self._next_ancilla)
                max_val += v
                min_val -= v

            self.add_constraint_eq_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=True
            )
            self._pop_constraint("eq")

        return self

    def add_constraint_lt_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_lt_zero.

        Enforce that ``P < 0`` by penalizing invalid solutions with ``lam``.
        See Notes below for more details.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P < 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... < 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_lt_zero({(0,): 1, (1,): 2, (2,): -.5, (): .4})
          >>> test_sol = {0: 0, 1: 0, 2: 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\min_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\min_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c < 2`.

        >>> H = PCBO().add_constraint_lt_zero(
                {('a', 'b', 'c'): -1, ('a',): 1,
                 ('a', 'b'): -4, ('c',): 3, (): -2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', '__a0', 'c', 'a'): -2,
         ('b', 'c', 'a', '__a1'): -4, ('a',): -3, ('b', 'a'): 24,
         ('c', 'a'): 6, ('__a0', 'a'): 2, ('a', '__a1'): 4,
         ('b', '__a0', 'a'): -8, ('b', 'a', '__a1'): -16, ('c',): -3,
         ('__a0', 'c'): 6, ('c', '__a1'): 12, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a0', '__a1'): 4}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("lt", P)
        if not lam:
            return self

        min_val, max_val = _get_bounds(P, bounds)

        if min_val >= 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam * P
        elif max_val < 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        else:
            # copy P, don't do +=
            P = P + 1
            min_val += 1
            max_val += 1
            self.add_constraint_le_zero(
                P, lam=lam, log_trick=log_trick,
                bounds=(min_val, max_val), suppress_warnings=True
            )
            self._pop_constraint("le")

        return self

    def add_constraint_le_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_le_zero.

        Enforce that ``P <= 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P <= 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... \leq 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_le_zero({(0,): 1, (1,): 2, (2,): -1.5, (): .4})
          >>> H
          {(0,): 1.7999999999999998, (0, 1): 4, (0, 2): -3.0, (0, '__a0'): 2,
           (1,): 5.6, (1, 2): -6.0, (1, '__a0'): 4, (2,): 1.0499999999999998,
           (2, '__a0'): -3.0, (): 0.16000000000000003, ('__a0',): 1.8}
          >>> test_sol = {0: 0, 1: 0, 2: 1, '__a0': 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\min_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\min_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c \leq 2`.

        >>> H = PCBO().add_constraint_le_zero(
                {('a', 'b', 'c'): -1, ('a',): 1,
                 ('a', 'b'): -4, ('c',): 3, (): -2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', 'c', 'a', '__a0'): -2,
         ('__a1', 'b', 'c', 'a'): -4, ('__a2', 'b', 'c', 'a'): -8, ('a',): -3,
         ('b', 'a'): 24, ('c', 'a'): 6, ('a', '__a0'): 2, ('__a1', 'a'): 4,
         ('__a2', 'a'): 8, ('b', 'a', '__a0'): -8, ('__a1', 'b', 'a'): -16,
         ('__a2', 'b', 'a'): -32, ('c',): -3, ('c', '__a0'): 6,
         ('__a1', 'c'): 12, ('__a2', 'c'): 24, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a1', '__a0'): 4, ('__a2', '__a0'): 8,
         ('__a2', '__a1'): 16}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("le", P)
        if not lam:
            return self

        bounds = min_val, max_val = _get_bounds(P, bounds)
        if _special_constraints_le_zero(self, P, lam, log_trick, bounds):
            return self

        if min_val > 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam * P
        elif max_val <= 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        else:
            # don't mutate the P that we put in self._constraints
            P = P.copy()
            if min_val:
                for i in range(num_bits(-min_val, log_trick)):
                    v = pow(2, i) if log_trick else 1
                    P[(self._next_ancilla,)] += v
                    max_val += v

            self.add_constraint_eq_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=True
            )
            self._pop_constraint("eq")

        return self

    def add_constraint_gt_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_gt_zero.

        Enforce that ``P > 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P > 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... > 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_gt_zero({(0,): -1, (1,): -2, (2,): .5, (): -.4})
          >>> test_sol = {0: 0, 1: 0, 2: 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\max_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\max_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`x_a x_b x_c - x_a + 4x_a x_b - 3x_c > -2`.

        >>> H = PCBO().add_constraint_gt_zero(
                {('a', 'b', 'c'): 1, ('a',): -1,
                 ('a', 'b'): 4, ('c',): -3, (): 2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', '__a0', 'c', 'a'): -2,
         ('b', 'c', 'a', '__a1'): -4, ('a',): -3, ('b', 'a'): 24,
         ('c', 'a'): 6, ('__a0', 'a'): 2, ('a', '__a1'): 4,
         ('b', '__a0', 'a'): -8, ('b', 'a', '__a1'): -16, ('c',): -3,
         ('__a0', 'c'): 6, ('c', '__a1'): 12, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a0', '__a1'): 4}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("gt", P)
        if not lam:
            return self
        min_val, max_val = _get_bounds(P, bounds)
        bounds = -max_val, -min_val
        self.add_constraint_lt_zero(
            -P, lam=lam, log_trick=log_trick,
            bounds=bounds, suppress_warnings=suppress_warnings
        )
        self._pop_constraint("lt")
        return self

    def add_constraint_ge_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_ge_zero.

        Enforce that ``P >= 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P >= 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... \geq 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_ge_zero({(0,): -1, (1,): -2, (2,):1.5, (): -.4})
          >>> H
          {(0,): 1.7999999999999998, (0, 1): 4, (0, 2): -3.0, (0, '__a0'): 2,
           (1,): 5.6, (1, 2): -6.0, (1, '__a0'): 4, (2,): 1.0499999999999998,
           (2, '__a0'): -3.0, (): 0.16000000000000003, ('__a0',): 1.8}
          >>> test_sol = {0: 0, 1: 0, 2: 1, '__a0': 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\max_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\max_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`x_a x_b x_c - x_a + 4x_a x_b - 3x_c \geq -2`.

        >>> H = PCBO().add_constraint_ge_zero(
                {('a', 'b', 'c'): 1, ('a',): -1,
                 ('a', 'b'): 4, ('c',): -3, (): 2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', 'c', 'a', '__a0'): -2,
         ('__a1', 'b', 'c', 'a'): -4, ('__a2', 'b', 'c', 'a'): -8, ('a',): -3,
         ('b', 'a'): 24, ('c', 'a'): 6, ('a', '__a0'): 2, ('__a1', 'a'): 4,
         ('__a2', 'a'): 8, ('b', 'a', '__a0'): -8, ('__a1', 'b', 'a'): -16,
         ('__a2', 'b', 'a'): -32, ('c',): -3, ('c', '__a0'): 6,
         ('__a1', 'c'): 12, ('__a2', 'c'): 24, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a1', '__a0'): 4, ('__a2', '__a0'): 8,
         ('__a2', '__a1'): 16}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("ge", P)
        if not lam:
            return self
        min_val, max_val = _get_bounds(P, bounds)
        bounds = -max_val, -min_val
        self.add_constraint_le_zero(
            -P, lam=lam, log_trick=log_trick,
            bounds=bounds, suppress_warnings=suppress_warnings
        )
        self._pop_constraint("le")
        return self

    def add_constraint_eq_AND(self, a, *variables, lam=1):
        r"""add_constraint_eq_AND.

        Add a penalty to the PCBO that enforces that
        :math:`a = v_0 \land v_1 \land v_2 \land ...`,
        with a penalty factor
        ``lam``, where ``v_1 = variables[0]``, ``v_2 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_AND('a', 'b', 'c')  # enforce a == b AND c

        >>> H = PCBO()
        >>> # enforce (a AND b AND c AND d) == 'e'
        >>> H.add_constraint_eq_AND('e', 'a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> H = PCBO()
        >>> # enforce that a == b AND c
        >>> H.add_constraint_eq_AND(a, b, c)

        References
        ----------
        https://arxiv.org/pdf/1307.8041.pdf equation 6.

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to AND. "
                             "See ``add_constraint_eq_BUFFER`` for less.")

        a = BUFFER(a)
        b, c = 1, 1
        for v in variables[:n // 2]:
            b *= BUFFER(v)
        for v in variables[n // 2:]:
            c *= BUFFER(v)

        P = 3 * a + b * c - 2 * a * (b + c)

        return self.add_constraint_eq_zero(P, lam, bounds=(0, 3))

    def add_constraint_eq_OR(self, a, *variables, lam=1):
        r"""add_constraint_eq_OR.

        Add a penalty to the PCBO that enforces that
        :math:`v_0 \lor v_1 \lor v_2 \lor ... == a`,
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_OR('a', 'b', 'c')  # enforce a == b OR c

        >>> H = PCBO()
        >>> # enforce a == b OR c OR d
        >>> H.add_constraint_eq_OR('a', 'b', 'c', 'd')

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to OR.")

        a = BUFFER(a)
        if n == 2:
            b, c = BUFFER(variables[0]), BUFFER(variables[1])
            P = a + b + c + b * c - 2 * a * (b + c)
            bounds = 0, 3
        else:
            P = PCBO().add_constraint_NOR(*variables) - a
            bounds = -1, 1

        return self.add_constraint_eq_zero(P, lam=lam, bounds=bounds)

    def add_constraint_eq_XOR(self, a, *variables, lam=1):
        r"""add_constraint_eq_XOR.

        Add a penalty to the PCBO that enforces that
        :math:`v_0 \oplus v_1 \oplus ... == a`
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_XOR('a', 'b', 'c')  # enforce a == b XOR c

        >>> H = PCBO()
        >>> # enforce a == b XOR c XOR d
        >>> H.add_constraint_eq_XOR('a', 'b', 'c', 'd')

        """
        P = PCBO().add_constraint_XNOR(*variables) - BUFFER(a)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_eq_BUFFER(self, a, b, lam=1):
        r"""add_constraint_eq_BUFFER.

        Add a penalty to the PCBO that enforces that :math:`a == b` with
        a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variables ``a``, or its PUBO representation.
        b : any hashable object or a dict.
            The label for boolean variables ``b``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_BUFFER('a', 'b')  # enforce a == b

        >>> from qubovert import PCBO, boolean_var
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> H = PCBO()
        >>> H.add_constraint_eq_BUFFER(a, b)  # enforce a == b

        """
        P = BUFFER(a) - BUFFER(b)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_eq_NAND(self, a, *variables, lam=1):
        r"""add_constraint_eq_NAND.

        Add a penalty to the PCBO that enforces that
        :math:`\lnot (v_0 \land v_1 \land v_2 \land ...) == a`, with a
        penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variables ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_NAND('a', 'b', 'c')  # enforce a == b NAND c

        >>> H = PCBO()
        >>> # enforce a == b NAND c NAND d
        >>> H.add_constraint_eq_NAND('a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> # enforce a == b NAND c
        >>> H = PCBO().add_constraint_eq_NAND(a, b, c)

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to NAND. "
                             "See ``add_constraint_eq_NOT`` for less.")
        b, c = 1, 1
        for v in variables[:n // 2]:
            b *= BUFFER(v)
        for v in variables[n // 2:]:
            c *= BUFFER(v)

        P = NOT(a) * (3 - 2 * (b + c)) + b * c
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 3))

    def add_constraint_eq_NOR(self, a, *variables, lam=1):
        r"""add_constraint_eq_NOR.

        Add a penalty to the PCBO that enforces that
        :math:`\lnot(v_0 \lor v_1 \lor v_2 \lor ...) == a`,
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_NOR('a', 'b', 'c')  # enforce a == b NOR c

        >>> H = PCBO()
        >>> # enforce a == b NOR c NOR d
        >>> H.add_constraint_eq_NOR('a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> # enforce a == b NOR c
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> H.add_constraint_eq_NOR(a, b, c)

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to NOR.")

        a = BUFFER(a)
        if n == 2:
            b, c = BUFFER(variables[0]), BUFFER(variables[1])
            P = 1 - a - b - c + b * c + 2 * a * (b + c)
            bounds = 0, 3
        else:
            P = PCBO().add_constraint_OR(*variables) - a
            bounds = -1, 1

        return self.add_constraint_eq_zero(P, lam=lam, bounds=bounds)

    def add_constraint_eq_XNOR(self, a, *variables, lam=1):
        r"""add_constraint_eq_XNOR.

        Add a penalty to the PCBO that enforces that
        :math:\lnot(`v_0 \oplus v_1 \oplus ...) == a`
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_XNOR('a', 'b', 'c')  # enforce a == b XNOR c

        >>> H = PCBO()
        >>> # enforce a == b XNOR c XNOR d
        >>> H.add_constraint_eq_XNOR('a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> # enforce a == b XNOR c
        >>> H = PCBO().add_constraint_eq_XNOR(a, b, c)

        """
        P = PCBO().add_constraint_XOR(*variables) - BUFFER(a)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_eq_NOT(self, a, b, lam=1):
        r"""NOT.

        Add a penalty to the PCBO that enforces that :math:`\lnot a == b` with
        a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        b : any hashable object or a dict.
            The label for boolean variable ``b``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_NOT('a', 'b')  # enforce NOT(a) == b

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> H.add_constraint_eq_NOT(a, b)  # enforce NOT(a) == b

        """
        P = PCBO().add_constraint_BUFFER(a) - BUFFER(b)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_AND(self, *variables, lam=1):
        r"""add_constraint_AND.

