Intelligent Counting Under Information Imprecision: Applications to Intelligent Systems and Decision Support

294 Pages·2013·4.495 MB·English

by  Maciej Wygralak (auth.)

Studies in Fuzziness and Soft Computing
Maciej Wygralak

Intelligent Counting
under Information
Imprecision
Applications to Intelligent Systems
and Decision Support
Studies in Fuzziness and Soft Computing 292
Editor-in-Chief
Prof.JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
01-447Warsaw
Poland
E-mail:[email protected]
Forfurthervolumes:
http://www.springer.com/series/2941
Maciej Wygralak
Intelligent Counting
under Information
Imprecision
Applications to Intelligent Systems
and Decision Support
ABC
MaciejWygralak,Assoc.Prof.
FacultyofMathematicsandComputerScience
AdamMickiewiczUniversity
Umultowska87
61-614Poznan´
Poland
E-mail:[email protected]
ISSN1434-9922 ISSN1860-0808 (electronic)
ISBN978-3-642-34684-2 ISBN978-3-642-34685-9 (eBook)
DOI10.1007/978-3-642-34685-9
SpringerHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2013937947
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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To my four ladies:
Renata, Karolina, Agata, and Alicja
Preface
This monograph begins with a presentation of the concept of and selected issues in
fuzzy sets and fuzzy logic, interval-valued fuzzy sets, and I-fuzzy sets - Atanassov’s
intuitionistic fuzzy sets. However, that study is only of preparatory character. The
main subject of the book will be intelligent counting under imprecision of information
(about the objects of counting). Why is this worth investigating and deliberating?
It seems that counting belongs to the most basic and frequent mental activities of
human beings as its results are a basis for coming to a decision in a lot of situations.
One should distinguish, however, between two very different cases occurring in
practice. First, the objects of counting can be precisely specified and, then, the
counting process collapses to the trivial task of counting in a set by means of the
natural numbers. Second, those objects can be imprecisely (fuzzily) specified and just
this much more advanced and sophisticated case of counting requiring intelligence
will be the subject of our investigations. Speaking formally, that intelligent counting
collapses to counting in a fuzzy set or - whenever imprecision is combined with
incompleteness of information - to counting in an interval-valued fuzzy set or I-fuzzy
set. Theoretical aspects as well as applications of intelligent counting will be
discussed. Especially, we mean applications to intelligent systems and decision
support. The emphasis will be on showing that the presented methods of intelligent
counting and the resulting cardinalities are human-consistent, i.e. are reflections and
formalizations of real, human counting methods under imprecision possibly combined
with incompleteness of information. It is self-evident that our main interest will be in
counting in finite fuzzy sets, finite interval-valued fuzzy sets and I-fuzzy sets.
Nevertheless, for completeness, the infinite case will be concisely discussed, too.
The monograph is divided into two parts and eleven chapters. The first one is of
introductory character, whereas Chapters 2-6, forming Part I, are devoted to those
elements of fuzzy sets and fuzzy logic which are relevant from the viewpoint of the
main aim of this book. We will present operations on and basic characteristics of
fuzzy sets, negations, triangular norms and conorms, fuzzy numbers and linguistic
variables, aggregation operators, fuzzy relations, and an introduction to approximate
reasoning and fuzzy rule-based systems. Moreover, interval-valued fuzzy sets and
I-fuzzy sets will be discussed as tools for modeling incompletely known fuzzy sets.
Chapters 7-11 constitute Part II, the key part of this book, devoted to methods of
intelligent counting and related cardinalities of fuzzy sets, interval-valued fuzzy sets
and I-fuzzy sets. Both the scalar and fuzzy approaches to these questions will be
discussed in detail, including human-consistency and giving the reader a novel and
up-to-date image of the subject matter.
The presentation is self-contained as much as possible and equipped with many
figures, examples, and references to the source literature. Our general principle is
“motivations and ideas before technical details”. Nevertheless, the reader should have
viii Preface
some basic knowledge of mathematics with special reference to set theory, mathe-
matical logic, analysis, and general algebra.
This monograph is intended for computer and information scientists, researchers,
engineers and practitioners, applied mathematicians, and postgraduate students
interested in dealing with information imprecision and incompleteness.
The most pleasant moment of each book project is to write acknowledgments. I
