Log24

Monday, May 15, 2023

Boolean Functions Review

Filed under: General — Tags: , — m759 @ 1:25 pm

The previous post included an illustration by Solomon Golomb
from his 1959 paper "On the Classification of Boolean Functions."

This suggests a review of some later work in this area —

This post was suggested by the word "Boolean" in a May 10
ChatGPT response —

In the above, "Boolean algebras" should be "Boolean functions,"
as indicated by Harrison's 1964 remarks.

Sunday, February 18, 2024

Vocabulary Note

Filed under: General — m759 @ 10:23 am

The URLs powerset.space and powerset.group are now operative.

Related vocabulary:  See Boolean Functions Review (Log24, May 15, 2023).

Sunday, October 15, 2023

Efficient Packend

Filed under: General — Tags: , — m759 @ 11:00 pm

"Stencils" from a 1959 paper by Golomb —

Boolean functions illustration by Golomb, 1959

These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).

This  journal on Beltane (May 1), 2016 —

Saturday, September 30, 2023

Underlying Symbolic Structures

Filed under: General — Tags: — m759 @ 8:27 pm

"As McCarthy peers through the screen, or veil, of technological modernity
to reveal the underlying symbolic structures of human experience, 
The Making of Incarnation  weaves a set of stories one inside the other,
rings within rings, a perpetual motion machine."

— Amazon.com description of a novel published on All Souls' Day 
    (Dia de los Muertos), 2021.

See also the underlying symbolic structures of Boolean functions . . .
as discussed, for instance, on Sept. 23 at medium.com

Tuesday, June 27, 2023

The Representation of Minus One…

Filed under: General — Tags: , — m759 @ 3:09 pm

Continued from February 6, 2014.

This flashback to 2014 was prompted by the following search history —

Related logic —

Related material — "Boolean Functions" in this journal.

Thursday, June 15, 2023

Michaelmas 2019

Filed under: General — Tags: — m759 @ 1:06 pm

Transcribed from a PDF:

Received September 29, 2019, accepted October 15, 2019,
date of publication October 24, 2019, date of current version
November 7, 2019.

Digital Object Identifier 10.1109/ACCESS.2019.2949310

A Method for Determining
the Affine Equivalence of Boolean Functions

ZIYU WANG1 , XIAO ZENG1 , JINZHAO WU2,3, AND
GUOWU YANG1

1Big Data Research Center, School of Computer Science
and Engineering, University of Electronic Science and Technology
of China, Chengdu 611731, China

2Guangxi Key Laboratory of Hybrid Computation and
IC Design Analysis, Guangxi University for Nationalities,
Nanning 530006, China

3School of Computer and Electronic Information,
Guangxi University, Nanning 530004, China

Corresponding authors:
Jinzhao Wu (gxmdwjzh@aliyun.com) and
Guowu Yang (ygwuestc@163.com)

This work was supported in part by the National Natural Science Foundation
of China under Grant 61772006 and Grant 61572109, in part by the
State Key Laboratory of Information Security, Institute of Information Engineering, 
Chinese Academy of Sciences, Beijing, in part by the Science and Technology
Program of Guangxi  under Grant AB17129012, in part by the Science and
Technology Major Project of Guangxi under Grant AA17204096, in part by 
the Special Fund for Scientific and Technological Bases and Talents
of Guangxi under Grant 2016AD05050, and in part by the Special Fund for
Bagui Scholars of Guangxi, in part by the Open fund of State Key Laboratory 
of Information Security.

ABSTRACT 
Determining the affine equivalence of Boolean functions
has significant applications in circuit and cryptography.
Previous methods for determining this require a large
amount of computation when Boolean functions are bent
functions or when the truth table is sparse. This paper
presents a new method to determine the affine equivalence
based on matrix algebra. By transforming Boolean function
to the corresponding matrix representation, we first propose
and prove the congruent standard form of Boolean function.
It lays the foundation for the determination of equivalence
because affine Boolean functions must have the same
standard form. Then we find the generators of orthogonal
matrix group and symplectic matrix group, which greatly
reduce the search space. The computation complexity of
our method is o (2r2/2+n∗(nr) ), where is the number of
bit operations, and r  is the rank of the matrix, which is
the product of Boolean-1 matrix of the test Boolean function
and its transposition. The experimental results show that our
method is useful when the test Boolean function is no more
than 7 bits and r  is greater than 2.

INDEX TERMS  Logic synthesis, Boolean functions,
affine equivalence, matrix group, algorithm.

Meanwhile . . .

Wednesday, June 14, 2023

From Mysticism to Mathematics…

Filed under: General — Tags: , — m759 @ 10:58 am

Continued from October 6, 2022

A paper from an August 2017 Melbourne conference
on artificial intelligence —

See as well a Log24 search for Boolean functions.