        Add a penalty to the PCBO that is only zero when
        :math:`a \land b \land c \land ...` is True, with a penalty factor
        ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_AND('a', 'b')  # enforce a AND b
        >>> H
        {('b', 'a'): -1, (): 1}

        >>> H = PCBO()
        >>> H.add_constraint_AND('a', 'b', 'c', 'd')
        >>> # enforce a AND b AND c AND d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a AND b
        >>> H = PCBO().add_constraint_AND(a, b)

        """
        return self.add_constraint_BUFFER(AND(*variables), lam=lam)

    def add_constraint_OR(self, *variables, lam=1):
        r"""add_constraint_OR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`a \lor b \lor c \lor d \lor ...` is True, with a penalty factor
        ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_OR('a', 'b')  # enforce a OR b
        >>> H
        {('a',): -1, ('b',): -1, ('b', 'a'): 1, (): 1}

        >>> H = PCBO()
        >>> H.add_constraint_OR('a', 'b', 'c', 'd')  # enforce a OR b OR c OR d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a OR b
        >>> H = PCBO().add_constraint_OR(a, b)

        """
        P = 1 - OR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_XOR(self, *variables, lam=1):
        r"""add_constraint_XOR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`v_0 \oplus v_1 \oplus ... \oplus v_n` is True, with a penalty
        factor ``lam``, where ``v_0 = variables[0]``, ``v_1 = variables[1]``,
        ..., ``v_n = variables[-1]``. See ``qubovert.sat.XOR`` for the
        XOR convention that qubovert uses for more than two inputs.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_XOR('a', 'b')  # enforce a XOR b

        >>> H = PCBO()
        >>> H.add_constraint_XOR('a', 'b', 'c')  # enforce a XOR b XOR c

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a XOR b
        >>> H = PCBO().add_constraint_XOR(a, b)

        """
        P = 1 - XOR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_BUFFER(self, a, lam=1):
        r"""add_constraint_BUFFER.

        Add a penalty to the PCBO that is only nonzero when :math:`a == 1` is
        True, with a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_BUFFER('a')  # enforce a

        >>> from qubovert import boolean_var, PCBO
        >>> a = boolean_var('a')
        >>> # enforce a
        >>> H = PCBO().add_constraint_BUFFER(a)

        """
        return self.add_constraint_eq_zero(NOT(a), lam=lam, bounds=(0, 1))

    def add_constraint_NAND(self, *variables, lam=1):
        r"""add_constraint_NAND.

        Add a penalty to the PCBO that is only zero when
        :math:`\lnot (a \land b \land c \land ...)` is True, with a penalty
        factor ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_NAND('a', 'b')  # enforce a NAND b

        >>> H = PCBO()
        >>> H.add_constraint_NAND('a', 'b', 'c', 'd')
        >>> # enforce a NAND b NAND c NAND d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a NAND b
        >>> H = PCBO().add_constraint_NAND(a, b)

        """
        return self.add_constraint_NOT(AND(*variables), lam=lam)

    def add_constraint_NOR(self, *variables, lam=1):
        r"""add_constraint_NOR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`\lnot(a \lor b \lor c \lor d \lor ...)` is True, with a penalty
        factor ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_NOR('a', 'b')  # enforce a NOR b

        >>> H = PCBO()
        >>> H.add_constraint_NOR('a', 'b', 'c', 'd')
        >>> # enforce a NOR b NOR c NOR d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a NOR b
        >>> H = PCBO().add_constraint_NOR(a, b)

        """
        P = 1 - PCBO().add_constraint_OR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_XNOR(self, *variables, lam=1):
        r"""add_constraint_XNOR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`\lnot(v_0 \oplus v_1 \oplus ... \oplus v_n)` is True, with a
        penalty  factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ..., ``v_n = variables[-1]``. See
        ``qubovert.sat.XNOR`` for the XNOR convention that qubovert uses for
        more than two inputs.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_XNOR('a', 'b')  # enforce a XNOR b

        >>> H = PCBO()
        >>> H.add_constraint_XNOR('a', 'b', 'c')  # enforce a XNOR b XNOR c

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a XNOR b
        >>> H = PCBO().add_constraint_XNOR(a, b)

        """
        P = 1 - PCBO().add_constraint_XOR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_NOT(self, a, lam=1):
        r"""add_constraint_NOT.

        Add a penalty to the PCBO that is only nonzero when
        :math:`\lnot a` is True, with a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variables ``a``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_NOT('a')  # enforce not a
        >>> H
        {('a',): 1}

        >>> from qubovert import boolean_var, PCBO
        >>> a = boolean_var('a')
        >>> # enforce not a
        >>> H = PCBO().add_constraint_NOT(a)

        """
        return self.add_constraint_eq_zero(BUFFER(a), lam=lam, bounds=(0, 1))

#   Copyright 2020 <NAME>
#
#   Licensed under the Apache License, Version 2.0 (the "License");
#   you may not use this file except in compliance with the License.
#   You may obtain a copy of the License at
#
#       http://www.apache.org/licenses/LICENSE-2.0
#
#   Unless required by applicable law or agreed to in writing, software
#   distributed under the License is distributed on an "AS IS" BASIS,
#   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
#   See the License for the specific language governing permissions and
#   limitations under the License.

"""_pcbo.py.

Contains the PCBO class. See ``help(qubovert.PCBO)``.

"""

from . import PUBO
from .utils import QUBOVertWarning, num_bits, approximate_pubo_extrema
from .sat import OR, XOR, BUFFER, NOT, AND


__all__ = 'PCBO', 'boolean_var', 'integer_var'


# special constraint forms

def _special_constraints_eq_zero(pcbo, P, lam):
    """_special_constraints_eq_zero.

    See if the constraint that ``P == 0`` matches any special forms.

    Parameters
    ----------
    pcbo : PCBO object.
        The PCBO to add the constraint to.
    P : PUBO object.
        The PUBO constraint such that ``P == 0``.
        are present.
    lam : float > 0 or sympy.Symbol.
        Langrange multiplier to penalize violations of the constraint.

    Return
    ------
    success : bool.
        True if a special constraint was found and added to ``pcbo``, else
        False.

    """
    keys, values = tuple(P.keys()), tuple(P.values())

    # if P is of the form z == x * y. ie z == AND(x, y)
    if (
            not P.offset and P.num_binary_variables == 3 and
            P.num_terms == 2 and values[0] == -values[1] and
            {len(keys[0]), len(keys[1])} == {1, 2}
    ):
        a, = keys[0] if len(keys[0]) == 1 else keys[1]
        b, c = keys[0] if len(keys[0]) == 2 else keys[1]
        pcbo += PCBO().add_constraint_eq_AND(a, b, c, lam=lam)
        return True

    # if P is of the form z == 1 - x * y. ie z == NAND(x, y)

    # if P is of the form z == x * y. ie z == OR(x, y)

    # if P is of the form z == x * y. ie z == NOR(x, y)

    return False


def _special_constraints_le_zero(pcbo, P, lam, log_trick, bounds):
    """_special_constraints_le_zero.

    See if the constraint that ``P <= 0`` matches any special forms.

    Parameters
    ----------
    pcbo : PCBO object.
        The PCBO to add the constraint to.
    P : PUBO object.
        The PUBO constraint such that ``P <= 0``.
        are present.
    lam : float > 0 or sympy.Symbol.
        Langrange multiplier to penalize violations of the constraint.
    log_trick : bool.
        Whether or not to use the log trick to enforce the inequality
        constraint.
    bounds : two element tuple.
        A tuple ``(min, max)``, the minimum and maximum values that the
        PUBO ``P`` can take.

    Return
    ------
    success : bool.
        True if a special constraint was found and added to ``pcbo``, else
        False.

    """
    min_val, max_val = bounds

    # P without offset defined for convenience
    P_wo_offset = P - P.offset

    # if P is of the form sum(x_i) <= 1.
    if P.offset == -1 and all(x == 1 for x in P_wo_offset.values()):
        pcbo += lam * P * P_wo_offset / 2
        return True

    # if P is of the form P_wo_offset <= -P.offset and P_wo_offset is always
    # >= 0, and (important!) log_trick is False!!
    elif (
            not log_trick and not min_val - P.offset and
            P.offset <= 0 and min_val
    ):
        # We have that P_wo_offset <= P.offset and min(P_wo_offset) = 0. So we
        # can do a penalty lam(P_wo_offset - sum(ancillas))**2

        # create ancillas
        ancillas = PUBO()
        for i in range(num_bits(-P.offset, log_trick)):
            ancillas[(pcbo._next_ancilla,)] += 1

        diff = P_wo_offset - ancillas
        pcbo += lam * diff * diff

        return True

    # if P is of the form 1 <= x + y, then it is the same as OR(x, y)
    elif (P.offset == 1 and len(P_wo_offset) == 2 and
          set(P_wo_offset.values()) == {-1}):
        variables = tuple(P_wo_offset.keys())
        x, y = AND(*variables[0]), AND(*variables[1])
        pcbo += PCBO().add_constraint_OR(x, y, lam=lam)
        return True

    # if P is of the form x <= y
    elif not P.offset and len(P) == 2 and set(P.values()) == {1, -1}:
        coef = {v: k for k, v in P.items()}
        x, y = AND(*coef[1]), AND(*coef[-1])
        pcbo += lam * x * (1 - y)
        return True

    return False


# helpers

def _get_bounds(P, bounds):
    """_get_bounds.

    Compute the missing bounds of ``P``.

    Parameters
    ----------
    P : PUBO object.
    bounds : two element tuple or None.
        A tuple ``(min, max)``, the minimum and maximum values that the
        PUBO ``P`` can take. If ``bounds`` is None, then they will be
        calculated (approximately), or if either of the elements of
        ``bounds`` is None, then that element will be calculated
        (approximately).

    Return
    ------
    res : two element tuple.
        The bounds of ``P``.

    """
    if bounds is None or bounds == (None, None):
        bounds = approximate_pubo_extrema(P)
    elif bounds[0] is None:
        bounds = approximate_pubo_extrema(P)[0], bounds[1]
    elif bounds[1] is None:
        bounds = bounds[0], approximate_pubo_extrema(P)[1]

    return bounds


def boolean_var(name):
    """boolean_var.

    Create a PCBO (see ``qubovert.PCBO``) from a single boolean variable.

    Parameters
    ----------
    name : any hashable object.
        Name of the boolean variable.

    Return
    ------
    pcbo : qubovert.PCBO object.
        The model representing the boolean variable.

    Examples
    --------
    >>> from qubovert import boolean_var, PCBO
    >>>
    >>> x0 = boolean_var("x0")
    >>> print(x0)
    {('x0',): 1}
    >>> print(isinstance(x0, PCBO))
    True
    >>> print(x0.name)
    x0

    >>> x = [boolean_var('x{}'.format(i)) for i in range(5)]
    >>> pcbo = sum(x)
    >>> print(pcbo)
    {('x0',): 1, ('x1',): 1, ('x2',): 1, ('x3',): 1, ('x4',): 1}
    >>> pcbo **= 2
    >>> print(pcbo)
    {('x0',): 1, ('x0', 'x1'): 2, ('x2', 'x0'): 2, ('x3', 'x0'): 2,
     ('x4', 'x0'): 2, ('x1',): 1, ('x2', 'x1'): 2, ('x3', 'x1'): 2,
     ('x4', 'x1'): 2, ('x2',): 1, ('x2', 'x3'): 2, ('x2', 'x4'): 2, ('x3',): 1,
     ('x4', 'x3'): 2, ('x4',): 1}
    >>> pcbo *= -1
    >>> print(pcbo.solve_bruteforce())
    {'x0': 1, 'x1': 1, 'x2': 1, 'x3': 1, 'x4': 1}
    >>> pcbo.add_constraint_eq_zero(x[0] + x[1])
    >>> print(pcbo.solve_bruteforce())
    {'x0': 0, 'x1': 0, 'x2': 1, 'x3': 1, 'x4': 1}

    Notes
    -----
    ``qubovert.boolean_var(name)`` is equivalent to
    ``qubovert.PCBO.create_var(name)``.

    """
    return PCBO.create_var(name)


def integer_var(prefix, num_bits, log_trick=True):
    """integer_var.

    Return a PCBO object representing an integer variable with `num_bits`
    bits.

    Parameters
    ----------
    prefix : str.
        The prefix for the boolean variable names.
    num_bits : int.
        Number of bits to represent the integer variable with.
    log_trick : bool (optional, defaults to True).
        Whether or not to use a log encoding for the integer.

    Return
    ------
    i : qubovert.PCBO object.

    Example
    -------
    >>> from qubovert import integer_var
    >>> var = integer_var('a', 4)
    >>> print(var)
    {('a0',): 1, ('a1',): 2, ('a2',): 4, ('a3',): 8}
    >>> print(var.name)
    a

    >>> from qubovert import integer_var
    >>> var = integer_var('a', 4, log_trick=False)
    >>> print(var)
    {('a0',): 1, ('a1',): 1, ('a2',): 1, ('a3',): 1}

    """
    var = PCBO()
    var.name = prefix
    for i in range(num_bits):
        var[(str(prefix) + str(i),)] = pow(2, i) if log_trick else 1
    return var


# main class

class PCBO(PUBO):
    """PCBO.

    This class deals with Polynomial Constrained Boolean Optimization problems.
    PCBO inherits some methods and attributes from the ``PUBO`` class. See
    ``help(qubovert.PUBO)``.

    ``PCBO`` has all the same methods as ``PUBO``, but adds some constraint
    methods; namely

    - ``add_constraint_eq_zero(P, lam=1, ...)`` enforces that ``P == 0`` by
      penalizing with ``lam``,
    - ``add_constraint_ne_zero(P, lam=1, ...)`` enforces that ``P != 0`` by
      penalizing with ``lam``,
    - ``add_constraint_lt_zero(P, lam=1, ...)`` enforces that ``P < 0`` by
      penalizing with ``lam``,
    - ``add_constraint_le_zero(P, lam=1, ...)`` enforces that ``P <= 0`` by
      penalizing with ``lam``,
    - ``add_constraint_gt_zero(P, lam=1, ...)`` enforces that ``P > 0`` by
      penalizing with ``lam``, and
    - ``add_constraint_ge_zero(P, lam=1, ...)`` enforces that ``P >= 0`` by
      penalizing with ``lam``.

    Each of these takes in a PUBO ``P`` and a lagrange multiplier ``lam`` that
    defaults to 1. See each of their docstrings for important details on their
    implementation.

    We then implement logical operations:

    - ``add_constraint_AND``, ``add_constraint_NAND``,
    - ``add_constraint_eq_AND``, ``add_constraint_eq_NAND``,
    - ``add_constraint_OR``, ``add_constraint_NOR```,
    - ``add_constraint_eq_OR``, ``add_constraint_eq_NOR``,
    - ``add_constraint_XOR``, ``add_constraint_XNOR``,
    - ``add_constraint_eq_XOR``, ``add_constraint_eq_XNOR``,
    - ``add_constraint_BUFFER``, ``add_constraint_NOT``,
    - ``add_constraint_eq_BUFFER``, ``add_constraint_eq_NOT``.

