r"""This module contains the :term:`algebra of clans` and related functionality.
A :term:`clan` is also a :term:`set` (of :term:`relation`\s), and inherits all operations
of the :term:`algebra of sets`. These are provided in :mod:`~.algebras.sets`.
"""
# $Id: clans.py 22702 2015-07-28 20:20:56Z jaustell $
# Copyright Algebraix Data Corporation 2015 - $Date: 2015-07-28 15:20:56 -0500 (Tue, 28 Jul 2015) $
#
# This file is part of algebraixlib <http://github.com/AlgebraixData/algebraixlib>.
#
# algebraixlib is free software: you can redistribute it and/or modify it under the terms of version
# 3 of the GNU Lesser General Public License as published by the Free Software Foundation.
#
# algebraixlib is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without
# even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License along with algebraixlib.
# If not, see <http://www.gnu.org/licenses/>.
# --------------------------------------------------------------------------------------------------
import functools as _functools
import algebraixlib.algebras.relations as _relations
import algebraixlib.algebras.sets as _sets
import algebraixlib.mathobjects as _mo
import algebraixlib.extension as _extension
import algebraixlib.structure as _structure
from algebraixlib.undef import make_or_raise_undef as _make_or_raise_undef
# --------------------------------------------------------------------------------------------------
[docs]class Algebra:
"""Provide the operations and relations that are members of the :term:`algebra of clans`.
This class contains only static member functions. Its main purpose is to provide a namespace for
and highlight the operations and relations that belong to the algebra of clans. All member
functions are also available at the enclosing module scope.
"""
# ----------------------------------------------------------------------------------------------
# Unary algebra operations.
@staticmethod
[docs] def transpose(clan: 'PP(M x M)', _checked=True) -> 'PP(M x M)':
"""Return a clan where all relations have their left and right components swapped.
:return: The :term:`unary extension` of :term:`transposition` from the :term:`algebra of
relations` to the :term:`algebra of clans`, applied to ``clan``, or `Undef()` if
``clan`` is not a :term:`clan`.
"""
if _checked:
if not is_member(clan):
return _make_or_raise_undef()
else:
assert is_member(clan)
result = _extension.unary_extend(clan, _functools.partial(
_relations.transpose, _checked=False), _checked=False).cache_is_clan(True)
if not result.is_empty:
if clan.cached_is_functional or clan.cached_is_not_functional:
result.cache_is_right_functional(clan.cached_is_functional)
if clan.cached_is_right_functional or clan.cached_is_not_right_functional:
result.cache_is_functional(clan.cached_is_right_functional)
return result
# ----------------------------------------------------------------------------------------------
# Binary algebra operations.
@staticmethod
[docs] def compose(clan1: 'PP(M x M)', clan2: 'PP(M x M)', _checked=True) -> 'PP(M x M)':
r"""Return the composition of ``clan1`` with ``clan2``.
:return: The :term:`binary extension` of :term:`composition` from the
:term:`algebra of relations` to the :term:`algebra of clans`, applied to ``clan1``
and ``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are not :term:`clan`\s.
"""
if _checked:
if not is_member(clan1):
return _make_or_raise_undef()
if not is_member(clan2):
return _make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
return _extension.binary_extend(clan1, clan2, _functools.partial(
_relations.compose, _checked=False), _checked=False).cache_is_clan(True)
@staticmethod
[docs] def cross_union(clan1: 'PP(M x M)', clan2: 'PP(M x M)', _checked=True) -> 'PP(M x M)':
r"""Return the cross-union of ``clan1`` and ``clan2``.
The :term:`cross-union` of two :term:`clan`\s is a clan that contains the result of
unioning every :term:`relation` from one clan with every relation from the other clan.
:return: The :term:`binary extension` of :term:`union` from the :term:`algebra of relations`
(which inherits it from the :term:`algebra of sets`) to the :term:`algebra of clans`
applied to ``clan1`` and ``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are not
clans.
"""
if _checked:
if not is_member(clan1):
return _make_or_raise_undef()
if not is_member(clan2):
return _make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
return _extension.binary_extend(
clan1, clan2, _functools.partial(
_sets.union, _checked=False), _checked=False).cache_is_clan(True)
@staticmethod
[docs] def cross_functional_union(clan1: 'PP(M x M)', clan2: 'PP(M x M)',
_checked=True) -> 'PP(M x M)':
r"""Return the cross-functional union of ``clan1`` and ``clan2``.
The :term:`cross-functional union` of two :term:`clan`\s is the :term:`cross-union` of
these clans, but removing all resulting :term:`relation`\s that are not :term:`function`\s.
