International Conference on
Data Intelligence
and Neutrosophic Sets with
Applications (DI-NSA 2019),
Chairman Prof. Dr. Xiaohong Zhang,
Shaanxi University of Science
and Technology,
Xi’an, P. R. China
December 20 - 22, 2019
On
Overview of Neutrosophic Theories and Applications
Prof. Dr. Florentin Smarandache, Postdoc
University of New Mexico
Mathematics Department
705 Gurley Ave., Gallup, NM 87301, USA
Abstract
This presentation is an Overview on the Foundation and Development of Neutrosophic
Theories and their Applications for a period of more than two decades
(1995-2019) since they were defined and studied, together with their applications, and links to many
open-source articles and books that the attendees can download.
Neutrosophic
Set is a Generalization of
Intuitionistic Fuzzy Set,
Inconsistent Intuitionistic
Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set (Atanassov’s Intuitionistic
Fuzzy Set of second type), q-Rung Orthopair
Fuzzy Set, Spherical Fuzzy Set, and n-HyperSpherical
Fuzzy Set, while
Neutrosophication is a
Generalization of Regret Theory, Grey System Theory, and Three-Ways Decision.
https://arxiv.org/ftp/arxiv/papers/1911/1911.07333.pdf
http://fs.unm.edu/Raspunsatan.pdf
Zadeh introduced the degree of membership/truth (T) in 1965 and defined the
fuzzy set.
Atanassov introduced the degree of nonmembership/falsehood (F) in 1986 and
defined the intuitionistic fuzzy set.
Smarandache introduced the degree of indeterminacy/neutrality (I) as independent
component in 1995 (published in 1998) and he defined the neutrosophic set on
three components:
(T, I, F) = (Truth, Indeterminacy, Falsehood), where in general T, I, F are
subsets of the interval [0, 1]; in particular T, I, F may be intervals, hesitant
sets, single-values, etc.; see
F. Smarandache, Neutrosophy / Neutrosophic probability, set, and logic",
Proquest, Michigan, USA, 1998,
https://arxiv.org/ftp/math/papers/0101/0101228.pdf
http://fs.unm.edu/eBook-Neutrosophics6.pdf;
reviewed in Zentralblatt fuer Mathematik (Berlin, Germany):
https://zbmath.org/?q=an:01273000
and cited by Denis Howe in The Free Online Dictionary of Computing, England,
1999.
Neutrosophic Set and Logic are generalizations of classical, fuzzy, and
intuitionistic fuzzy set and logic:
https://arxiv.org/ftp/math/papers/0404/0404520.pdf
https://arxiv.org/ftp/math/papers/0303/0303009.pdf
Etymology.
The words “neutrosophy” and “neutrosophic” were coined/invented by F.
Smarandache in his 1998 book.
Neutrosophy: A branch of philosophy, introduced by F. Smarandache in 1980, which
studies the origin, nature, and scope of neutralities, as well as their
interactions with different ideational spectra. Neutrosophy considers a
proposition, theory, event, concept, or entity <A> in relation to its opposite <antiA>,
and with their neutral <neutA>.
Neutrosophy (as dynamic of opposites and their neutrals) is an extension of the
Dialectics and Yin Yang (which are the dynamic of opposites only).
Neutrosophy is the basis of neutrosophic set, neutrosophic logic, neutrosophic
measure, neutrosophic probability,
neutrosophic statistics etc.
https://arxiv.org/ftp/math/papers/0010/0010099.pdf
Neutrosophic Logic is a general framework for unification of many existing
logics, such as fuzzy logic (especially intuitionistic fuzzy logic),
paraconsistent logic, intuitionistic logic, etc. The main idea of NL is to
characterize each logical statement in a 3D-Neutrosophic Space, where each
dimension of the space represents respectively the truth (T), the falsehood (F),
and the indeterminacy (I) of the statement under consideration, where T, I, F
are standard or non-standard real subsets of ]-0, 1+[ with not necessarily any
connection between them.
For software engineering proposals the classical unit interval [0, 1] may be
used.
While Neutrosophic Probability and Statistics are generalizations of classical
and imprecise probability and classical statistics respectively.