    See each of their docstrings for important details on their implementation.

    Notes
    -----
    - Variables names that begin with ``"__a"`` should not be used since they
      are used internally to deal with some ancilla variables to enforce
      constraints.
    - The ``self.solve_bruteforce`` method will solve the PCBO ensuring that
      all the inputted constraints are satisfied. Whereas
      ``qubovert.utils.solve_pubo_bruteforce(self)`` or
      ``qubovert.utils.solve_pubo_bruteforce(self.to_pubo())`` will solve the
      PUBO created from the PCBO. If the inputted constraints are not enforced
      strong enough (ie too small lagrange multipliers) then these may not give
      the correct result, whereas ``self.solve_bruteforce()`` will always give
      the correct result (ie one that satisfies all the constraints).

    Examples
    --------
    See ``qubovert.PUBO`` for more examples of using PCBO without constraints.

    >>> H = PCBO()
    >>> H.add_constraint_eq_zero({('a', 1): 2, (1, 2): -1, (): -1})
    >>> H
    {('a', 1, 2): -4, (1, 2): 3, (): 1}
    >>> H -= 1
    >>> H
    {('a', 1, 2): -4, (1, 2): 3}

    >>> H = PCBO()
    >>>
    >>> # minimize -x_0 - x_1 - x_2
    >>> for i in (0, 1, 2):
    >>>     H[(i,)] -= 1
    >>>
    >>> # subject to constraints
    >>> H.add_constraint_eq_zero(  # enforce that x_0 x_1 - x_2 == 0
    >>>     {(0, 1): 1, (2,): -1}
    >>> ).add_constraint_lt_zero(  # enforce that x_1 x_2 + x_0 < 1
    >>>     {(1, 2): 1, (0,): 1, (): -1}
    >>> )
    >>> print(H)
    >>> # {(1,): -2, (2,): -1, (0, 1): 2, (1, 2): 2, (0, 1, 2): 2}
    >>>
    >>> print(H.solve_bruteforce(all_solutions=True))
    >>> # [{0: 0, 1: 1, 2: 0}]
    >>>
    >>> Q = H.to_qubo()
    >>> solutions = [H.convert_solution(sol)
    >>>              for sol in Q.solve_bruteforce(all_solutions=True)]
    >>> print(solutions)
    >>> # [{0: 0, 1: 1, 2: 0}]  # matches the PCBO solution!
    >>>
    >>> L = H.to_quso()
    >>> solutions = [H.convert_solution(sol)
    >>>              for sol in L.solve_bruteforce(all_solutions=True)]
    >>> print(solutions)
    >>> # [{0: 0, 1: 1, 2: 0}]  # matches the PCBO solution!

    Enforce that c == a b

    >>> H = PCBO().add_constraint_eq_AND('c', 'a', 'b')
    >>> H
    {('c',): 3, ('b', 'a'): 1, ('c', 'a'): -2, ('c', 'b'): -2}

    >>> from any_module import qubo_solver
    >>> # or from qubovert.utils import solve_qubo_bruteforce as qubo_solver
    >>> H = PCBO()
    >>>
    >>> # make it favorable to AND variables a and b, and variables b and c
    >>> H.add_constraint_AND('a', 'b').add_constraint_AND('b', 'c')
    >>>
    >>> # make it favorable to OR variables b and c
    >>> H.add_constraint_OR('b', 'c')
    >>>
    >>> # make it favorable to (a AND b) OR (c AND d)
    >>> H.add_constraint_OR(['a', 'b'], ['c', 'd'])
    >>>
    >>> H
    {('b', 'a'): -2, (): 4, ('b',): -1, ('c',): -1,
     ('c', 'd'): -1, ('c', 'd', 'b', 'a'): 1}
    >>> Q = H.to_qubo()
    >>> Q
    {(): 4, (0,): -1, (2,): -1, (2, 3): 1, (4,): 6, (0, 4): -4,
     (1, 4): -4, (5,): 6, (2, 5): -4, (3, 5): -4, (4, 5): 1}
    >>> obj_value, sol = qubo_solver(Q)
    >>> sol
    {0: 1, 1: 1, 2: 1, 3: 0, 4: 1, 5: 0}
    >>> solution = H.convert_solution(sol)
    >>> solution
    {'b': 1, 'a': 1, 'c': 1, 'd': 0}

    """

    def __init__(self, *args, **kwargs):
        """__init__.

        This class deals with polynomial constrained boolean optimization.
        Note that it is generally more efficient to initialize an empty PCBO
        object and then build the PCBO, rather than initialize a PCBO object
        with an already built dict.

        Parameters
        ----------
        arguments : define a dictionary with ``dict(*args, **kwargs)``.
            The dictionary will be initialized to follow all the convensions of
            the class. Alternatively, ``args[0]`` can be a PCBO object.

        Examples
        --------
        >>> pcbo = PCBO()
        >>> pcbo[('a',)] += 5
        >>> pcbo[(0, 'a')] -= 2
        >>> pcbo -= 1.5
        >>> pcbo
        {('a',): 5, ('a', 0): -2, (): -1.5}
        >>> pcbo.add_constraint_eq_zero({('a',): 1}, lam=5)
        >>> pcbo
        {('a',): 10, ('a', 0): -2, (): -1.5}

        >>> pcbo = PCBO({('a',): 5, (0, 'a', 1): -2, (): -1.5})
        >>> pcbo
        {('a',): 5, ('a', 0, 1): -2, (): -1.5}

        """
        # use self.__class__ here because PCSO uses this code as well.
        super(self.__class__, self).__init__(*args, **kwargs)
        if len(args) == 1 and isinstance(args[0], self.__class__):
            self._constraints = args[0].constraints
            self._ancilla = args[0].num_ancillas
        else:
            self._ancilla, self._constraints = 0, {}

    def update(self, *args, **kwargs):
        """update.

        Update the PCBO but following all the conventions of this class.

        Parameters
        ----------
        arguments : defines a dictionary or PCBO.
            Ie ``d = dict(*args, **kwargs)``.
            Each element in d will be added in place to this instance following
            all the required convensions.

        """
        # use self.__class__ here because PCSO uses this code as well.
        super(self.__class__, self).update(*args, **kwargs)
        if len(args) == 1 and isinstance(args[0], self.__class__):
            for k, v in args[0]._constraints.items():
                self._constraints.setdefault(k, []).extend(v)

    @property
    def constraints(self):
        """constraints.

        Return the constraints of the PCBO.

        Return
        ------
        res : dict.
            The keys of ``res`` are some or all of
            ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, and ``'ge'``.
            The values are lists of ``qubovert.PUBO`` objects. For a
            given key, value pair ``k, v``, the ``v[i]`` element represents
            the PUBO ``v[i]`` being == 0 if ``k == 'eq'``,
            != 0 if ``k == 'ne'``,
            < 0 if ``k == 'lt'``, <= 0 if ``k == 'le'``,
            > 0 if ``k == 'gt'``, >= 0 if ``k == 'ge'``.

        """
        return {k: [x.copy() for x in v] for k, v in self._constraints.items()}

    def _append_constraint(self, key, constraint):
        """_append_constraint.

        Internal method to add a constraint to the constraints dictionary.

        Parameters
        ----------
        key : str.
            One of ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, or
            ``'ge'``.
        constraint : qubovert.PUBO object.

        """
        self._constraints.setdefault(key, []).append(constraint)

    def _pop_constraint(self, key):
        """_append_constraint.

        Internal method to remove the most recently added constraint.

        Parameters
        ----------
        key : str.
            One of ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, or
            ``'ge'``.

        """
        if self._constraints.get(key, []):
            self._constraints[key].pop()
            if not self._constraints[key]:
                self._constraints.pop(key)

    @property
    def num_ancillas(self):
        """num_ancillas.

        Return the number of ancilla variables introduced to the PCBO in
        order to enforce the inputted constraints.

        Returns
        -------
        num : int.
            Number of ancillas in the PCBO.

        """
        return self._ancilla

    @property
    def _next_ancilla(self):
        """_next_ancilla.

        Get the next available ancilla bit and increment.

        Return
        ------
        a : int.
            Ancilla bit label.

        """
        self._ancilla += 1
        return "__a%d" % (self._ancilla - 1)

    @classmethod
    def remove_ancilla_from_solution(cls, solution):
        """remove_ancilla_from_solution.

        Take a solution to the PCBO and remove all the ancilla variables, (
        represented by `_a` prefixes).

        Parameters
        ----------
        solution : dict.
            Must be the solution in terms of the original variables. Thus if
            ``solution`` is the solution to the ``self.to_pubo``,
            ``self.to_qubo``, ``self.to_puso``, or ``self.to_quso``
            formulations, then you should first call ``self.convert_solution``.
            See ``help(self.convert_solution)``.

        Return
        ------
        res : dict.
            The same as ``solution`` but with all the ancilla bits removed.

        """
        return {k: v for k, v in solution.items() if str(k)[:3] != "__a"}

    def is_solution_valid(self, solution):
        """is_solution_valid.

        Finds whether or not the given solution satisfies the constraints.

        Parameters
        ----------
        solution : dict.
            Must be the solution in terms of the original variables. Thus if
            ``solution`` is the solution to the ``self.to_pubo``,
            ``self.to_qubo``, ``self.to_puso``, or ``self.to_quso``
            formulations, then you should first call ``self.convert_solution``.
            See ``help(self.convert_solution)``.

        Return
        ------
        valid : bool.
            Whether or not the given solution satisfies the constraints.

        """
        if any(v.value(solution) != 0
               for v in self._constraints.get('eq', [])):
            return False

        if any(v.value(solution) == 0
               for v in self._constraints.get('ne', [])):
            return False

        if any(v.value(solution) >= 0
               for v in self._constraints.get("lt", [])):
            return False

        if any(v.value(solution) > 0
               for v in self._constraints.get("le", [])):
            return False

        if any(v.value(solution) <= 0
               for v in self._constraints.get("gt", [])):
            return False

        if any(v.value(solution) < 0
               for v in self._constraints.get("ge", [])):
            return False

        return True

    # override
    def __round__(self, ndigits=None):
        """round.

        Round values of the PCBO object.

        Parameters
        ----------
        ndigits : int.
            Number of decimal digits to round to.

        Returns
        -------
        res : PCBO object.
            Copy of self but with each value rounded to ``ndigits`` decimal
            digits. Each value has a type according to the docstring
            specifications of ``round``, see ``help(round)``.

        """
        # use self.__class__ here because PCSO uses this code as well.
        d = super(self.__class__, self).__round__(ndigits)
        d._constraints = self.constraints
        return d

    # override
    def subs(self, *args, **kwargs):
        """subs.

        Replace any ``sympy`` symbols that are used in the dict with values.
        Please see ``help(sympy.Symbol.subs)`` for more info.

        Parameters
        ----------
        arguments : substitutions.
            Same parameters as are inputted into ``sympy.Symbol.subs``.

        Returns
        -------
        res : PCBO object.
            Same as ``self`` but with all the symbols replaced with values.

        """
        # use self.__class__ here because PCSO uses this code as well.
        d = super(self.__class__, self).subs(*args, **kwargs)
        d._constraints = {
            k: [P.subs(*args, **kwargs) for P in v]
            for k, v in self._constraints.items()
        }
        return d

    def add_constraint_eq_zero(self,
                               P, lam=1,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_eq_zero.

        Enforce that ``P == 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P == 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they may be
            calculated (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        The following enforces that :math:`\prod_{i=0}^{3} x_i == 0`.

        >>> H = PCBO()
        >>> H.add_constraint_eq_zero({(0, 1, 2, 3): 1})
        >>> H
        {(0, 1, 2, 3): 1}

        The following enforces that :math:`\sum_{i=1}^{3} i x_i x_{i+1} == 0`.

        >>> H = PCBO()
        >>> H.add_constraint_eq_zero({(1, 2): 1, (2, 3): 2, (3, 4): 3})
        >>> H
        {(1, 2): 1, (1, 2, 3): 4, (1, 2, 3, 4): 6,
         (2, 3): 4, (2, 3, 4): 12, (3, 4): 9}

        Here we show how operations can be strung together.

        >>> H = PCBO()
        >>> H.add_constraint_eq_zero(
                {(0, 1): 1}
            ).add_constraint_eq_zero(
                {(1, 2): 1, (): -1}
            )
        >>> H
        {(0, 1): 1, (1, 2): -1, (): 1}

        """
        P = PUBO(P)
        self._append_constraint("eq", P)
        if not lam:
            return self

        if _special_constraints_eq_zero(self, P, lam):
            return self

        min_val, max_val = _get_bounds(P, bounds)

        if min_val == max_val == 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        elif min_val > 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam * P
        elif max_val < 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self -= lam * P
        elif min_val == 0:
            self += lam * P
        elif max_val == 0:
            self -= lam * P
        else:
            self += lam * P * P

        return self

    def add_constraint_ne_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_ne_zero.

        Enforce that ``P != 0`` by penalizing invalid solutions with ``lam``.
        See Notes below for more details.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P != 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... != 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function.
        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\min_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\min_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c != 2`.

