Boolean-Cayley-graphs / boolean_cayley_graphs / __pycache__ / boolean_function_extended_translate_classification.cpython-310.pyc
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Classification of boolean functions within an extended translation class
========================================================================
The ``boolean_function_extended_translate_classification`` module defines:
* the ``BooleanFunctionExtendedTranslateClassification`` class;
which represents the classification of the general linear classes
within the extended translation class of a boolean function; and
* the ``BooleanFunctionExtendedTranslateClassPart`` class,
which represents part of an extended translate classification.
AUTHORS:
- Paul Leopardi (2023-01-26): initial version
EXAMPLES:
::
The classification of the boolean function defined by the polynomial x2 + x1*x2.
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassification as BooleanFunctionETC)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c = BooleanFunctionETC.from_function(f)
sage: dict(sorted(c.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x1,
'boolean_function_index_matrix': [0 2 1 3]
[1 3 0 2]
[2 0 3 1]
[3 1 2 0],
'boolean_function_list': [Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables],
'general_linear_class_index_matrix': [0 0 1 0]
[1 0 0 0]
[0 0 0 1]
[0 1 0 0],
'general_linear_class_list': [Boolean function with 2 variables,
Boolean function with 2 variables]}
REFERENCES:
The extended translation equivalence class and the general linear equivalence class
of a boolean function are defined by Leopardi [Leo2017]_.
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Partial classification of the general linear classes within the
extended translation equivalence class of a boolean function.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassPart as BooleanFunctionETCPart)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETCPart.from_function(f, c_stop=1)
sage: print(c1)
BooleanFunctionExtendedTranslateClassPart.from_function(BooleanFunctionImproved(x0*x1 + x0 + x1, c_start=0, c_stop=1))
sage: latex(c1)
\text{\texttt{BooleanFunctionExtendedTranslateClassPart.from{\char`\_}function(BooleanFunctionImproved(x0*x1{ }+{ }x0{ }+{ }x1,{ }c{\char`\_}start=0,{ }c{\char`\_}stop=1))}}
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Constructor from an object or from class attributes.
INPUT:
- ``algebraic_normal_form`` -- a polynomial of the type
returned by ``BooleanFunction.algebraic_normal_form()``,
representing the ``BooleanFunction`` whose classification this is.
- ``boolean_function_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``boolean_function_list`` representing the
distinct boolean functions.
- ``boolean_function_list`` -- a list of boolean functions.
- ``general_linear_class_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``general_linear_class_list`` representing the
general linear equivalence classes.
- ``general_linear_class_list`` -- a list of matrices representing the
general linear equivalence classes.
- ``c_start`` -- an integer representing the index of
the first row of each matrix.
OUTPUT:
None.
EFFECT:
The current object ``self`` is initialized as follows.
Each of
- ``algebraic_normal_form``
- ``boolean_function_index_matrix``
- ``boolean_function_list``
- ``general_linear_class_index_matrix``
- ``general_linear_class_list``
- ``c_start``
is set to the corresponding input parameter.
EXAMPLES:
The partial classification of the boolean function defined by the polynomial
:math:`x_1 + x_2 + x_1 x_2` is copied from `c1` to `c2`.
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassPart as BooleanFunctionETCPart)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETCPart.from_function(f, c_stop=1)
sage: c2 = BooleanFunctionETCPart(c1)
sage: print(c1 == c2)
True
r �algebraic_normal_form� boolean_function_index_matrix�boolean_function_list�!general_linear_class_index_matrix�general_linear_class_list�c_startN)r r r r r r �pop��self�args�kwargs�sobj� r" �l/home/user/Boolean-Cayley-graphs/boolean_cayley_graphs/boolean_function_extended_translate_classification.py�__init__s s6 8����� �z2BooleanFunctionExtendedTranslateClassPart.__init__c C sF | j | j�� }t| �jd t| j� d t| j � d t|� d S )a%
Sage string representation.
INPUT:
- ``self`` -- the current object.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassPart as BooleanFunctionETCPart)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETCPart.from_function(f, c_stop=1)
sage: print(c1)
BooleanFunctionExtendedTranslateClassPart.from_function(BooleanFunctionImproved(x0*x1 + x0 + x1, c_start=0, c_stop=1))
�'.from_function(BooleanFunctionImproved(z
, c_start=z , c_stop=�)))r r �nrows�type�__name__�reprr )r �c_stopr" r" r# �_repr_� s" ��������z0BooleanFunctionExtendedTranslateClassPart._repr_r NFc s� t j}t j}|�� }d| }|dkr|}nt||�}|�� } |dkp'|dko'|}
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Constructor from the ``BooleanFunction`` ``boolf``.
INPUT:
- ``boolf`` -- an object of class ``BooleanFunction``.
- ``c_start`` -- integer (default: 0).
The smallest value of `c` to use for extended translates.
- ``c_stop`` -- integer (default: ``None``).
One more than largest value of `c` to use for extended
translates. ``None`` means use all remaining values.
- ``limited_memory`` -- boolean (default: ``False``).
A flag indicating whether the classification might be
too large to fit into memory.