The Most Important Books and Papers
on the Advancement of Neutrosophics
1995-1998 – Smarandache extended the dialectics (which is a
particular case of Chinese philosophy Yin Yan) to neutrosophy;
introduced the neutrosophic set/logic/probability/statistics;
introduces the single-valued neutrosophic set (pp. 7-8);
https://arxiv.org/ftp/math/papers/0101/0101228.pdf (fourth edition)
http://fs.unm.edu/eBook-Neutrosophics6.pdf (online
sixth edition)
1998, 20019 - Introduction of Nonstandard Neutrosophic Logic, Set, Probability
https://arxiv.org/ftp/arxiv/papers/1903/1903.04558.pdf
2002 – Introduction of corner cases of sets / probabilities / statistics /
logics, such as:
- Neutrosophic intuitionistic set (different from intuitionistic fuzzy set),
neutrosophic paraconsistent set, neutrosophic faillibilist set, neutrosophic
paradoxist set, neutrosophic pseudo-paradoxist set, neutrosophic tautological
set, neutrosophic nihilist set, neutrosophic dialetheist set, neutrosophic
trivialist set;
- Neutrosophic intuitionistic probability and statistics, neutrosophic
paraconsistent probability and statistics, neutrosophic faillibilist probability
and statistics, neutrosophic paradoxist probability and statistics, neutrosophic
pseudo-paradoxist probability and statistics, neutrosophic tautological
probability and statistics, neutrosophic nihilist probability and statistics,
neutrosophic dialetheist probability and statistics,neutrosophic trivialist
probability and statistics;
- Neutrosophic paradoxist logic (or paradoxism), neutrosophic pseudo-paradoxist
logic (or neutrosophic pseudo-paradoxism), neutrosophic tautological logic (or
neutrosophic tautologism):
https://arxiv.org/ftp/math/papers/0301/0301340.pdf
http://fs.unm.edu/DefinitionsDerivedFromNeutrosophics.pdf
2003 – Introduction by Kandasamy and Smarandache of
Neutrosophic Numbers (a+bI, where I = indeterminacy, I^2 = I),
I-Neutrosophic Algebraic Structures
and Neutrosophic Cognitive Maps
https://arxiv.org/ftp/math/papers/0311/0311063.pdf
http://fs.unm.edu/NCMs.pdf
2005 - Introduction of Interval Neutrosophic Set/Logic
https://arxiv.org/pdf/cs/0505014.pdf
http://fs.unm.edu/INSL.pdf
2006 – Introduction of Degree of Dependence and Degree of Independence
between the Neutrosophic Components T, I, F.
For single valued neutrosophic logic, the sum of the components is:
0 ≤ t+i+f ≤ 3 when all three components are independent;
0 ≤ t+i+f ≤ 2 when two components are dependent, while the third one is
independent from them;
0 ≤ t+i+f ≤ 1 when all three components are dependent.
When three or two of the components T, I, F are independent, one leaves room for
incomplete information (sum < 1), paraconsistent and contradictory information
(sum > 1), or complete information (sum = 1).
If all three components T, I, F are dependent, then similarly one leaves room
for incomplete information (sum < 1), or complete information (sum = 1).
In general, the sum of two components x and y that vary in the unitary interval
[0, 1] is:
0 ≤ x + y ≤ 2 - d°(x, y), where d°(x, y) is the degree of dependence between x
and y, while
d°(x, y) is the degree of independence between x and y.
Degrees of Dependence and Independence between Neutrosophic Components
T, I, F are independent components, leaving room for incomplete information
(when their superior sum < 1), paraconsistent and contradictory information
(when the superior sum > 1), or complete information (sum of components = 1).
For software engineering proposals the classical unit interval [0, 1] is used.
https://doi.org/10.5281/zenodo.571359
http://fs.unm.edu/eBook-Neutrosophics6.pdf (p. 92)
http://fs.unm.edu/NSS/DegreeOfDependenceAndIndependence.pdf
2007 – The Neutrosophic Set was extended [Smarandache, 2007] to Neutrosophic
Overset (when some neutrosophic component is > 1), since he observed that, for
example, an employee working overtime deserves a degree of membership > 1, with
respect to an employee that only works regular full-time and whose degree of
membership = 1;
and to Neutrosophic Underset (when some neutrosophic component is < 0), since,
for example, an employee making more damage than benefit to his company deserves
a degree of membership < 0, with respect to an employee that produces benefit to
the company and has the degree of membership > 0;
and to and to Neutrosophic Offset (when some neutrosophic components are off the
interval [0, 1], i.e. some neutrosophic component > 1 and some neutrosophic
component < 0).
Then, similarly, the Neutrosophic Logic/Measure/Probability/Statistics etc. were
extended to respectively Neutrosophic Over-/Under-/Off- Logic, Measure,
Probability, Statistics etc.