        >>> H = PCBO().add_constraint_ne_zero(
                {('a', 'b', 'c'): -1, ('a',): 2,
                ('a', 'b'): -4, ('c',): 3, (): -2}
            )
        >>> print(H)
        {('c', 'a', 'b'): -19, ('__a0', 'c', 'a', 'b'): -4,
         ('__a0', 'c', '__a1', 'a', 'b'): -4, ('c', '__a1', 'a', 'b'): 2,
         ('__a0', 'c', '__a2', 'a', 'b'): -8, ('c', '__a2', 'a', 'b'): 4,
         ('__a0', '__a3', 'c', 'a', 'b'): -16, ('__a3', 'c', 'a', 'b'): 8,
         ('__a0', 'c', '__a4', 'a', 'b'): -32, ('c', '__a4', 'a', 'b'): 16,
         ('a',): -8, ('c', 'a'): 12, ('__a0', 'a'): 8,
         ('__a0', '__a1', 'a'): 8, ('__a1', 'a'): -4,
         ('__a0', '__a2', 'a'): 16, ('__a2', 'a'): -8,
         ('__a0', '__a3', 'a'): 32, ('__a3', 'a'): -16,
         ('__a0', '__a4', 'a'): 64, ('__a4', 'a'): -32, ('a', 'b'): 24,
         ('__a0', 'a', 'b'): -16, ('__a0', '__a1', 'a', 'b'): -16,
         ('__a1', 'a', 'b'): 8, ('__a0', '__a2', 'a', 'b'): -32,
         ('__a2', 'a', 'b'): 16, ('__a0', '__a3', 'a', 'b'): -64,
         ('__a3', 'a', 'b'): 32, ('__a0', '__a4', 'a', 'b'): -128,
         ('__a4', 'a', 'b'): 64, ('__a0', 'c'): 12, ('__a0', 'c', '__a1'): 12,
         ('c', '__a1'): -6, ('__a0', 'c', '__a2'): 24, ('c', '__a2'): -12,
         ('__a0', '__a3', 'c'): 48, ('__a3', 'c'): -24,
         ('__a0', 'c', '__a4'): 96, ('c', '__a4'): -48, ('c',): -9, (): 9,
         ('__a0',): -8, ('__a0', '__a1'): -8, ('__a1',): 7,
         ('__a0', '__a2'): -16, ('__a2',): 16, ('__a3',): 40,
         ('__a0', '__a4'): -64, ('__a4',): 112, ('__a2', '__a1'): 4,
         ('__a3', '__a1'): 8, ('__a4', '__a1'): 16, ('__a3', '__a2'): 16,
         ('__a4', '__a2'): 32, ('__a0', '__a3'): -32, ('__a3', '__a4'): 64}
        >>> print(H.is_solution_valid({'b': 0, 'c': 0, 'a': 1}))
        False
        >>> print(H.is_solution_valid({'b': 0, 'c': 1, 'a': 1}))
        True

        """
        P = PUBO(P)
        self._append_constraint("ne", P)
        if not lam:
            return self

        min_val, max_val = _get_bounds(P, bounds)

        if min_val == max_val == 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam
        elif min_val > 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        elif max_val < 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        elif min_val == 0:
            self.add_constraint_gt_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=suppress_warnings
            )
            self._pop_constraint('gt')
        elif max_val == 0:
            self.add_constraint_lt_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=suppress_warnings
            )
            self._pop_constraint('lt')

        else:
            # don't mutate the P that we put in self._constraints
            P = P.copy()
            sign = 2 * boolean_var(self._next_ancilla) - 1
            P += sign
            max_val += 1
            min_val -= 1
            for i in range(num_bits(max_val - min_val - 1, log_trick)):
                v = pow(2, i) if log_trick else 1
                P += sign * v * boolean_var(self._next_ancilla)
                max_val += v
                min_val -= v

            self.add_constraint_eq_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=True
            )
            self._pop_constraint("eq")

        return self

    def add_constraint_lt_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_lt_zero.

        Enforce that ``P < 0`` by penalizing invalid solutions with ``lam``.
        See Notes below for more details.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P < 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... < 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_lt_zero({(0,): 1, (1,): 2, (2,): -.5, (): .4})
          >>> test_sol = {0: 0, 1: 0, 2: 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\min_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\min_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c < 2`.

        >>> H = PCBO().add_constraint_lt_zero(
                {('a', 'b', 'c'): -1, ('a',): 1,
                 ('a', 'b'): -4, ('c',): 3, (): -2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', '__a0', 'c', 'a'): -2,
         ('b', 'c', 'a', '__a1'): -4, ('a',): -3, ('b', 'a'): 24,
         ('c', 'a'): 6, ('__a0', 'a'): 2, ('a', '__a1'): 4,
         ('b', '__a0', 'a'): -8, ('b', 'a', '__a1'): -16, ('c',): -3,
         ('__a0', 'c'): 6, ('c', '__a1'): 12, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a0', '__a1'): 4}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("lt", P)
        if not lam:
            return self

        min_val, max_val = _get_bounds(P, bounds)

        if min_val >= 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam * P
        elif max_val < 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        else:
            # copy P, don't do +=
            P = P + 1
            min_val += 1
            max_val += 1
            self.add_constraint_le_zero(
                P, lam=lam, log_trick=log_trick,
                bounds=(min_val, max_val), suppress_warnings=True
            )
            self._pop_constraint("le")

        return self

    def add_constraint_le_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_le_zero.

        Enforce that ``P <= 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P <= 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... \leq 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_le_zero({(0,): 1, (1,): 2, (2,): -1.5, (): .4})
          >>> H
          {(0,): 1.7999999999999998, (0, 1): 4, (0, 2): -3.0, (0, '__a0'): 2,
           (1,): 5.6, (1, 2): -6.0, (1, '__a0'): 4, (2,): 1.0499999999999998,
           (2, '__a0'): -3.0, (): 0.16000000000000003, ('__a0',): 1.8}
          >>> test_sol = {0: 0, 1: 0, 2: 1, '__a0': 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\min_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\min_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c \leq 2`.

        >>> H = PCBO().add_constraint_le_zero(
                {('a', 'b', 'c'): -1, ('a',): 1,
                 ('a', 'b'): -4, ('c',): 3, (): -2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', 'c', 'a', '__a0'): -2,
         ('__a1', 'b', 'c', 'a'): -4, ('__a2', 'b', 'c', 'a'): -8, ('a',): -3,
         ('b', 'a'): 24, ('c', 'a'): 6, ('a', '__a0'): 2, ('__a1', 'a'): 4,
         ('__a2', 'a'): 8, ('b', 'a', '__a0'): -8, ('__a1', 'b', 'a'): -16,
         ('__a2', 'b', 'a'): -32, ('c',): -3, ('c', '__a0'): 6,
         ('__a1', 'c'): 12, ('__a2', 'c'): 24, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a1', '__a0'): 4, ('__a2', '__a0'): 8,
         ('__a2', '__a1'): 16}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("le", P)
        if not lam:
            return self

        bounds = min_val, max_val = _get_bounds(P, bounds)
        if _special_constraints_le_zero(self, P, lam, log_trick, bounds):
            return self

        if min_val > 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint cannot be satisfied")
            self += lam * P
        elif max_val <= 0:
            if not suppress_warnings:
                QUBOVertWarning.warn("Constraint is always satisfied")
        else:
            # don't mutate the P that we put in self._constraints
            P = P.copy()
            if min_val:
                for i in range(num_bits(-min_val, log_trick)):
                    v = pow(2, i) if log_trick else 1
                    P[(self._next_ancilla,)] += v
                    max_val += v

            self.add_constraint_eq_zero(
                P, lam=lam,
                bounds=(min_val, max_val), suppress_warnings=True
            )
            self._pop_constraint("eq")

        return self

    def add_constraint_gt_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_gt_zero.

        Enforce that ``P > 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P > 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... > 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_gt_zero({(0,): -1, (1,): -2, (2,): .5, (): -.4})
          >>> test_sol = {0: 0, 1: 0, 2: 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\max_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\max_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`x_a x_b x_c - x_a + 4x_a x_b - 3x_c > -2`.

        >>> H = PCBO().add_constraint_gt_zero(
                {('a', 'b', 'c'): 1, ('a',): -1,
                 ('a', 'b'): 4, ('c',): -3, (): 2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', '__a0', 'c', 'a'): -2,
         ('b', 'c', 'a', '__a1'): -4, ('a',): -3, ('b', 'a'): 24,
         ('c', 'a'): 6, ('__a0', 'a'): 2, ('a', '__a1'): 4,
         ('b', '__a0', 'a'): -8, ('b', 'a', '__a1'): -16, ('c',): -3,
         ('__a0', 'c'): 6, ('c', '__a1'): 12, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a0', '__a1'): 4}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("gt", P)
        if not lam:
            return self
        min_val, max_val = _get_bounds(P, bounds)
        bounds = -max_val, -min_val
        self.add_constraint_lt_zero(
            -P, lam=lam, log_trick=log_trick,
            bounds=bounds, suppress_warnings=suppress_warnings
        )
        self._pop_constraint("lt")
        return self

    def add_constraint_ge_zero(self,
                               P, lam=1, log_trick=True,
                               bounds=None, suppress_warnings=False):
        r"""add_constraint_ge_zero.

        Enforce that ``P >= 0`` by penalizing invalid solutions with ``lam``.

        Parameters
        ----------
        P : dict representing a PUBO.
            The PUBO constraint such that P >= 0. Note that ``P`` will be
            converted to a ``qubovert.PUBO`` object if it is not already, thus
            it must follow the conventions, see ``help(qubovert.PUBO)``.
            Please note that if ``P`` contains any symbols, then ``bounds``
            must be supplied, since they cannot be determined when symbols
            are present.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the constraint.
        log_trick : bool (optional, defaults to True).
            Whether or not to use the log trick to enforce the inequality
            constraint. See Notes below for more details.
        bounds : two element tuple (optional, defaults to None).
            A tuple ``(min, max)``, the minimum and maximum values that the
            PUBO ``P`` can take. If ``bounds`` is None, then they will be
            calculated (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately), or if either of the elements of
            ``bounds`` is None, then that element will be calculated
            (approximately).
        suppress_warnings : bool (optional, defaults to False).
            Whether or not to surpress warnings.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Notes
        -----
        - There is no general way to enforce non integer inequality
          constraints. Thus this function is only guarenteed to work for
          integer inequality constraints (ie constraints of the form
          :math:`x_0 + 2x_1 + ... \geq 0`). However, it can be used for non
          integer inequality constraints, but it is recommended that the value
          of ``lam`` be set small, since valid solutions may still recieve a
          penalty to the objective function. For example,

          >>> H = PCBO()
          >>> H.add_constraint_ge_zero({(0,): -1, (1,): -2, (2,):1.5, (): -.4})
          >>> H
          {(0,): 1.7999999999999998, (0, 1): 4, (0, 2): -3.0, (0, '__a0'): 2,
           (1,): 5.6, (1, 2): -6.0, (1, '__a0'): 4, (2,): 1.0499999999999998,
           (2, '__a0'): -3.0, (): 0.16000000000000003, ('__a0',): 1.8}
          >>> test_sol = {0: 0, 1: 0, 2: 1, '__a0': 1}
          >>> H.is_solution_valid(test_sol)
          True
          >>> H.value(test_sol)
          0.01

          {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still
          cause a nonzero penalty to be added to the objective function.

        - To enforce the inequality constraint, ancilla bits will be
          introduced (labels with `_a`). If ``log_trick`` is ``True``, then
          approximately :math:`\log_2 |\max_x \text{P.value(x)}|`
          ancilla bits will be used. If ``log_trick`` is ``False``, then
          approximately :math:`|\max_x \text{P.value(x)}|` ancilla
          bits will be used.

        Examples
        --------
        Enforce that :math:`x_a x_b x_c - x_a + 4x_a x_b - 3x_c \geq -2`.

        >>> H = PCBO().add_constraint_ge_zero(
                {('a', 'b', 'c'): 1, ('a',): -1,
                 ('a', 'b'): 4, ('c',): -3, (): 2}
            )
        >>> H
        {('b', 'c', 'a'): -19, ('b', 'c', 'a', '__a0'): -2,
         ('__a1', 'b', 'c', 'a'): -4, ('__a2', 'b', 'c', 'a'): -8, ('a',): -3,
         ('b', 'a'): 24, ('c', 'a'): 6, ('a', '__a0'): 2, ('__a1', 'a'): 4,
         ('__a2', 'a'): 8, ('b', 'a', '__a0'): -8, ('__a1', 'b', 'a'): -16,
         ('__a2', 'b', 'a'): -32, ('c',): -3, ('c', '__a0'): 6,
         ('__a1', 'c'): 12, ('__a2', 'c'): 24, (): 4, ('__a0',): -3,
         ('__a1',): -4, ('__a1', '__a0'): 4, ('__a2', '__a0'): 8,
         ('__a2', '__a1'): 16}
        >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})
        True
        >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})
        False

        """
        P = PUBO(P)
        self._append_constraint("ge", P)
        if not lam:
            return self
        min_val, max_val = _get_bounds(P, bounds)
        bounds = -max_val, -min_val
        self.add_constraint_le_zero(
            -P, lam=lam, log_trick=log_trick,
            bounds=bounds, suppress_warnings=suppress_warnings
        )
        self._pop_constraint("le")
        return self

    def add_constraint_eq_AND(self, a, *variables, lam=1):
        r"""add_constraint_eq_AND.

        Add a penalty to the PCBO that enforces that
        :math:`a = v_0 \land v_1 \land v_2 \land ...`,
        with a penalty factor
        ``lam``, where ``v_1 = variables[0]``, ``v_2 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_AND('a', 'b', 'c')  # enforce a == b AND c

        >>> H = PCBO()
        >>> # enforce (a AND b AND c AND d) == 'e'
        >>> H.add_constraint_eq_AND('e', 'a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> H = PCBO()
        >>> # enforce that a == b AND c
        >>> H.add_constraint_eq_AND(a, b, c)

        References
        ----------
        https://arxiv.org/pdf/1307.8041.pdf equation 6.

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to AND. "
                             "See ``add_constraint_eq_BUFFER`` for less.")

        a = BUFFER(a)
        b, c = 1, 1
        for v in variables[:n // 2]:
            b *= BUFFER(v)
        for v in variables[n // 2:]:
            c *= BUFFER(v)

        P = 3 * a + b * c - 2 * a * (b + c)

        return self.add_constraint_eq_zero(P, lam, bounds=(0, 3))

    def add_constraint_eq_OR(self, a, *variables, lam=1):
        r"""add_constraint_eq_OR.