OUTPUT:
An object of class BooleanFunctionExtendedTranslateClassPart,
initialized as follows.
- ``algebraic_normal_form`` is set to ``boolf.algebraic_normal_form()``,
- ``boolean_function_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``boolean_function_list`` representing the
distinct boolean functions.
- ``boolean_function_list`` -- a list of boolean functions.
- ``general_linear_class_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``general_linear_class_list`` representing the
general linear equivalence classes.
- ``general_linear_class_list`` -- a list of matrices representing the
general linear equivalence classes.
- ``c_start`` is set to smallest value of `c` used for extended translates.
Each entry ``boolean_function_index_matrix[c-c_start,b]`` corresponds to
the boolean function
:math:`x \mapsto \mathtt{boolf}(x+b) + \langle c, x \rangle + \mathtt{boolf}(b)`.
This enumerates all of the extended translates of ``boolf`` having ``c``
from ``c_start`` to but not including ``c_stop``.
EXAMPLES:
A partial classification of the boolean function defined by the polynomial
:math:`x_1 + x_2 + x_1 x_2`.
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassPart as BooleanFunctionETCPart)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETCPart.from_function(f,c_start=2,c_stop=4)
sage: dict(sorted(c1.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x0 + x1,
'boolean_function_index_matrix': [0 2 1 3]
[1 3 0 2],
'boolean_function_list': [Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables],
'c_start': 2,
'general_linear_class_index_matrix': [0 1 0 0]
[0 0 0 1],
'general_linear_class_list': [Boolean function with 2 variables,
Boolean function with 2 variables]}
� N� � )�endc 3 s � | ]}� |�V qd S )Nr" )�.0�x��fr" r# � <genexpr>L s � zJBooleanFunctionExtendedTranslateClassPart.from_function.<locals>.<genexpr>)r r r r r r )�controls�checking�timing�
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close_dict� remove_dict)�cls�boolfr r+ �limited_memoryr7 r8 �dim�vr Z
use_shelveZboolean_function_bijection�c_lenr r r �b�c�ttZbf_ttZ bf_tt_indexZbf_etcZ glc_indexr r" r3 r# �
from_function� sh I
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�z7BooleanFunctionExtendedTranslateClassPart.from_functionc C sT |du rdS | j |j ko)| j|jko)| j|jko)| j|jko)| j|jko)| j|jkS )a�
Test for equality between partial classifications.
WARNING:
This test is for strict equality rather than mathematical equivalence.
INPUT:
- ``other`` - BooleanFunctionExtendedTranslateClassPart: another partial classification.
OUTPUT:
A Boolean value indicating whether ``self`` strictly equals ``other``.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassPart as BooleanFunctionETCPart)
sage: R2.<x0,x1> = BooleanPolynomialRing(2)
sage: p = x0*x1
sage: f1 = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETCPart.from_function(f1, c_stop=1)
sage: f2 = BooleanFunctionImproved([0,0,0,1])
sage: c2 = BooleanFunctionETCPart.from_function(f2, c_stop=1)
sage: print(c2.algebraic_normal_form)
x0*x1
sage: print(c1 == c2)
True
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__module__� __qualname__�__doc__r$ r, � classmethodrQ rT r" r" r" r# r \ s O" � r c @ sF e Zd ZdZdZdZdZdd� Zdd� Ze
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Classification of the Cayley graphs within the
extended translation equivalence class of a boolean function.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassification as BooleanFunctionETC)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETC.from_function(f)
sage: print(c1)
BooleanFunctionExtendedTranslateClassification.from_function(BooleanFunctionImproved(x0*x1 + x0 + x1))
sage: latex(c1)
\text{\texttt{BooleanFunctionExtendedTranslateClassification.from{\char`\_}function(BooleanFunctionImproved(x0*x1{ }+{ }x0{ }+{ }x1))}}
z_bent_function.csvz_cg_class_list.csvz
_matrices.csvc O s� z|d }|j | _ |j| _|j| _|j| _|j| _W dS |�d�| _ |�d�| _|�d�| _|�d�| _|�d�| _Y dS )aO
Constructor from an object or from class attributes.
INPUT:
- ``sobj`` -- BooleanFunctionExtendedTranslateClassification: object to copy.
- ``algebraic_normal_form`` -- a polynomial of the type
returned by ``BooleanFunction.algebraic_normal_form()``,
representing the ``BooleanFunctionImproved`` whose classification this is.
- ``boolean_function_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``boolean_function_list`` representing the
distinct boolean functions.
- ``boolean_function_list`` -- a list of boolean functions.
- ``general_linear_class_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``general_linear_class_list`` representing the
general linear equivalence classes.
- ``general_linear_class_list`` -- a list of matrices representing the
general linear equivalence classes.
OUTPUT:
None.
EFFECT:
The current object ``self`` is initialized as follows.
Each of
- ``algebraic_normal_form``
- ``boolean_function_index_matrix``
- ``boolean_function_list``
- ``general_linear_class_index_matrix``
- ``general_linear_class_list``
is set to the corresponding input parameter.
EXAMPLES:
The classification of the boolean function defined by the polynomial
:math:`x_1 + x_2 + x_1 x_2` is copied from `c1` to `c2`.