https://arxiv.org/ftp/arxiv/papers/1607/1607.00234.pdf
http://fs.unm.edu/NeutrosophicOversetUndersetOffset.pdf
http://fs.unm.edu/SVNeutrosophicOverset-JMI.pdf
http://fs.unm.edu/IV-Neutrosophic-Overset-Underset-Offset.pdf
2007 – Smarandache introduced the
Neutrosophic Tripolar Set and Neutrosophic Multipolar Set
and consequently
the Neutrosophic Tripolar Graph and Neutrosophic Multipolar Graph
http://fs.unm.edu/eBook-Neutrosophics6.pdf (p. 93)
http://fs.unm.edu/IFS-generalized.pdf
2009 – Introduction of N-norm and N-conorm
https://arxiv.org/ftp/arxiv/papers/0901/0901.1289.pdf
http://fs.unm.edu/N-normN-conorm.pdf
2013 - Development of Neutrosophic Measure and Neutrosophic Probability
(chance that an event occurs, indeterminate chance of occurrence,
chance that the event does not occur)
https://arxiv.org/ftp/arxiv/papers/1311/1311.7139.pdf
http://fs.unm.edu/NeutrosophicMeasureIntegralProbability.pdf
2013 – Smarandache Refined/Split the Neutrosophic Components (T, I, F) into
Neutrosophic Subcomponents (T1, T2,
...; I1, I2, ...; F1, F2, ...;):
https://arxiv.org/ftp/arxiv/papers/1407/1407.1041.pdf
http://fs.unm.edu/n-ValuedNeutrosophicLogic-PiP.pdf
2014 – Introduction of the Law of Included Multiple Middle
(<A>; <neut1A>, <neut2A>, …; <antiA>)
http://fs.unm.edu/LawIncludedMultiple-Middle.pdf
2014 - Development of Neutrosophic Statistics (indeterminacy is introduced into
classical statistics with respect to the sample/population, or with respect to
the individuals that only partially belong to a sample/population)
https://arxiv.org/ftp/arxiv/papers/1406/1406.2000.pdf
http://fs.unm.edu/NeutrosophicStatistics.pdf
2015 - Introduction of Neutrosophic Precalculus and Neutrosophic Calculus
https://arxiv.org/ftp/arxiv/papers/1509/1509.07723.pdf
http://fs.unm.edu/NeutrosophicPrecalculusCalculus.pdf
2015 – Refined Neutrosophic Numbers (a+ b1I1 + b2I2 + … + bnIn), where I1, I2,
…, In are subindeterminacies of indeterminacy I;
2015 – (t,i,f)-neutrosophic graphs;
2015 - Thesis-Antithesis-Neutrothesis, and Neutrosynthesis, Neutrosophic
Axiomatic System, neutrosophic dynamic systems, symbolic neutrosophic logic, (t,
i, f)-Neutrosophic Structures, I-Neutrosophic Structures, Refined Literal
Indeterminacy, Quadruple Neutrosophic Algebraic Structures, Multiplication Law
of Subindeterminacies:
https://arxiv.org/ftp/arxiv/papers/1512/1512.00047.pdf
http://fs.unm.edu/SymbolicNeutrosophicTheory.pdf
2015 – Introduction of the Subindeterminacies of the form (I0)n = k/0, for k ∈ {0, 1, 2, …, n-1}, into the ring of modulo integers Zn - called natural neutrosophic indeterminacies (Vasantha-Smarandache)
http://fs.unm.edu/MODNeutrosophicNumbers.pdf
2015 – Introduction of Neutrosophic Crisp Set and Topology (Salama & Smarandache)
http://fs.unm.edu/NeutrosophicCrispSetTheory.pdf
2016 – Introduction of Neutrosophic Multisets (as generalization of classical
multisets)
http://fs.unm.edu/NeutrosophicMultisets.htm
2016 – Introduction of Neutrosophic Triplet Structures and m-valued refined
neutrosophic triplet structures [Smarandache - Ali]
http://fs.unm.edu/NeutrosophicTriplets.htm
2016 – Introduction of Neutrosophic Duplet Structures
http://fs.unm.edu/NeutrosophicDuplets.htm
2017 - In biology Smarandache introduced the Theory of Neutrosophic Evolution:
Degrees of Evolution, Indeterminacy or Neutrality, and Involution
http://fs.unm.edu/neutrosophic-evolution-PP-49-13.pdf
2017 - Introduction by F. Smarandache of Plithogeny (as generalization of
Dialectics and Neutrosophy), and Plithogenic Set/Logic/Probability/Statistics
(as generalization of fuzzy, intuitionistic fuzzy, neutrosophic
set/logic/probability/statistics)
https://arxiv.org/ftp/arxiv/papers/1808/1808.03948.pdf
http://fs.unm.edu/Plithogeny.pdf
http://fs.unm.edu/NSS/PlithogenicSetAnExtensionOfCrisp.pdf
2017 - Enunciation of the Law that It Is Easier to Break
from Inside than from Outside a Neutrosophic Dynamic System (Smarandache
- Vatuiu):
http://fs.unm.edu/EasierMaiUsor.pdf
2018 – Introduction to Neutrosophic Psychology (Neutropsyche, Refined
Neutrosophic Memory: conscious, aconscious, unconscious, Neutropsychic
Personality, Eros / Aoristos / Thanatos, Neutropsychic Crisp Personality):
http://fs.unm.edu/NeutropsychicPersonality-ed3.pdf
2019 - Theory of Spiral Neutrosophic Human Evolution (Smarandache - Vatuiu):
http://fs.unm.edu/SpiralNeutrosophicEvolution.pdf
2019 - Introduction to Neutrosophic Sociology (Neutrosociology) [neutrosophic
concept, or (T, I, F)-concept, is a concept that is T% true,I% indeterminate,
and F% false]:
http://fs.unm.edu/Neutrosociology.pdf
Applications in:
Artificial Intelligence, Information Systems, Computer Science, Cybernetics,
Theory Methods, Mathematical Algebraic Structures, Applied Mathematics,
Automation, Control Systems, Big Data, Engineering, Electrical, Electronic,
Philosophy, Social Science, Psychology, Biology, Biomedical, Engineering,
Medical Informatics, Operational Research, Management Science, Imaging Science,
Photographic Technology, Instruments, Instrumentation, Physics, Optics,
Economics, Mechanics, Neurosciences, Radiology Nuclear, Medicine, Medical
Imaging, Interdisciplinary Applications, Multidisciplinary Sciences etc.