:return: The :term:`binary extension` of the :term:`functional union` from the
:term:`algebra of relations` to the :term:`algebra of clans`, applied to ``clan1`` and
``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are not clans.
"""
if _checked:
if not is_member(clan1):
return _make_or_raise_undef()
if not is_member(clan2):
return _make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
return _extension.binary_extend(
clan1, clan2, _functools.partial(_relations.functional_union, _checked=False),
_checked=False).cache_is_clan(True).cache_is_functional(True)
@staticmethod
[docs] def lhs_cross_functional_union(lhs: 'PP( MxM )', rhs: 'PP( MxM )',
_checked=True) -> 'PP(M x M)':
"""Return the :term:`lhs-cross-functional union` ('left join') of ``lhs`` and ``rhs``.
This operation results in a :term:`clan` that contains every :term:`relation` of a
:term:`cross-functional union`, but also contains all relations in ``lhs`` that
are not already part of one of the resulting relations.
:param lhs: All relations in this clan are guaranteed to be represented in the result.
:return: The resulting clan or `Undef()` if ``lhs`` or ``rhs`` are not clans.
"""
if _checked:
if not is_member(lhs):
return _make_or_raise_undef()
if not is_member(rhs):
return _make_or_raise_undef()
else:
assert is_member(lhs)
assert is_member(rhs)
result = _sets.union(
cross_functional_union(lhs, rhs, _checked=False),
_mo.Set(lhs_elem for lhs_elem in lhs if cross_functional_union(
_mo.Set(lhs_elem, direct_load=True), rhs).is_empty), _checked=False)
if lhs.cached_is_functional:
result.cache_is_functional(True)
return result
@staticmethod
[docs] def cross_right_functional_union(clan1: 'PP(M x M)', clan2: 'PP(M x M)',
_checked=True) -> 'PP(M x M)':
r"""Return the cross-right-functional union of ``clan1`` and ``clan2``.
The :term:`cross-right-functional union` of two :term:`clan`\s is the :term:`cross-union`
of these clans, but removing all resulting :term:`relation`\s that are not
:term:`right-functional`.
:return: The :term:`binary extension` of the :term:`right-functional union` from the
:term:`algebra of relations` to the :term:`algebra of clans`, applied to ``clan1`` and
``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are not :term:`clan`\s.
"""
if _checked:
if not is_member(clan1):
return _make_or_raise_undef()
if not is_member(clan2):
return _make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
return _extension.binary_extend(
clan1, clan2, _functools.partial(_relations.right_functional_union, _checked=False),
_checked=False).cache_is_clan(True).cache_is_right_functional(True)
@staticmethod
[docs] def cross_intersect(clan1: 'PP(M x M)', clan2: 'PP(M x M)', _checked=True) -> 'PP(M x M)':
r"""Return the cross-intersection of ``clan1`` and ``clan2``.
The :term:`cross-intersection` of two :term:`clan`\s is a clan that contains the result of
intersecting every :term:`relation` from one clan with every relation from the other clan.
:return: The :term:`binary extension` of :term:`intersection` from the :term:`algebra of
relations` (which inherits it from the :term:`algebra of sets`) to the :term:`algebra of
clans` applied to ``clan1`` and ``clan2``, or `Undef()` if ``clan1`` or ``clan2`` are
not :term:`clan`\s.
"""
if _checked:
if not is_member(clan1):
return _make_or_raise_undef()
if not is_member(clan2):
return _make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
return _extension.binary_extend(
clan1, clan2, _functools.partial(
_sets.intersect, _checked=False), _checked=False).cache_is_clan(True)
@staticmethod
[docs] def substrict(clan1: 'PP(M x M)', clan2: 'PP(M x M)', _checked=True) -> 'PP(M x M)':
r"""Return the substriction of ``clan1`` and ``clan2``.
The :term:`substriction` of two :term:`clan`\s is a clan that contains all
:term:`relation`\s from ``clan1`` that are a :term:`subset` of a relation from ``clan2``.
:return: The :term:`binary extension` of :term:`substriction` from the :term:`algebra of
sets` to the :term:`algebra of clans` applied to ``clan1`` and ``clan2``, or `Undef()`
if ``clan1`` or ``clan2`` are not clans.