        Add a penalty to the PCBO that enforces that
        :math:`v_0 \lor v_1 \lor v_2 \lor ... == a`,
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_OR('a', 'b', 'c')  # enforce a == b OR c

        >>> H = PCBO()
        >>> # enforce a == b OR c OR d
        >>> H.add_constraint_eq_OR('a', 'b', 'c', 'd')

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to OR.")

        a = BUFFER(a)
        if n == 2:
            b, c = BUFFER(variables[0]), BUFFER(variables[1])
            P = a + b + c + b * c - 2 * a * (b + c)
            bounds = 0, 3
        else:
            P = PCBO().add_constraint_NOR(*variables) - a
            bounds = -1, 1

        return self.add_constraint_eq_zero(P, lam=lam, bounds=bounds)

    def add_constraint_eq_XOR(self, a, *variables, lam=1):
        r"""add_constraint_eq_XOR.

        Add a penalty to the PCBO that enforces that
        :math:`v_0 \oplus v_1 \oplus ... == a`
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_XOR('a', 'b', 'c')  # enforce a == b XOR c

        >>> H = PCBO()
        >>> # enforce a == b XOR c XOR d
        >>> H.add_constraint_eq_XOR('a', 'b', 'c', 'd')

        """
        P = PCBO().add_constraint_XNOR(*variables) - BUFFER(a)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_eq_BUFFER(self, a, b, lam=1):
        r"""add_constraint_eq_BUFFER.

        Add a penalty to the PCBO that enforces that :math:`a == b` with
        a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variables ``a``, or its PUBO representation.
        b : any hashable object or a dict.
            The label for boolean variables ``b``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_BUFFER('a', 'b')  # enforce a == b

        >>> from qubovert import PCBO, boolean_var
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> H = PCBO()
        >>> H.add_constraint_eq_BUFFER(a, b)  # enforce a == b

        """
        P = BUFFER(a) - BUFFER(b)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_eq_NAND(self, a, *variables, lam=1):
        r"""add_constraint_eq_NAND.

        Add a penalty to the PCBO that enforces that
        :math:`\lnot (v_0 \land v_1 \land v_2 \land ...) == a`, with a
        penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variables ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_NAND('a', 'b', 'c')  # enforce a == b NAND c

        >>> H = PCBO()
        >>> # enforce a == b NAND c NAND d
        >>> H.add_constraint_eq_NAND('a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> # enforce a == b NAND c
        >>> H = PCBO().add_constraint_eq_NAND(a, b, c)

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to NAND. "
                             "See ``add_constraint_eq_NOT`` for less.")
        b, c = 1, 1
        for v in variables[:n // 2]:
            b *= BUFFER(v)
        for v in variables[n // 2:]:
            c *= BUFFER(v)

        P = NOT(a) * (3 - 2 * (b + c)) + b * c
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 3))

    def add_constraint_eq_NOR(self, a, *variables, lam=1):
        r"""add_constraint_eq_NOR.

        Add a penalty to the PCBO that enforces that
        :math:`\lnot(v_0 \lor v_1 \lor v_2 \lor ...) == a`,
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_NOR('a', 'b', 'c')  # enforce a == b NOR c

        >>> H = PCBO()
        >>> # enforce a == b NOR c NOR d
        >>> H.add_constraint_eq_NOR('a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> # enforce a == b NOR c
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> H.add_constraint_eq_NOR(a, b, c)

        """
        n = len(variables)
        if n < 2:
            raise ValueError("Must supply at least two variables to NOR.")

        a = BUFFER(a)
        if n == 2:
            b, c = BUFFER(variables[0]), BUFFER(variables[1])
            P = 1 - a - b - c + b * c + 2 * a * (b + c)
            bounds = 0, 3
        else:
            P = PCBO().add_constraint_OR(*variables) - a
            bounds = -1, 1

        return self.add_constraint_eq_zero(P, lam=lam, bounds=bounds)

    def add_constraint_eq_XNOR(self, a, *variables, lam=1):
        r"""add_constraint_eq_XNOR.

        Add a penalty to the PCBO that enforces that
        :math:\lnot(`v_0 \oplus v_1 \oplus ...) == a`
        with a penalty factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ...

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_XNOR('a', 'b', 'c')  # enforce a == b XNOR c

        >>> H = PCBO()
        >>> # enforce a == b XNOR c XNOR d
        >>> H.add_constraint_eq_XNOR('a', 'b', 'c', 'd')

        >>> from qubovert import boolean_var, PCBO
        >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')
        >>> # enforce a == b XNOR c
        >>> H = PCBO().add_constraint_eq_XNOR(a, b, c)

        """
        P = PCBO().add_constraint_XOR(*variables) - BUFFER(a)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_eq_NOT(self, a, b, lam=1):
        r"""NOT.

        Add a penalty to the PCBO that enforces that :math:`\lnot a == b` with
        a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        b : any hashable object or a dict.
            The label for boolean variable ``b``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_eq_NOT('a', 'b')  # enforce NOT(a) == b

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> H.add_constraint_eq_NOT(a, b)  # enforce NOT(a) == b

        """
        P = PCBO().add_constraint_BUFFER(a) - BUFFER(b)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(-1, 1))

    def add_constraint_AND(self, *variables, lam=1):
        r"""add_constraint_AND.

        Add a penalty to the PCBO that is only zero when
        :math:`a \land b \land c \land ...` is True, with a penalty factor
        ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_AND('a', 'b')  # enforce a AND b
        >>> H
        {('b', 'a'): -1, (): 1}

        >>> H = PCBO()
        >>> H.add_constraint_AND('a', 'b', 'c', 'd')
        >>> # enforce a AND b AND c AND d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a AND b
        >>> H = PCBO().add_constraint_AND(a, b)

        """
        return self.add_constraint_BUFFER(AND(*variables), lam=lam)

    def add_constraint_OR(self, *variables, lam=1):
        r"""add_constraint_OR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`a \lor b \lor c \lor d \lor ...` is True, with a penalty factor
        ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_OR('a', 'b')  # enforce a OR b
        >>> H
        {('a',): -1, ('b',): -1, ('b', 'a'): 1, (): 1}

        >>> H = PCBO()
        >>> H.add_constraint_OR('a', 'b', 'c', 'd')  # enforce a OR b OR c OR d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a OR b
        >>> H = PCBO().add_constraint_OR(a, b)

        """
        P = 1 - OR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_XOR(self, *variables, lam=1):
        r"""add_constraint_XOR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`v_0 \oplus v_1 \oplus ... \oplus v_n` is True, with a penalty
        factor ``lam``, where ``v_0 = variables[0]``, ``v_1 = variables[1]``,
        ..., ``v_n = variables[-1]``. See ``qubovert.sat.XOR`` for the
        XOR convention that qubovert uses for more than two inputs.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_XOR('a', 'b')  # enforce a XOR b

        >>> H = PCBO()
        >>> H.add_constraint_XOR('a', 'b', 'c')  # enforce a XOR b XOR c

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a XOR b
        >>> H = PCBO().add_constraint_XOR(a, b)

        """
        P = 1 - XOR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_BUFFER(self, a, lam=1):
        r"""add_constraint_BUFFER.

        Add a penalty to the PCBO that is only nonzero when :math:`a == 1` is
        True, with a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variable ``a``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_BUFFER('a')  # enforce a

        >>> from qubovert import boolean_var, PCBO
        >>> a = boolean_var('a')
        >>> # enforce a
        >>> H = PCBO().add_constraint_BUFFER(a)

        """
        return self.add_constraint_eq_zero(NOT(a), lam=lam, bounds=(0, 1))

    def add_constraint_NAND(self, *variables, lam=1):
        r"""add_constraint_NAND.

        Add a penalty to the PCBO that is only zero when
        :math:`\lnot (a \land b \land c \land ...)` is True, with a penalty
        factor ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_NAND('a', 'b')  # enforce a NAND b

        >>> H = PCBO()
        >>> H.add_constraint_NAND('a', 'b', 'c', 'd')
        >>> # enforce a NAND b NAND c NAND d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a NAND b
        >>> H = PCBO().add_constraint_NAND(a, b)

        """
        return self.add_constraint_NOT(AND(*variables), lam=lam)

    def add_constraint_NOR(self, *variables, lam=1):
        r"""add_constraint_NOR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`\lnot(a \lor b \lor c \lor d \lor ...)` is True, with a penalty
        factor ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_NOR('a', 'b')  # enforce a NOR b

        >>> H = PCBO()
        >>> H.add_constraint_NOR('a', 'b', 'c', 'd')
        >>> # enforce a NOR b NOR c NOR d

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a NOR b
        >>> H = PCBO().add_constraint_NOR(a, b)

        """
        P = 1 - PCBO().add_constraint_OR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_XNOR(self, *variables, lam=1):
        r"""add_constraint_XNOR.

        Add a penalty to the PCBO that is only nonzero when
        :math:`\lnot(v_0 \oplus v_1 \oplus ... \oplus v_n)` is True, with a
        penalty  factor ``lam``, where ``v_0 = variables[0]``,
        ``v_1 = variables[1]``, ..., ``v_n = variables[-1]``. See
        ``qubovert.sat.XNOR`` for the XNOR convention that qubovert uses for
        more than two inputs.

        Parameters
        ----------
        *variables : arguments.
            Each element of variables is a hashable object or a dict
            (its PUBO representation). They are the label of the boolean
            variables.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_XNOR('a', 'b')  # enforce a XNOR b

        >>> H = PCBO()
        >>> H.add_constraint_XNOR('a', 'b', 'c')  # enforce a XNOR b XNOR c

        >>> from qubovert import boolean_var, PCBO
        >>> a, b = boolean_var('a'), boolean_var('b')
        >>> # enforce a XNOR b
        >>> H = PCBO().add_constraint_XNOR(a, b)

        """
        P = 1 - PCBO().add_constraint_XOR(*variables)
        return self.add_constraint_eq_zero(P, lam=lam, bounds=(0, 1))

    def add_constraint_NOT(self, a, lam=1):
        r"""add_constraint_NOT.

        Add a penalty to the PCBO that is only nonzero when
        :math:`\lnot a` is True, with a penalty factor ``lam``.

        Parameters
        ----------
        a : any hashable object or a dict.
            The label for boolean variables ``a``, or its PUBO representation.
        lam : float > 0 or sympy.Symbol (optional, defaults to 1).
            Langrange multiplier to penalize violations of the clause.

        Return
        ------
        self : PCBO.
            Updates the PCBO in place, but returns ``self`` so that operations
            can be strung together.

        Examples
        --------
        >>> H = PCBO()
        >>> H.add_constraint_NOT('a')  # enforce not a
        >>> H
        {('a',): 1}

        >>> from qubovert import boolean_var, PCBO
        >>> a = boolean_var('a')
        >>> # enforce not a
        >>> H = PCBO().add_constraint_NOT(a)

        """
        return self.add_constraint_eq_zero(BUFFER(a), lam=lam, bounds=(0, 1))