[ Xindong Peng and Jingguo Dai, A bibliometric analysis of neutrosophic set: two
decades review from 1998 to 2017, Artificial Intelligence Review, Springer, 18
August 2018;
http://fs.unm.edu/BibliometricNeutrosophy.pdf ]
Important
Neutrosophic Researchers:
There are about 1,000 neutrosophic researchers, within 62
countries around the globe, that have produced about 2,000 publications within
more than two decades. Many neutrosophic researchers got specialized into
various fields of neutrosophics:
Xiaohong Zhang &
Yingcang Ma (neutrosophic triplet
and quadruple algebraic structures), Yanhui Guo (neutrosophic image processing),
Jun Ye & Peide Liu &
Jianqiang Wang (neutrosophic
optimization),
Xindong Peng & Jingguo Dai (neutrosophic bibliometrics),
Jianqiang
Wang, Guiwu Wei,
Donghai Liu, Xiaohong Chen, Dan Peng, Jiongmei Mo, Han-Liang
Huang, Victor Chang, Hongjun Guan, Shuang Guan, Aiwu Zhao, Wen-Hua Cui, Xiaofei
Yang, Xin Zhou, G.L. Tang, W.L. Liu, Wen Jiang, Zihan Zhang, Xinyang Deng,
Changxing Fan, Sheng Feng, En Fan, Keli Hu,
Xingsen Li, Xin Zhou, Ping Li;
Rajab
Ali Borzooei & Young Bae Jun (neutrosophic BCK/BCI-algebras), Arsham Borumand
Saeid (neutrosophic structures), Saied Jafari (neutrosophic topology), Prem
Kumar Singh (Neutrosophic MCDM), Maikel Leyva-Vazquez (neutrosophic cognitive
maps);
Amira S. Ashour, Muhammad Aslam (neutrosophic statistics), Nguyen
Xuan Thao (neutrosophic similarity measures), Le Hoang Son, Vakkas Ulucay &
Memet Sahin (neutrosophic quadruple structures), Irfan Deli,
Madad Khan (neutrosophic algebraic structures),
Said Broumi & Muhammad Akram
(neutrosophic graphs), Mohamed Abdel-Baset (neutrosophic
linear and non-linear programming), Ahmed Mostafa Khalil,
Ahmed Salama (neutrosophic crisp topology),
etc.
Neutrosophic Sets and Systems (NSS) international journal started in 2013 and it
is indexed by Scopus, Web of Science (ESCI), DOAJ, Index Copernicus, Redalyc -
Universidad Autonoma del Estado de Mexico (IberoAmerica), Publons, CNKI
(Beijing, China), Chinese Baidu Scholar etc. (
http://fs.unm.edu/NSS/ ).
Submit papers on neutrosophic set/logic/probability/statistics and their
applications to the editor-in-chief: smarand@unm.edu.
Encyclopedia of Neutrosophic Researchers
The authors who have published or presented papers on neutrosophics and are not
included in the Encyclopedia of Neutrosophic Researchers (ENR), vols. 1, 2, and
3,
http://fs.unm.edu/EncyclopediaNeutrosophicResearchers.pdf
http://fs.unm.edu/EncyclopediaNeutrosophicResearchers2.pdf
http://fs.unm.edu/EncyclopediaNeutrosophicResearchers3.pdf
are pleased to send their CV, photo, and List of Neutrosophic Publications to
smarand@unm.edu in order to be included into the fourth volume of ENR.
References
University of New Mexico (USA)’s web sites:
http://fs.unm.edu/neutrosophy.htm
http://fs.unm.edu/NSS/Articles.htm
http://fs.unm.edu/CR/CR-Articles.htm