"""
if _checked:
if not is_member(clan1):
return _make_or_raise_undef()
if not is_member(clan2):
return _make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
result = _extension.binary_extend(
clan1, clan2, _functools.partial(
_sets.substrict, _checked=False), _checked=False).cache_is_clan(True)
# The subset of clan1 that is returned has all properties of clan1
if not result.is_empty:
result.copy_flags(clan1)
return result
@staticmethod
[docs] def superstrict(clan1: 'PP(M x M)', clan2: 'PP(M x M)', _checked=True) -> 'PP(M x M)':
r"""Return the superstriction of ``clan1`` and ``clan2``.
The :term:`superstriction` of two :term:`clan`\s is a clan that contains all
:term:`relation`\s from ``clan1`` that are a :term:`superset` of a relation from ``clan2``.
:return: The :term:`binary extension` of :term:`superstriction` from the :term:`algebra of
sets` to the :term:`algebra of clans` applied to ``clan1`` and ``clan2``, or `Undef()`
if ``clan1`` or ``clan2`` are not clans.
"""
if _checked:
if not is_member(clan1):
return _make_or_raise_undef()
if not is_member(clan2):
return _make_or_raise_undef()
else:
assert is_member(clan1)
assert is_member(clan2)
result = _extension.binary_extend(clan1, clan2, _functools.partial(
_sets.superstrict, _checked=False)).cache_is_clan(True)
# The subset of clan1 that is returned has all properties of clan1
if not result.is_empty:
result.copy_flags(clan1)
return result
# For convenience, make the members of class Algebra (they are all static functions) available at
# the module level.
#: Convenience redirection to `Algebra.transpose`.
transpose = Algebra.transpose
#: Convenience redirection to `Algebra.compose`.
compose = Algebra.compose
#: Convenience redirection to `Algebra.cross_union`.
cross_union = Algebra.cross_union
#: Convenience redirection to `Algebra.cross_functional_union`.
cross_functional_union = Algebra.cross_functional_union
#: Convenience redirection to `Algebra.lhs_cross_functional_union`.
lhs_cross_functional_union = Algebra.lhs_cross_functional_union
#: Convenience redirection to `Algebra.cross_right_functional_union`.
cross_right_functional_union = Algebra.cross_right_functional_union
#: Convenience redirection to `Algebra.cross_intersect`.
cross_intersect = Algebra.cross_intersect
#: Convenience redirection to `Algebra.substrict`.
substrict = Algebra.substrict
#: Convenience redirection to `Algebra.superstrict`.
superstrict = Algebra.superstrict
# --------------------------------------------------------------------------------------------------
# Metadata functions.
[docs]def get_name() -> str:
"""Return the name and :term:`ground set` of this :term:`algebra` in string form."""
return 'Clans(M): {ground_set}'.format(ground_set=str(get_ground_set()))
[docs]def get_ground_set() -> _structure.Structure:
"""Return the :term:`ground set` of this :term:`algebra`."""
return _structure.PowerSet(_relations.get_ground_set())
[docs]def get_absolute_ground_set() -> _structure.Structure:
"""Return the :term:`absolute ground set` of this :term:`algebra`."""
return _structure.PowerSet(_relations.get_absolute_ground_set())
[docs]def is_member(obj: _mo.MathObject) -> bool:
"""Return whether ``obj`` is a member of the :term:`ground set` of this :term:`algebra`.
.. note:: This function may call :meth:`~.MathObject.get_ground_set` on ``obj``. The result of
this operation is cached.
"""
_mo.raise_if_not_mathobject(obj)
if not obj.cached_is_clan and not obj.cached_is_not_clan:
obj.cache_is_clan(obj.get_ground_set().is_subset(get_ground_set()))
return obj.cached_is_clan
[docs]def is_absolute_member(obj: _mo.MathObject) -> bool:
"""Return whether ``obj`` is a member of the :term:`absolute ground set` of this algebra.
:return: Whether ``obj`` is an :term:`absolute clan`.
.. note:: This function calls :meth:`~.MathObject.get_ground_set` on ``obj``.
"""
_mo.raise_if_not_mathobject(obj)
return obj.get_ground_set().is_subset(get_absolute_ground_set())
# --------------------------------------------------------------------------------------------------
# Related operations, not formally part of the algebra.
[docs]def get_lefts(clan: 'PP(M x M)', _checked=True) -> 'P( M )':
r"""Return the set of the left components of all couplets in all relations in ``clan``.
:return: The :term:`union` of the :term:`left set`\s of all :term:`relation`\s in ``clan`` or
`Undef()` if ``clan`` is not a :term:`clan`.