A check on the date of the above paper's presentation —

From this  journal on that date

Happy 10th birthday to the hashtag.

Monday, May 22, 2023

Triangular Hyperplane Arrangement

Filed under: General — Tags: , — m759 @ 11:07 am

Abstract: Boolean functions on a triangular grid.

Note: It seems that the above rearrangement of a square array
of hyperplanes to a triangular array of hyperplanes, which was
rather arbitrarily constructed to have nice symmetries, will
answer a question posed here on Dec. 15, 2015.

See a check of the rearrangement.

Wednesday, May 17, 2023

British Combinatorics 2023

Filed under: General — Tags: — m759 @ 12:35 pm

"Martin told us about his work on machine learning…."

Peter J. Cameron reports on last week's (May 10 and 11)
London Combinatorics Colloquia 2023.

Related material — Boolean Functions in this journal and in a book
from May 16, 2011 . . .

Some historical background on machine learning —

https://tripleampersand.org/kernelled-connections-perceptron-diagram/ .

Monday, May 15, 2023

Chatbot Review

Filed under: General — Tags: , , — m759 @ 9:49 pm
 

By ChatGPT* — 

What is the Cullinane diamond theorem? 
5/10/2023, revised as below on 5/15/2023

The Cullinane Diamond Theorem, also known as the Diamond Theorem, is a result in combinatorial mathematics proposed by Steven H. Cullinane. The theorem explores connections between finite geometries, algebraic structures, and combinatorial designs. 

The theorem is primarily concerned with four related structures:
 
1. Finite geometries: Specifically, the theorem relates to finite projective geometries over finite fields, such as projective planes and projective spaces. 

2. Vector spaces: The theorem connects finite vector spaces over finite fields to the aforementioned finite geometries. 

3. Boolean functions:* The theorem establishes relationships between the finite geometries and certain Boolean functions derived from them. 

4. Symmetry groups: The theorem involves symmetry groups associated with the finite geometries.
 
The precise details and implications of the Cullinane Diamond Theorem are beyond the scope of this response. It involves abstract algebra, group theory, combinatorial designs, and finite fields. The theorem has been the subject of various publications and research discussions in mathematical literature. If you are interested in delving deeper into the Cullinane Diamond Theorem, I recommend referring to the original works by Steven H. Cullinane and related research articles on the topic.

* With one revision by Cullinane: "Boolean functions" replaces "Boolean algebras." See (for instance) M. A. Harrison, “On the classification of Boolean functions by the general linear and affine groups,” Journal of the Society for Industrial and Applied Mathematics  1964 12:2, 285-299.

Death on Beltane

Filed under: General — Tags: , , — m759 @ 2:46 am

"Stencils" from a 1959 paper by Golomb —

Boolean functions illustration by Golomb, 1959

These 15 figures also represent the 15 points of a finite geometry
(Cullinane diamond theorem, February 1979).

This  journal on Beltane (May 1), 2016 —

Thursday, October 6, 2022

From Mysticism to Mathematics

Filed under: General — Tags: , — m759 @ 2:48 pm

[Klein, 1983] S. Klein.
"Analogy and Mysticism and the Structure of Culture
(and Comments & Reply)
"
Current Anthropology , 24 (2):151–180, 1983.

The citation above is from a 2017 paper —

"Analogy-preserving Functions:
A Way to Extend Boolean Samples
,"
by M. Couceiro, N. Hug, H. Prade, G, Richard.
26th International Joint Conference on Artificial Intelligence
(IJCAI 2017), Aug. 2017, Melbourne, Australia. pp.1-7, ff.

That 2017 paper discusses Boolean functions .

Some more-recent remarks on these functions
as pure  mathematics —

"On the Number of Affine Equivalence Classes
of Boolean Functions,
" by Xiang-dong Hou,
arXiv:2007.12308v2 [math.CO]. Rev. Aug. 18, 2021.

See also other posts now tagged Analogy and Mysticism.

Thursday, July 19, 2018

The Corrections (Continued)

Filed under: General — m759 @ 8:33 pm

A search for the phrase "nonlinear Boolean algebra" yields few results.

"Nonlinear Boolean functions " seems to be the phrase intended.

On the mathematics of nonlinear Boolean functions —

A memorable death —

For those who prefer narrative to mathematics —

A related image —

Tuesday, January 12, 2016

Lechner’s Beginning

Filed under: General — m759 @ 12:00 am

Professionally, at least

Click image to enlarge.

See also the previous post, Lechner's End.

For a more up-to-date look at harmonic analysis
and switching functions (i.e., Boolean functions),
see Ryan O'Donnell, Analysis of Boolean Functions ,
Cambridge U. Press, 2014.  Page 40 gives an
informative overview of the history of this field.

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