#   Copyright 2020 <NAME> # #   Licensed under the Apache License, Version 2.0 (the "License"); #   you may not use this file except in compliance with the License. #   You may obtain a copy of the License at # #       http://www.apache.org/licenses/LICENSE-2.0 # #   Unless required by applicable law or agreed to in writing, software #   distributed under the License is distributed on an "AS IS" BASIS, #   WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. #   See the License for the specific language governing permissions and #   limitations under the License. _pcbo.py.  Contains the PCBO class. See ``help(qubovert.PCBO)``. # special constraint forms _special_constraints_eq_zero.      See if the constraint that ``P == 0`` matches any special forms.      Parameters     ----------     pcbo : PCBO object.         The PCBO to add the constraint to.     P : PUBO object.         The PUBO constraint such that ``P == 0``.         are present.     lam : float > 0 or sympy.Symbol.         Langrange multiplier to penalize violations of the constraint.      Return     ------     success : bool.         True if a special constraint was found and added to ``pcbo``, else         False. # if P is of the form z == x * y. ie z == AND(x, y) # if P is of the form z == 1 - x * y. ie z == NAND(x, y) # if P is of the form z == x * y. ie z == OR(x, y) # if P is of the form z == x * y. ie z == NOR(x, y) _special_constraints_le_zero.      See if the constraint that ``P <= 0`` matches any special forms.      Parameters     ----------     pcbo : PCBO object.         The PCBO to add the constraint to.     P : PUBO object.         The PUBO constraint such that ``P <= 0``.         are present.     lam : float > 0 or sympy.Symbol.         Langrange multiplier to penalize violations of the constraint.     log_trick : bool.         Whether or not to use the log trick to enforce the inequality         constraint.     bounds : two element tuple.         A tuple ``(min, max)``, the minimum and maximum values that the         PUBO ``P`` can take.      Return     ------     success : bool.         True if a special constraint was found and added to ``pcbo``, else         False. # P without offset defined for convenience # if P is of the form sum(x_i) <= 1. # if P is of the form P_wo_offset <= -P.offset and P_wo_offset is always # >= 0, and (important!) log_trick is False!! # We have that P_wo_offset <= P.offset and min(P_wo_offset) = 0. So we # can do a penalty lam(P_wo_offset - sum(ancillas))**2 # create ancillas # if P is of the form 1 <= x + y, then it is the same as OR(x, y) # if P is of the form x <= y # helpers _get_bounds.      Compute the missing bounds of ``P``.      Parameters     ----------     P : PUBO object.     bounds : two element tuple or None.         A tuple ``(min, max)``, the minimum and maximum values that the         PUBO ``P`` can take. If ``bounds`` is None, then they will be         calculated (approximately), or if either of the elements of         ``bounds`` is None, then that element will be calculated         (approximately).      Return     ------     res : two element tuple.         The bounds of ``P``. boolean_var.      Create a PCBO (see ``qubovert.PCBO``) from a single boolean variable.      Parameters     ----------     name : any hashable object.         Name of the boolean variable.      Return     ------     pcbo : qubovert.PCBO object.         The model representing the boolean variable.      Examples     --------     >>> from qubovert import boolean_var, PCBO     >>>     >>> x0 = boolean_var("x0")     >>> print(x0)     {('x0',): 1}     >>> print(isinstance(x0, PCBO))     True     >>> print(x0.name)     x0      >>> x = [boolean_var('x{}'.format(i)) for i in range(5)]     >>> pcbo = sum(x)     >>> print(pcbo)     {('x0',): 1, ('x1',): 1, ('x2',): 1, ('x3',): 1, ('x4',): 1}     >>> pcbo **= 2     >>> print(pcbo)     {('x0',): 1, ('x0', 'x1'): 2, ('x2', 'x0'): 2, ('x3', 'x0'): 2,      ('x4', 'x0'): 2, ('x1',): 1, ('x2', 'x1'): 2, ('x3', 'x1'): 2,      ('x4', 'x1'): 2, ('x2',): 1, ('x2', 'x3'): 2, ('x2', 'x4'): 2, ('x3',): 1,      ('x4', 'x3'): 2, ('x4',): 1}     >>> pcbo *= -1     >>> print(pcbo.solve_bruteforce())     {'x0': 1, 'x1': 1, 'x2': 1, 'x3': 1, 'x4': 1}     >>> pcbo.add_constraint_eq_zero(x[0] + x[1])     >>> print(pcbo.solve_bruteforce())     {'x0': 0, 'x1': 0, 'x2': 1, 'x3': 1, 'x4': 1}      Notes     -----     ``qubovert.boolean_var(name)`` is equivalent to     ``qubovert.PCBO.create_var(name)``. integer_var.      Return a PCBO object representing an integer variable with `num_bits`     bits.      Parameters     ----------     prefix : str.         The prefix for the boolean variable names.     num_bits : int.         Number of bits to represent the integer variable with.     log_trick : bool (optional, defaults to True).         Whether or not to use a log encoding for the integer.      Return     ------     i : qubovert.PCBO object.      Example     -------     >>> from qubovert import integer_var     >>> var = integer_var('a', 4)     >>> print(var)     {('a0',): 1, ('a1',): 2, ('a2',): 4, ('a3',): 8}     >>> print(var.name)     a      >>> from qubovert import integer_var     >>> var = integer_var('a', 4, log_trick=False)     >>> print(var)     {('a0',): 1, ('a1',): 1, ('a2',): 1, ('a3',): 1} # main class PCBO.      This class deals with Polynomial Constrained Boolean Optimization problems.     PCBO inherits some methods and attributes from the ``PUBO`` class. See     ``help(qubovert.PUBO)``.      ``PCBO`` has all the same methods as ``PUBO``, but adds some constraint     methods; namely      - ``add_constraint_eq_zero(P, lam=1, ...)`` enforces that ``P == 0`` by       penalizing with ``lam``,     - ``add_constraint_ne_zero(P, lam=1, ...)`` enforces that ``P != 0`` by       penalizing with ``lam``,     - ``add_constraint_lt_zero(P, lam=1, ...)`` enforces that ``P < 0`` by       penalizing with ``lam``,     - ``add_constraint_le_zero(P, lam=1, ...)`` enforces that ``P <= 0`` by       penalizing with ``lam``,     - ``add_constraint_gt_zero(P, lam=1, ...)`` enforces that ``P > 0`` by       penalizing with ``lam``, and     - ``add_constraint_ge_zero(P, lam=1, ...)`` enforces that ``P >= 0`` by       penalizing with ``lam``.      Each of these takes in a PUBO ``P`` and a lagrange multiplier ``lam`` that     defaults to 1. See each of their docstrings for important details on their     implementation.      We then implement logical operations:      - ``add_constraint_AND``, ``add_constraint_NAND``,     - ``add_constraint_eq_AND``, ``add_constraint_eq_NAND``,     - ``add_constraint_OR``, ``add_constraint_NOR```,     - ``add_constraint_eq_OR``, ``add_constraint_eq_NOR``,     - ``add_constraint_XOR``, ``add_constraint_XNOR``,     - ``add_constraint_eq_XOR``, ``add_constraint_eq_XNOR``,     - ``add_constraint_BUFFER``, ``add_constraint_NOT``,     - ``add_constraint_eq_BUFFER``, ``add_constraint_eq_NOT``.      See each of their docstrings for important details on their implementation.      Notes     -----     - Variables names that begin with ``"__a"`` should not be used since they       are used internally to deal with some ancilla variables to enforce       constraints.     - The ``self.solve_bruteforce`` method will solve the PCBO ensuring that       all the inputted constraints are satisfied. Whereas       ``qubovert.utils.solve_pubo_bruteforce(self)`` or       ``qubovert.utils.solve_pubo_bruteforce(self.to_pubo())`` will solve the       PUBO created from the PCBO. If the inputted constraints are not enforced       strong enough (ie too small lagrange multipliers) then these may not give       the correct result, whereas ``self.solve_bruteforce()`` will always give       the correct result (ie one that satisfies all the constraints).      Examples     --------     See ``qubovert.PUBO`` for more examples of using PCBO without constraints.      >>> H = PCBO()     >>> H.add_constraint_eq_zero({('a', 1): 2, (1, 2): -1, (): -1})     >>> H     {('a', 1, 2): -4, (1, 2): 3, (): 1}     >>> H -= 1     >>> H     {('a', 1, 2): -4, (1, 2): 3}      >>> H = PCBO()     >>>     >>> # minimize -x_0 - x_1 - x_2     >>> for i in (0, 1, 2):     >>>     H[(i,)] -= 1     >>>     >>> # subject to constraints     >>> H.add_constraint_eq_zero(  # enforce that x_0 x_1 - x_2 == 0     >>>     {(0, 1): 1, (2,): -1}     >>> ).add_constraint_lt_zero(  # enforce that x_1 x_2 + x_0 < 1     >>>     {(1, 2): 1, (0,): 1, (): -1}     >>> )     >>> print(H)     >>> # {(1,): -2, (2,): -1, (0, 1): 2, (1, 2): 2, (0, 1, 2): 2}     >>>     >>> print(H.solve_bruteforce(all_solutions=True))     >>> # [{0: 0, 1: 1, 2: 0}]     >>>     >>> Q = H.to_qubo()     >>> solutions = [H.convert_solution(sol)     >>>              for sol in Q.solve_bruteforce(all_solutions=True)]     >>> print(solutions)     >>> # [{0: 0, 1: 1, 2: 0}]  # matches the PCBO solution!     >>>     >>> L = H.to_quso()     >>> solutions = [H.convert_solution(sol)     >>>              for sol in L.solve_bruteforce(all_solutions=True)]     >>> print(solutions)     >>> # [{0: 0, 1: 1, 2: 0}]  # matches the PCBO solution!      Enforce that c == a b      >>> H = PCBO().add_constraint_eq_AND('c', 'a', 'b')     >>> H     {('c',): 3, ('b', 'a'): 1, ('c', 'a'): -2, ('c', 'b'): -2}      >>> from any_module import qubo_solver     >>> # or from qubovert.utils import solve_qubo_bruteforce as qubo_solver     >>> H = PCBO()     >>>     >>> # make it favorable to AND variables a and b, and variables b and c     >>> H.add_constraint_AND('a', 'b').add_constraint_AND('b', 'c')     >>>     >>> # make it favorable to OR variables b and c     >>> H.add_constraint_OR('b', 'c')     >>>     >>> # make it favorable to (a AND b) OR (c AND d)     >>> H.add_constraint_OR(['a', 'b'], ['c', 'd'])     >>>     >>> H     {('b', 'a'): -2, (): 4, ('b',): -1, ('c',): -1,      ('c', 'd'): -1, ('c', 'd', 'b', 'a'): 1}     >>> Q = H.to_qubo()     >>> Q     {(): 4, (0,): -1, (2,): -1, (2, 3): 1, (4,): 6, (0, 4): -4,      (1, 4): -4, (5,): 6, (2, 5): -4, (3, 5): -4, (4, 5): 1}     >>> obj_value, sol = qubo_solver(Q)     >>> sol     {0: 1, 1: 1, 2: 1, 3: 0, 4: 1, 5: 0}     >>> solution = H.convert_solution(sol)     >>> solution     {'b': 1, 'a': 1, 'c': 1, 'd': 0} __init__.          This class deals with polynomial constrained boolean optimization.         Note that it is generally more efficient to initialize an empty PCBO         object and then build the PCBO, rather than initialize a PCBO object         with an already built dict.          Parameters         ----------         arguments : define a dictionary with ``dict(*args, **kwargs)``.             The dictionary will be initialized to follow all the convensions of             the class. Alternatively, ``args[0]`` can be a PCBO object.          Examples         --------         >>> pcbo = PCBO()         >>> pcbo[('a',)] += 5         >>> pcbo[(0, 'a')] -= 2         >>> pcbo -= 1.5         >>> pcbo         {('a',): 5, ('a', 0): -2, (): -1.5}         >>> pcbo.add_constraint_eq_zero({('a',): 1}, lam=5)         >>> pcbo         {('a',): 10, ('a', 0): -2, (): -1.5}          >>> pcbo = PCBO({('a',): 5, (0, 'a', 1): -2, (): -1.5})         >>> pcbo         {('a',): 5, ('a', 0, 1): -2, (): -1.5} # use self.__class__ here because PCSO uses this code as well. update.          Update the PCBO but following all the conventions of this class.          Parameters         ----------         arguments : defines a dictionary or PCBO.             Ie ``d = dict(*args, **kwargs)``.             Each element in d will be added in place to this instance following             all the required convensions. # use self.__class__ here because PCSO uses this code as well. constraints.          Return the constraints of the PCBO.          Return         ------         res : dict.             The keys of ``res`` are some or all of             ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, and ``'ge'``.             The values are lists of ``qubovert.PUBO`` objects. For a             given key, value pair ``k, v``, the ``v[i]`` element represents             the PUBO ``v[i]`` being == 0 if ``k == 'eq'``,             != 0 if ``k == 'ne'``,             < 0 if ``k == 'lt'``, <= 0 if ``k == 'le'``,             > 0 if ``k == 'gt'``, >= 0 if ``k == 'ge'``. _append_constraint.          Internal method to add a constraint to the constraints dictionary.          Parameters         ----------         key : str.             One of ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, or             ``'ge'``.         constraint : qubovert.PUBO object. _append_constraint.          Internal method to remove the most recently added constraint.          Parameters         ----------         key : str.             One of ``'eq'``, ``'ne'``, ``'lt'``, ``'le'``, ``'gt'``, or             ``'ge'``. num_ancillas.          Return the number of ancilla variables introduced to the PCBO in         order to enforce the inputted constraints.          Returns         -------         num : int.             Number of ancillas in the PCBO. _next_ancilla.          Get the next available ancilla bit and increment.          Return         ------         a : int.             Ancilla bit label. remove_ancilla_from_solution.          Take a solution to the PCBO and remove all the ancilla variables, (         represented by `_a` prefixes).          Parameters         ----------         solution : dict.             Must be the solution in terms of the original variables. Thus if             ``solution`` is the solution to the ``self.to_pubo``,             ``self.to_qubo``, ``self.to_puso``, or ``self.to_quso``             formulations, then you should first call ``self.convert_solution``.             See ``help(self.convert_solution)``.          Return         ------         res : dict.             The same as ``solution`` but with all the ancilla bits removed. is_solution_valid.          Finds whether or not the given solution satisfies the constraints.          Parameters         ----------         solution : dict.             Must be the solution in terms of the original variables. Thus if             ``solution`` is the solution to the ``self.to_pubo``,             ``self.to_qubo``, ``self.to_puso``, or ``self.to_quso``             formulations, then you should first call ``self.convert_solution``.             See ``help(self.convert_solution)``.          Return         ------         valid : bool.             Whether or not the given solution satisfies the constraints. # override round.          Round values of the PCBO object.          Parameters         ----------         ndigits : int.             Number of decimal digits to round to.          Returns         -------         res : PCBO object.             Copy of self but with each value rounded to ``ndigits`` decimal             digits. Each value has a type according to the docstring             specifications of ``round``, see ``help(round)``. # use self.__class__ here because PCSO uses this code as well. # override subs.          Replace any ``sympy`` symbols that are used in the dict with values.         Please see ``help(sympy.Symbol.subs)`` for more info.          Parameters         ----------         arguments : substitutions.             Same parameters as are inputted into ``sympy.Symbol.subs``.          Returns         -------         res : PCBO object.             Same as ``self`` but with all the symbols replaced with values. # use self.__class__ here because PCSO uses this code as well. add_constraint_eq_zero.          Enforce that ``P == 0`` by penalizing invalid solutions with ``lam``.          Parameters         ----------         P : dict representing a PUBO.             The PUBO constraint such that P == 0. Note that ``P`` will be             converted to a ``qubovert.PUBO`` object if it is not already, thus             it must follow the conventions, see ``help(qubovert.PUBO)``.             Please note that if ``P`` contains any symbols, then ``bounds``             must be supplied, since they cannot be determined when symbols             are present.