"""
if _checked:
if not is_member(clan):
return _make_or_raise_undef()
else:
assert is_member(clan)
if clan.is_empty:
# The left set of an empty set is the empty set
return clan
clan_itr = iter(clan)
left_set = _relations.get_lefts(next(clan_itr), _checked=False)
for rel in clan_itr:
left_set = _sets.union(
_relations.get_lefts(rel, _checked=False), left_set, _checked=False)
return left_set
[docs]def get_rights(clan: 'PP(M x M)', _checked=True) -> "P( M )":
r"""Return the set of the right components of all couplets in all relations in ``clan``.
:return: The :term:`union` of the :term:`right set`\s of all :term:`relation`\s in ``clan`` or
`Undef()` if ``clan`` is not a :term:`clan`.
"""
if _checked:
if not is_member(clan):
return _make_or_raise_undef()
else:
assert is_member(clan)
if clan.is_empty:
# The right set of an empty set is the empty set
return clan
clan_itr = iter(clan)
right_set = _relations.get_rights(next(clan_itr), _checked=False)
for rel in clan_itr:
right_set = _sets.union(
_relations.get_rights(rel, _checked=False), right_set, _checked=False)
return right_set
[docs]def is_regular(clan, _checked=True) -> bool:
"""Return whether ``clan`` is regular.
:return: ``True`` if ``clan`` is :term:`regular`, ``False`` if not or `Undef()` if ``clan``
is not a :term:`clan`.
"""
if _checked:
if not is_member(clan):
return _make_or_raise_undef()
else:
assert is_member(clan)
if not clan.cached_is_regular and not clan.cached_is_not_regular:
# NOTE: The empty case is handled in Set().__init__ via flags initialization
if clan.cached_is_not_functional:
clan.cache_is_regular(False)
return False
itr = iter(clan)
rel = next(itr)
if not rel.is_functional():
clan.cache_is_regular(False)
return False
left_set = rel.get_left_set()
regular = all(rel.is_functional() and left_set == rel.get_left_set() for rel in itr)
clan.cache_is_regular(regular)
return clan.cached_is_regular
[docs]def project(clan: 'PP(M x M)', *lefts) -> 'PP(M x M)':
r"""Return a clan that contains only the couplets with lefts from ``clan`` that match ``lefts``.
:param clan: The source data. Must be a :term:`clan`.
:param lefts: The names of the :term:`left component`\s to match. (If you want to pass in an
iterable, you need to prefix it with an asterisk ``*``.)
:return: The :term:`projection` of ``clan`` (a clan that contains only :term:`couplet`\s with
left components as indicated by ``lefts``), or `Undef()` if ``clan`` is not a clan.
"""
clan = compose(clan, diag(*lefts))
return clan
[docs]def from_set(left: '( M )', *values: '( M )') -> 'PP(M x M)':
r"""Return a clan where all relations contain a single couplet with the same left component.
:param left: The :term:`left component` of all :term:`couplet`\s in the returned :term:`clan`.
:param values: The :term:`right component`\s of the couplets in the returned clan. (If you want
to pass in an iterable, you need to prefix it with an asterisk ``*``.)
:return: A clan where every :term:`relation` consists of a single couplet with a left component
of ``left`` and a right component from ``values``.
"""
left_mo = _mo.auto_convert(left)
clan = _mo.Set((_mo.Set(_mo.Couplet(left_mo, _mo.auto_convert(right), direct_load=True),
direct_load=True).cache_is_relation(True)
for right in values), direct_load=True).cache_is_clan(True)
return clan
[docs]def from_dict(dict1: dict) -> 'PP(M x M)':
r"""Return a :term:`clan` with a single :term:`relation` where the :term:`couplet`\s are the
elements of ``dict1``."""
clan = _mo.Set(_mo.Set((_mo.Couplet(left, right) for left, right in dict1.items()),
direct_load=True).cache_is_relation(True),
direct_load=True).cache_is_clan(True)
return clan
[docs]def diag(*args, _checked=True) -> 'PP(M x M)':
"""Return a clan diagonal of the arguments.
:param args: Pass in the elements from which the :term:`clan diagonal` is formed. (If you want
to pass in an iterable, you need to prefix it with an asterisk ``*``.)
"""
clan = _mo.Set(_relations.diag(*args, _checked=_checked), direct_load=True).cache_is_clan(True)
return clan
[docs]def defined_at(clan, left, _checked=True):
r"""Return the :term:`relation`\s of ``clan`` that are defined for ``left``."""
clan = _extension.unary_extend(clan, _functools.partial(
_relations.defined_at, left=left, _checked=_checked), _checked=_checked).cache_is_clan(True)
return clan