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassification as BooleanFunctionETC)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETC.from_function(f)
sage: c2 = BooleanFunctionETC(c1)
sage: print(c1 == c2)
True
r r r r r r N)r r r r r r r r" r" r# r$ � s. 7���� �z7BooleanFunctionExtendedTranslateClassification.__init__c C s t | �jd t| j� d S )a
Sage string representation.
INPUT:
- ``self`` -- the current object.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassification as BooleanFunctionETC)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETC.from_function(f)
sage: print(c1)
BooleanFunctionExtendedTranslateClassification.from_function(BooleanFunctionImproved(x0*x1 + x0 + x1))
r% r& )r( r) r* r )r r" r" r# r, s ����z5BooleanFunctionExtendedTranslateClassification._repr_TFc C s t j||d�}| |�S )a�
Constructor from the ``BooleanFunctionImproved`` ``boolf``.
INPUT:
- ``boolf`` -- an object of class ``BooleanFunctionImproved``.
- ``list_dual_graphs`` -- boolean. a flag indicating
whether to list dual graphs.
- ``limited_memory`` -- boolean. A flag indicating
whether the classification might be too large
to fit into memory. Default is False.
OUTPUT:
An object of class BooleanFunctionExtendedTranslateClassification,
initialized as follows.
- ``algebraic_normal_form`` is set to ``boolf.algebraic_normal_form()``,
- ``boolean_function_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``boolean_function_list`` representing the
distinct boolean functions.
- ``boolean_function_list`` -- a list of boolean functions.
- ``general_linear_class_index_matrix`` -- a ``Matrix` of integers,
which are indices into ``general_linear_class_list`` representing the
general linear equivalence classes.
- ``general_linear_class_list`` -- a list of matrices representing the
general linear equivalence classes.
Each entry ``boolean_function_index_matrix[c,b]`` corresponds to
the Cayley graph of the boolean function
:math:`x \mapsto \mathtt{boolf}(x+b) + \langle c, x \rangle + \mathtt{boolf}(b)`.
This enumerates all of the extended translates of ``boolf``.
EXAMPLES:
The classification of the boolean function defined by the polynomial
:math:`x_1 + x_2 + x_1 x_2`.
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassification as BooleanFunctionETC)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c3 = BooleanFunctionETC.from_function(f)
sage: dict(sorted(c3.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x0 + x1,
'boolean_function_index_matrix': [0 2 1 3]
[1 3 0 2]
[2 0 3 1]
[3 1 2 0],
'boolean_function_list': [Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables],
'general_linear_class_index_matrix': [0 1 1 1]
[1 1 0 1]
[1 0 1 1]
[1 1 1 0],
'general_linear_class_list': [Boolean function with 2 variables,
Boolean function with 2 variables]}
TESTS:
The classification of the boolean function defined by the polynomial
:math:`x_1 + x_2 + x_1 x_2`, but with list_dual_graphs=False.
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassification as BooleanFunctionETC)
sage: R2.<x1,x2> = BooleanPolynomialRing(2)
sage: p = x1+x2+x1*x2
sage: f = BooleanFunctionImproved(p)
sage: c4 = BooleanFunctionETC.from_function(f,list_dual_graphs=False)
sage: dict(sorted(c4.__dict__.items()))
{'algebraic_normal_form': x0*x1 + x0 + x1,
'boolean_function_index_matrix': [0 2 1 3]
[1 3 0 2]
[2 0 3 1]
[3 1 2 0],
'boolean_function_list': [Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables,
Boolean function with 2 variables],
'general_linear_class_index_matrix': [0 1 1 1]
[1 1 0 1]
[1 0 1 1]
[1 1 1 0],
'general_linear_class_list': [Boolean function with 2 variables,
Boolean function with 2 variables]}
)rJ )r rQ )rH rI �list_dual_graphsrJ �cpr" r" r# rQ s
e�z<BooleanFunctionExtendedTranslateClassification.from_functionc C sH |du rdS | j |j ko#| j|jko#| j|jko#| j|jko#| j|jkS )ah
Test for equality between classifications.
WARNING:
This test is for strict equality rather than mathematical equivalence.
INPUT:
- ``other`` - BooleanFunctionExtendedTranslateClassification: another classification.
OUTPUT:
A Boolean value indicating whether ``self`` strictly equals ``other``.
EXAMPLES:
::
sage: from boolean_cayley_graphs.boolean_function_improved import BooleanFunctionImproved
sage: from boolean_cayley_graphs.boolean_function_extended_translate_classification import (
....: BooleanFunctionExtendedTranslateClassification as BooleanFunctionETC)
sage: R2.<x0,x1> = BooleanPolynomialRing(2)
sage: p = x0*x1
sage: f1 = BooleanFunctionImproved(p)
sage: c1 = BooleanFunctionETC.from_function(f1)
sage: f2 = BooleanFunctionImproved([0,0,0,1])
sage: c2 = BooleanFunctionETC.from_function(f2)
sage: print(c2.algebraic_normal_form)
x0*x1
sage: print(c1 == c2)
True
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