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the constraint.         bounds : two element tuple (optional, defaults to None).             A tuple ``(min, max)``, the minimum and maximum values that the             PUBO ``P`` can take. If ``bounds`` is None, then they may be             calculated (approximately).         suppress_warnings : bool (optional, defaults to False).             Whether or not to surpress warnings.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         The following enforces that :math:`\prod_{i=0}^{3} x_i == 0`.          >>> H = PCBO()         >>> H.add_constraint_eq_zero({(0, 1, 2, 3): 1})         >>> H         {(0, 1, 2, 3): 1}          The following enforces that :math:`\sum_{i=1}^{3} i x_i x_{i+1} == 0`.          >>> H = PCBO()         >>> H.add_constraint_eq_zero({(1, 2): 1, (2, 3): 2, (3, 4): 3})         >>> H         {(1, 2): 1, (1, 2, 3): 4, (1, 2, 3, 4): 6,          (2, 3): 4, (2, 3, 4): 12, (3, 4): 9}          Here we show how operations can be strung together.          >>> H = PCBO()         >>> H.add_constraint_eq_zero(                 {(0, 1): 1}             ).add_constraint_eq_zero(                 {(1, 2): 1, (): -1}             )         >>> H         {(0, 1): 1, (1, 2): -1, (): 1} add_constraint_ne_zero.          Enforce that ``P != 0`` by penalizing invalid solutions with ``lam``.         See Notes below for more details.          Parameters         ----------         P : dict representing a PUBO.             The PUBO constraint such that P != 0. Note that ``P`` will be             converted to a ``qubovert.PUBO`` object if it is not already, thus             it must follow the conventions, see ``help(qubovert.PUBO)``.             Please note that if ``P`` contains any symbols, then ``bounds``             must be supplied, since they cannot be determined when symbols             are present.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the constraint.         log_trick : bool (optional, defaults to True).             Whether or not to use the log trick to enforce the inequality             constraint. See Notes below for more details.         bounds : two element tuple (optional, defaults to None).             A tuple ``(min, max)``, the minimum and maximum values that the             PUBO ``P`` can take. If ``bounds`` is None, then they will be             calculated (approximately), or if either of the elements of             ``bounds`` is None, then that element will be calculated             (approximately).         suppress_warnings : bool (optional, defaults to False).             Whether or not to surpress warnings.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Notes         -----         - There is no general way to enforce non integer inequality           constraints. Thus this function is only guarenteed to work for           integer inequality constraints (ie constraints of the form           :math:`x_0 + 2x_1 + ... != 0`). However, it can be used for non           integer inequality constraints, but it is recommended that the value           of ``lam`` be set small, since valid solutions may still recieve a           penalty to the objective function.         - To enforce the inequality constraint, ancilla bits will be           introduced (labels with `_a`). If ``log_trick`` is ``True``, then           approximately :math:`\log_2 |\min_x \text{P.value(x)}|`           ancilla bits will be used. If ``log_trick`` is ``False``, then           approximately :math:`|\min_x \text{P.value(x)}|` ancilla           bits will be used.          Examples         --------         Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c != 2`.          >>> H = PCBO().add_constraint_ne_zero(                 {('a', 'b', 'c'): -1, ('a',): 2,                 ('a', 'b'): -4, ('c',): 3, (): -2}             )         >>> print(H)         {('c', 'a', 'b'): -19, ('__a0', 'c', 'a', 'b'): -4,          ('__a0', 'c', '__a1', 'a', 'b'): -4, ('c', '__a1', 'a', 'b'): 2,          ('__a0', 'c', '__a2', 'a', 'b'): -8, ('c', '__a2', 'a', 'b'): 4,          ('__a0', '__a3', 'c', 'a', 'b'): -16, ('__a3', 'c', 'a', 'b'): 8,          ('__a0', 'c', '__a4', 'a', 'b'): -32, ('c', '__a4', 'a', 'b'): 16,          ('a',): -8, ('c', 'a'): 12, ('__a0', 'a'): 8,          ('__a0', '__a1', 'a'): 8, ('__a1', 'a'): -4,          ('__a0', '__a2', 'a'): 16, ('__a2', 'a'): -8,          ('__a0', '__a3', 'a'): 32, ('__a3', 'a'): -16,          ('__a0', '__a4', 'a'): 64, ('__a4', 'a'): -32, ('a', 'b'): 24,          ('__a0', 'a', 'b'): -16, ('__a0', '__a1', 'a', 'b'): -16,          ('__a1', 'a', 'b'): 8, ('__a0', '__a2', 'a', 'b'): -32,          ('__a2', 'a', 'b'): 16, ('__a0', '__a3', 'a', 'b'): -64,          ('__a3', 'a', 'b'): 32, ('__a0', '__a4', 'a', 'b'): -128,          ('__a4', 'a', 'b'): 64, ('__a0', 'c'): 12, ('__a0', 'c', '__a1'): 12,          ('c', '__a1'): -6, ('__a0', 'c', '__a2'): 24, ('c', '__a2'): -12,          ('__a0', '__a3', 'c'): 48, ('__a3', 'c'): -24,          ('__a0', 'c', '__a4'): 96, ('c', '__a4'): -48, ('c',): -9, (): 9,          ('__a0',): -8, ('__a0', '__a1'): -8, ('__a1',): 7,          ('__a0', '__a2'): -16, ('__a2',): 16, ('__a3',): 40,          ('__a0', '__a4'): -64, ('__a4',): 112, ('__a2', '__a1'): 4,          ('__a3', '__a1'): 8, ('__a4', '__a1'): 16, ('__a3', '__a2'): 16,          ('__a4', '__a2'): 32, ('__a0', '__a3'): -32, ('__a3', '__a4'): 64}         >>> print(H.is_solution_valid({'b': 0, 'c': 0, 'a': 1}))         False         >>> print(H.is_solution_valid({'b': 0, 'c': 1, 'a': 1}))         True # don't mutate the P that we put in self._constraints add_constraint_lt_zero.          Enforce that ``P < 0`` by penalizing invalid solutions with ``lam``.         See Notes below for more details.          Parameters         ----------         P : dict representing a PUBO.             The PUBO constraint such that P < 0. Note that ``P`` will be             converted to a ``qubovert.PUBO`` object if it is not already, thus             it must follow the conventions, see ``help(qubovert.PUBO)``.             Please note that if ``P`` contains any symbols, then ``bounds``             must be supplied, since they cannot be determined when symbols             are present.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the constraint.         log_trick : bool (optional, defaults to True).             Whether or not to use the log trick to enforce the inequality             constraint. See Notes below for more details.         bounds : two element tuple (optional, defaults to None).             A tuple ``(min, max)``, the minimum and maximum values that the             PUBO ``P`` can take. If ``bounds`` is None, then they will be             calculated (approximately), or if either of the elements of             ``bounds`` is None, then that element will be calculated             (approximately).         suppress_warnings : bool (optional, defaults to False).             Whether or not to surpress warnings.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Notes         -----         - There is no general way to enforce non integer inequality           constraints. Thus this function is only guarenteed to work for           integer inequality constraints (ie constraints of the form           :math:`x_0 + 2x_1 + ... < 0`). However, it can be used for non           integer inequality constraints, but it is recommended that the value           of ``lam`` be set small, since valid solutions may still recieve a           penalty to the objective function. For example,            >>> H = PCBO()           >>> H.add_constraint_lt_zero({(0,): 1, (1,): 2, (2,): -.5, (): .4})           >>> test_sol = {0: 0, 1: 0, 2: 1}           >>> H.is_solution_valid(test_sol)           True           >>> H.value(test_sol)           0.01            {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still           cause a nonzero penalty to be added to the objective function.          - To enforce the inequality constraint, ancilla bits will be           introduced (labels with `_a`). If ``log_trick`` is ``True``, then           approximately :math:`\log_2 |\min_x \text{P.value(x)}|`           ancilla bits will be used. If ``log_trick`` is ``False``, then           approximately :math:`|\min_x \text{P.value(x)}|` ancilla           bits will be used.          Examples         --------         Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c < 2`.          >>> H = PCBO().add_constraint_lt_zero(                 {('a', 'b', 'c'): -1, ('a',): 1,                  ('a', 'b'): -4, ('c',): 3, (): -2}             )         >>> H         {('b', 'c', 'a'): -19, ('b', '__a0', 'c', 'a'): -2,          ('b', 'c', 'a', '__a1'): -4, ('a',): -3, ('b', 'a'): 24,          ('c', 'a'): 6, ('__a0', 'a'): 2, ('a', '__a1'): 4,          ('b', '__a0', 'a'): -8, ('b', 'a', '__a1'): -16, ('c',): -3,          ('__a0', 'c'): 6, ('c', '__a1'): 12, (): 4, ('__a0',): -3,          ('__a1',): -4, ('__a0', '__a1'): 4}         >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})         True         >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})         False # copy P, don't do += add_constraint_le_zero.          Enforce that ``P <= 0`` by penalizing invalid solutions with ``lam``.          Parameters         ----------         P : dict representing a PUBO.             The PUBO constraint such that P <= 0. Note that ``P`` will be             converted to a ``qubovert.PUBO`` object if it is not already, thus             it must follow the conventions, see ``help(qubovert.PUBO)``.             Please note that if ``P`` contains any symbols, then ``bounds``             must be supplied, since they cannot be determined when symbols             are present.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the constraint.         log_trick : bool (optional, defaults to True).             Whether or not to use the log trick to enforce the inequality             constraint. See Notes below for more details.         bounds : two element tuple (optional, defaults to None).             A tuple ``(min, max)``, the minimum and maximum values that the             PUBO ``P`` can take. If ``bounds`` is None, then they will be             calculated (approximately), or if either of the elements of             ``bounds`` is None, then that element will be calculated             (approximately).         suppress_warnings : bool (optional, defaults to False).             Whether or not to surpress warnings.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Notes         -----         - There is no general way to enforce non integer inequality           constraints. Thus this function is only guarenteed to work for           integer inequality constraints (ie constraints of the form           :math:`x_0 + 2x_1 + ... \leq 0`). However, it can be used for non           integer inequality constraints, but it is recommended that the value           of ``lam`` be set small, since valid solutions may still recieve a           penalty to the objective function. For example,            >>> H = PCBO()           >>> H.add_constraint_le_zero({(0,): 1, (1,): 2, (2,): -1.5, (): .4})           >>> H           {(0,): 1.7999999999999998, (0, 1): 4, (0, 2): -3.0, (0, '__a0'): 2,            (1,): 5.6, (1, 2): -6.0, (1, '__a0'): 4, (2,): 1.0499999999999998,            (2, '__a0'): -3.0, (): 0.16000000000000003, ('__a0',): 1.8}           >>> test_sol = {0: 0, 1: 0, 2: 1, '__a0': 1}           >>> H.is_solution_valid(test_sol)           True           >>> H.value(test_sol)           0.01            {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still           cause a nonzero penalty to be added to the objective function.          - To enforce the inequality constraint, ancilla bits will be           introduced (labels with `_a`). If ``log_trick`` is ``True``, then           approximately :math:`\log_2 |\min_x \text{P.value(x)}|`           ancilla bits will be used. If ``log_trick`` is ``False``, then           approximately :math:`|\min_x \text{P.value(x)}|` ancilla           bits will be used.          Examples         --------         Enforce that :math:`-x_a x_b x_c + x_a -4x_a x_b + 3x_c \leq 2`.          >>> H = PCBO().add_constraint_le_zero(                 {('a', 'b', 'c'): -1, ('a',): 1,                  ('a', 'b'): -4, ('c',): 3, (): -2}             )         >>> H         {('b', 'c', 'a'): -19, ('b', 'c', 'a', '__a0'): -2,          ('__a1', 'b', 'c', 'a'): -4, ('__a2', 'b', 'c', 'a'): -8, ('a',): -3,          ('b', 'a'): 24, ('c', 'a'): 6, ('a', '__a0'): 2, ('__a1', 'a'): 4,          ('__a2', 'a'): 8, ('b', 'a', '__a0'): -8, ('__a1', 'b', 'a'): -16,          ('__a2', 'b', 'a'): -32, ('c',): -3, ('c', '__a0'): 6,          ('__a1', 'c'): 12, ('__a2', 'c'): 24, (): 4, ('__a0',): -3,          ('__a1',): -4, ('__a1', '__a0'): 4, ('__a2', '__a0'): 8,          ('__a2', '__a1'): 16}         >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})         True         >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})         False # don't mutate the P that we put in self._constraints add_constraint_gt_zero.          Enforce that ``P > 0`` by penalizing invalid solutions with ``lam``.          Parameters         ----------         P : dict representing a PUBO.             The PUBO constraint such that P > 0. Note that ``P`` will be             converted to a ``qubovert.PUBO`` object if it is not already, thus             it must follow the conventions, see ``help(qubovert.PUBO)``.             Please note that if ``P`` contains any symbols, then ``bounds``             must be supplied, since they cannot be determined when symbols             are present.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the constraint.         log_trick : bool (optional, defaults to True).             Whether or not to use the log trick to enforce the inequality             constraint. See Notes below for more details.         bounds : two element tuple (optional, defaults to None).             A tuple ``(min, max)``, the minimum and maximum values that the             PUBO ``P`` can take. If ``bounds`` is None, then they will be             calculated (approximately), or if either of the elements of             ``bounds`` is None, then that element will be calculated             (approximately).         suppress_warnings : bool (optional, defaults to False).             Whether or not to surpress warnings.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Notes         -----         - There is no general way to enforce non integer inequality           constraints. Thus this function is only guarenteed to work for           integer inequality constraints (ie constraints of the form           :math:`x_0 + 2x_1 + ... > 0`). However, it can be used for non           integer inequality constraints, but it is recommended that the value           of ``lam`` be set small, since valid solutions may still recieve a           penalty to the objective function. For example,            >>> H = PCBO()           >>> H.add_constraint_gt_zero({(0,): -1, (1,): -2, (2,): .5, (): -.4})           >>> test_sol = {0: 0, 1: 0, 2: 1}           >>> H.is_solution_valid(test_sol)           True           >>> H.value(test_sol)           0.01            {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still           cause a nonzero penalty to be added to the objective function.          - To enforce the inequality constraint, ancilla bits will be           introduced (labels with `_a`). If ``log_trick`` is ``True``, then           approximately :math:`\log_2 |\max_x \text{P.value(x)}|`           ancilla bits will be used. If ``log_trick`` is ``False``, then           approximately :math:`|\max_x \text{P.value(x)}|` ancilla           bits will be used.          Examples         --------         Enforce that :math:`x_a x_b x_c - x_a + 4x_a x_b - 3x_c > -2`.          >>> H = PCBO().add_constraint_gt_zero(                 {('a', 'b', 'c'): 1, ('a',): -1,                  ('a', 'b'): 4, ('c',): -3, (): 2}             )         >>> H         {('b', 'c', 'a'): -19, ('b', '__a0', 'c', 'a'): -2,          ('b', 'c', 'a', '__a1'): -4, ('a',): -3, ('b', 'a'): 24,          ('c', 'a'): 6, ('__a0', 'a'): 2, ('a', '__a1'): 4,          ('b', '__a0', 'a'): -8, ('b', 'a', '__a1'): -16, ('c',): -3,          ('__a0', 'c'): 6, ('c', '__a1'): 12, (): 4, ('__a0',): -3,          ('__a1',): -4, ('__a0', '__a1'): 4}         >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})         True         >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})         False add_constraint_ge_zero.          Enforce that ``P >= 0`` by penalizing invalid solutions with ``lam``.          Parameters         ----------         P : dict representing a PUBO.             The PUBO constraint such that P >= 0. Note that ``P`` will be             converted to a ``qubovert.PUBO`` object if it is not already, thus             it must follow the conventions, see ``help(qubovert.PUBO)``.             Please note that if ``P`` contains any symbols, then ``bounds``             must be supplied, since they cannot be determined when symbols             are present.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the constraint.         log_trick : bool (optional, defaults to True).             Whether or not to use the log trick to enforce the inequality             constraint. See Notes below for more details.         bounds : two element tuple (optional, defaults to None).             A tuple ``(min, max)``, the minimum and maximum values that the             PUBO ``P`` can take. If ``bounds`` is None, then they will be             calculated (approximately), or if either of the elements of             ``bounds`` is None, then that element will be calculated             (approximately), or if either of the elements of             ``bounds`` is None, then that element will be calculated             (approximately).         suppress_warnings : bool (optional, defaults to False).             Whether or not to surpress warnings.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Notes         -----         - There is no general way to enforce non integer inequality           constraints. Thus this function is only guarenteed to work for           integer inequality constraints (ie constraints of the form           :math:`x_0 + 2x_1 + ... \geq 0`). However, it can be used for non           integer inequality constraints, but it is recommended that the value           of ``lam`` be set small, since valid solutions may still recieve a           penalty to the objective function. For example,            >>> H = PCBO()           >>> H.add_constraint_ge_zero({(0,): -1, (1,): -2, (2,):1.5, (): -.4})           >>> H           {(0,): 1.7999999999999998, (0, 1): 4, (0, 2): -3.0, (0, '__a0'): 2,            (1,): 5.6, (1, 2): -6.0, (1, '__a0'): 4, (2,): 1.0499999999999998,            (2, '__a0'): -3.0, (): 0.16000000000000003, ('__a0',): 1.8}           >>> test_sol = {0: 0, 1: 0, 2: 1, '__a0': 1}           >>> H.is_solution_valid(test_sol)           True           >>> H.value(test_sol)           0.01            {0: 0, 1: 0, 2: 1} is a valid solution to ``H``, but it will still           cause a nonzero penalty to be added to the objective function.          - To enforce the inequality constraint, ancilla bits will be           introduced (labels with `_a`). If ``log_trick`` is ``True``, then           approximately :math:`\log_2 |\max_x \text{P.value(x)}|`           ancilla bits will be used. If ``log_trick`` is ``False``, then           approximately :math:`|\max_x \text{P.value(x)}|` ancilla           bits will be used.          Examples         --------         Enforce that :math:`x_a x_b x_c - x_a + 4x_a x_b - 3x_c \geq -2`.          >>> H = PCBO().add_constraint_ge_zero(                 {('a', 'b', 'c'): 1, ('a',): -1,                  ('a', 'b'): 4, ('c',): -3, (): 2}             )         >>> H         {('b', 'c', 'a'): -19, ('b', 'c', 'a', '__a0'): -2,          ('__a1', 'b', 'c', 'a'): -4, ('__a2', 'b', 'c', 'a'): -8, ('a',): -3,          ('b', 'a'): 24, ('c', 'a'): 6, ('a', '__a0'): 2, ('__a1', 'a'): 4,          ('__a2', 'a'): 8, ('b', 'a', '__a0'): -8, ('__a1', 'b', 'a'): -16,          ('__a2', 'b', 'a'): -32, ('c',): -3, ('c', '__a0'): 6,          ('__a1', 'c'): 12, ('__a2', 'c'): 24, (): 4, ('__a0',): -3,          ('__a1',): -4, ('__a1', '__a0'): 4, ('__a2', '__a0'): 8,          ('__a2', '__a1'): 16}         >>> H.is_solution_valid({'b': 0, 'c': 0, 'a': 1})         True         >>> H.is_solution_valid({'b': 0, 'c': 1, 'a': 1})         False add_constraint_eq_AND.          Add a penalty to the PCBO that enforces that         :math:`a = v_0 \land v_1 \land v_2 \land ...`,         with a penalty factor         ``lam``, where ``v_1 = variables[0]``, ``v_2 = variables[1]``, ...          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variable ``a``, or its PUBO representation.         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_AND('a', 'b', 'c')  # enforce a == b AND c          >>> H = PCBO()         >>> # enforce (a AND b AND c AND d) == 'e'         >>> H.add_constraint_eq_AND('e', 'a', 'b', 'c', 'd')          >>> from qubovert import boolean_var, PCBO         >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')         >>> H = PCBO()         >>> # enforce that a == b AND c         >>> H.add_constraint_eq_AND(a, b, c)          References         ----------         https://arxiv.org/pdf/1307.8041.pdf equation 6. add_constraint_eq_OR.          Add a penalty to the PCBO that enforces that         :math:`v_0 \lor v_1 \lor v_2 \lor ... == a`,         with a penalty factor ``lam``, where ``v_0 = variables[0]``,         ``v_1 = variables[1]``, ...          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variable ``a``, or its PUBO representation.         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_OR('a', 'b', 'c')  # enforce a == b OR c          >>> H = PCBO()         >>> # enforce a == b OR c OR d         >>> H.add_constraint_eq_OR('a', 'b', 'c', 'd') add_constraint_eq_XOR.          Add a penalty to the PCBO that enforces that         :math:`v_0 \oplus v_1 \oplus ... == a`         with a penalty factor ``lam``, where ``v_0 = variables[0]``,         ``v_1 = variables[1]``, ...          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variable ``a``, or its PUBO representation.         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_XOR('a', 'b', 'c')  # enforce a == b XOR c          >>> H = PCBO()         >>> # enforce a == b XOR c XOR d         >>> H.add_constraint_eq_XOR('a', 'b', 'c', 'd') add_constraint_eq_BUFFER.          Add a penalty to the PCBO that enforces that :math:`a == b` with         a penalty factor ``lam``.          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variables ``a``, or its PUBO representation.         b : any hashable object or a dict.             The label for boolean variables ``b``, or its PUBO representation.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_BUFFER('a', 'b')  # enforce a == b          >>> from qubovert import PCBO, boolean_var         >>> a, b = boolean_var('a'), boolean_var('b')         >>> H = PCBO()         >>> H.add_constraint_eq_BUFFER(a, b)  # enforce a == b add_constraint_eq_NAND.          Add a penalty to the PCBO that enforces that         :math:`\lnot (v_0 \land v_1 \land v_2 \land ...) == a`, with a         penalty factor ``lam``, where ``v_0 = variables[0]``,         ``v_1 = variables[1]``, ...          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variables ``a``, or its PUBO representation.         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_NAND('a', 'b', 'c')  # enforce a == b NAND c          >>> H = PCBO()         >>> # enforce a == b NAND c NAND d         >>> H.add_constraint_eq_NAND('a', 'b', 'c', 'd')          >>> from qubovert import boolean_var, PCBO         >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')         >>> # enforce a == b NAND c         >>> H = PCBO().add_constraint_eq_NAND(a, b, c) add_constraint_eq_NOR.          Add a penalty to the PCBO that enforces that         :math:`\lnot(v_0 \lor v_1 \lor v_2 \lor ...) == a`,         with a penalty factor ``lam``, where ``v_0 = variables[0]``,         ``v_1 = variables[1]``, ...          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variable ``a``, or its PUBO representation.         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_NOR('a', 'b', 'c')  # enforce a == b NOR c          >>> H = PCBO()         >>> # enforce a == b NOR c NOR d         >>> H.add_constraint_eq_NOR('a', 'b', 'c', 'd')          >>> from qubovert import boolean_var, PCBO         >>> # enforce a == b NOR c         >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')         >>> H.add_constraint_eq_NOR(a, b, c) add_constraint_eq_XNOR.          Add a penalty to the PCBO that enforces that         :math:\lnot(`v_0 \oplus v_1 \oplus ...) == a`         with a penalty factor ``lam``, where ``v_0 = variables[0]``,         ``v_1 = variables[1]``, ...          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variable ``a``, or its PUBO representation.         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_XNOR('a', 'b', 'c')  # enforce a == b XNOR c          >>> H = PCBO()         >>> # enforce a == b XNOR c XNOR d         >>> H.add_constraint_eq_XNOR('a', 'b', 'c', 'd')          >>> from qubovert import boolean_var, PCBO         >>> a, b, c = boolean_var('a'), boolean_var('b'), boolean_var('c')         >>> # enforce a == b XNOR c         >>> H = PCBO().add_constraint_eq_XNOR(a, b, c) NOT.          Add a penalty to the PCBO that enforces that :math:`\lnot a == b` with         a penalty factor ``lam``.          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variable ``a``, or its PUBO representation.         b : any hashable object or a dict.             The label for boolean variable ``b``, or its PUBO representation.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_eq_NOT('a', 'b')  # enforce NOT(a) == b          >>> from qubovert import boolean_var, PCBO         >>> a, b = boolean_var('a'), boolean_var('b')         >>> H.add_constraint_eq_NOT(a, b)  # enforce NOT(a) == b add_constraint_AND.          Add a penalty to the PCBO that is only zero when         :math:`a \land b \land c \land ...` is True, with a penalty factor         ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.          Parameters         ----------         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_AND('a', 'b')  # enforce a AND b         >>> H         {('b', 'a'): -1, (): 1}          >>> H = PCBO()         >>> H.add_constraint_AND('a', 'b', 'c', 'd')         >>> # enforce a AND b AND c AND d          >>> from qubovert import boolean_var, PCBO         >>> a, b = boolean_var('a'), boolean_var('b')         >>> # enforce a AND b         >>> H = PCBO().add_constraint_AND(a, b) add_constraint_OR.          Add a penalty to the PCBO that is only nonzero when         :math:`a \lor b \lor c \lor d \lor ...` is True, with a penalty factor         ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.          Parameters         ----------         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_OR('a', 'b')  # enforce a OR b         >>> H         {('a',): -1, ('b',): -1, ('b', 'a'): 1, (): 1}          >>> H = PCBO()         >>> H.add_constraint_OR('a', 'b', 'c', 'd')  # enforce a OR b OR c OR d          >>> from qubovert import boolean_var, PCBO         >>> a, b = boolean_var('a'), boolean_var('b')         >>> # enforce a OR b         >>> H = PCBO().add_constraint_OR(a, b) add_constraint_XOR.          Add a penalty to the PCBO that is only nonzero when         :math:`v_0 \oplus v_1 \oplus ... \oplus v_n` is True, with a penalty         factor ``lam``, where ``v_0 = variables[0]``, ``v_1 = variables[1]``,         ..., ``v_n = variables[-1]``. See ``qubovert.sat.XOR`` for the         XOR convention that qubovert uses for more than two inputs.          Parameters         ----------         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_XOR('a', 'b')  # enforce a XOR b          >>> H = PCBO()         >>> H.add_constraint_XOR('a', 'b', 'c')  # enforce a XOR b XOR c          >>> from qubovert import boolean_var, PCBO         >>> a, b = boolean_var('a'), boolean_var('b')         >>> # enforce a XOR b         >>> H = PCBO().add_constraint_XOR(a, b) add_constraint_BUFFER.          Add a penalty to the PCBO that is only nonzero when :math:`a == 1` is         True, with a penalty factor ``lam``.          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variable ``a``, or its PUBO representation.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_BUFFER('a')  # enforce a          >>> from qubovert import boolean_var, PCBO         >>> a = boolean_var('a')         >>> # enforce a         >>> H = PCBO().add_constraint_BUFFER(a) add_constraint_NAND.          Add a penalty to the PCBO that is only zero when         :math:`\lnot (a \land b \land c \land ...)` is True, with a penalty         factor ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.          Parameters         ----------         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_NAND('a', 'b')  # enforce a NAND b          >>> H = PCBO()         >>> H.add_constraint_NAND('a', 'b', 'c', 'd')         >>> # enforce a NAND b NAND c NAND d          >>> from qubovert import boolean_var, PCBO         >>> a, b = boolean_var('a'), boolean_var('b')         >>> # enforce a NAND b         >>> H = PCBO().add_constraint_NAND(a, b) add_constraint_NOR.          Add a penalty to the PCBO that is only nonzero when         :math:`\lnot(a \lor b \lor c \lor d \lor ...)` is True, with a penalty         factor ``lam``, where ``a = variables[0]``, ``b = variables[1]``, etc.          Parameters         ----------         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_NOR('a', 'b')  # enforce a NOR b          >>> H = PCBO()         >>> H.add_constraint_NOR('a', 'b', 'c', 'd')         >>> # enforce a NOR b NOR c NOR d          >>> from qubovert import boolean_var, PCBO         >>> a, b = boolean_var('a'), boolean_var('b')         >>> # enforce a NOR b         >>> H = PCBO().add_constraint_NOR(a, b) add_constraint_XNOR.          Add a penalty to the PCBO that is only nonzero when         :math:`\lnot(v_0 \oplus v_1 \oplus ... \oplus v_n)` is True, with a         penalty  factor ``lam``, where ``v_0 = variables[0]``,         ``v_1 = variables[1]``, ..., ``v_n = variables[-1]``. See         ``qubovert.sat.XNOR`` for the XNOR convention that qubovert uses for         more than two inputs.          Parameters         ----------         *variables : arguments.             Each element of variables is a hashable object or a dict             (its PUBO representation). They are the label of the boolean             variables.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_XNOR('a', 'b')  # enforce a XNOR b          >>> H = PCBO()         >>> H.add_constraint_XNOR('a', 'b', 'c')  # enforce a XNOR b XNOR c          >>> from qubovert import boolean_var, PCBO         >>> a, b = boolean_var('a'), boolean_var('b')         >>> # enforce a XNOR b         >>> H = PCBO().add_constraint_XNOR(a, b) add_constraint_NOT.          Add a penalty to the PCBO that is only nonzero when         :math:`\lnot a` is True, with a penalty factor ``lam``.          Parameters         ----------         a : any hashable object or a dict.             The label for boolean variables ``a``, or its PUBO representation.         lam : float > 0 or sympy.Symbol (optional, defaults to 1).             Langrange multiplier to penalize violations of the clause.          Return         ------         self : PCBO.             Updates the PCBO in place, but returns ``self`` so that operations             can be strung together.          Examples         --------         >>> H = PCBO()         >>> H.add_constraint_NOT('a')  # enforce not a         >>> H         {('a',): 1}          >>> from qubovert import boolean_var, PCBO         >>> a = boolean_var('a')         >>> # enforce not a         >>> H = PCBO().add_constraint_NOT(a)