am grateful to all who supported me. Especially, I would like to thank my wife Renata
and our daughters Karolina and Agata for their continuous and reliable support,
understanding and patience.
As to the financial side, this book project has been partially supported by a
research grant from the National Science Centre (NCN), and this support is greatly
appreciated.
Last but not least, let me thank my collaborators, Dr. Krzysztof Dyczkowski and
Dr. Anna Stachowiak, for kind technical assistance and valuable suggestions.
Maciej Wygralak
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Information Imprecision and Fuzzy Sets . . . . . . . . . . . . . . 1
1.2 Adding the Incompleteness Factor . . . . . . . . . . . . . . . . . 7
1.3 Counting under Information Imprecision . . . . . . . . . . . . . 9
Part I Elements of Fuzzy Sets and Their Extensions
2 Basic Notions of the Language of Fuzzy Sets . . . . . . . . . . . . . 19
2.1 What Are Fuzzy Sets? . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.1 The Concept of a Fuzzy Set . . . . . . . . . . . . . . . . 19
2.1.2 Examples and Interpretations . . . . . . . . . . . . . . . . 21
2.1.3 Remarks on Many-Valued Roots of Fuzzy Sets . . . . . . 25
2.2 Operations on Fuzzy Sets ! The Standard Approach . . . . . . . 28
2.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . 28
2.2.2 Properties of Operations . . . . . . . . . . . . . . . . . . 30
2.2.3 Looking through Many-Valued Logic . . . . . . . . . . . 32
2.3 Main Characteristics of Fuzzy Sets . . . . . . . . . . . . . . . . 33
2.3.1 Core, Support, t-Cuts . . . . . . . . . . . . . . . . . . . . 33
2.3.2 Decompositions and Maps of Fuzzy Sets . . . . . . . . . . 35
2.3.3 Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.3.4 Fuzziness Measures . . . . . . . . . . . . . . . . . . . . . 40
2.4 Flexible Framework for Operations on Fuzzy Sets . . . . . . . . 45
2.4.1 Negations . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4.2 Complements Based on Negations . . . . . . . . . . . . . 48
2.4.3 Triangular Norms . . . . . . . . . . . . . . . . . . . . . . 49
2.4.4 Operations Based on Triangular Norms and Conorms . . . 55
2.4.5 Implication Operators . . . . . . . . . . . . . . . . . . . . 57
2.4.6 Logical Background . . . . . . . . . . . . . . . . . . . . 59
2.5 Fuzzy Numbers and Linguistic Variables . . . . . . . . . . . . . 59
2.5.1 Fuzzy Numbers and Their Types . . . . . . . . . . . . . . 60
2.5.2 The Extension Principle and Operations on Fuzzy Numbers 63
2.5.3 Comparisons of Fuzzy Numbers . . . . . . . . . . . . . . 66
2.5.4 Linguistic Variables . . . . . . . . . . . . . . . . . . . . . 68
x Contents
3 Further Aspects of Triangular Norms - A Study Inspired
by Flexible Querying . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.1 Flexible Querying in Databases . . . . . . . . . . . . . . . . . . 71
3.1.1 Constructing the Answer to a Flexible Query . . . . . . . 72
3.1.2 Unequally Important Elementary Conditions . . . . . . . . 76
3.2 The Case of Hotel Simenon ! More Advanced Aspects of
Triangular Norms. . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.2.1 Classes of Triangular Norms . . . . . . . . . . . . . . . . 79
3.2.2 Continuous and Archimedean Triangular Norms . . . . . . 81
3.2.3 Generators . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.2.4 Induced Negations and Complementarity . . . . . . . . . 87
4 Aggregation of Information and Aggregation Operators. . . . . . . 93
4.1 The Issue of Information Aggregation . . . . . . . . . . . . . . . 93
4.2 Aggregation Operators . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Aggregation Operators Involving Triangular Norms and Conorms 95
4.3.1 Compensatory Operators . . . . . . . . . . . . . . . . . . 95
4.3.2 Soft Triangular Norms and Conorms . . . . . . . . . . . . 96
4.4 Averaging Operators . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.1 Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4.2 Weighted Means . . . . . . . . . . . . . . . . . . . . . . 98
4.4.3 OWA Operators . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Conclusions and Systematization . . . . . . . . . . . . . . . . . 100
4.6 Applications to Decision Making in a Fuzzy Environment . . . . 101
4.6.1 Bellman-Zadeh Model . . . . . . . . . . . . . . . . . . . 102
4.6.2 Computational Examples . . . . . . . . . . . . . . . . . . 103
4.7 Two Issues Related to Aggregation . . . . . . . . . . . . . . . . 105
4.7.1 Mean Values of Aggregations . . . . . . . . . . . . . . . 105
4.7.2 Averaging with Respect to Triangular Norms and Conorms 106
5 Fuzzy Relations, Approximate Reasoning,
Fuzzy Rule-Based Systems . . . . . . . . . . . . . . . . . . . . . . . 111
5.1 Fuzzy Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.1 The Concept of a Fuzzy Relation . . . . . . . . . . . . . . 111
5.1.2 Composition of Fuzzy Relations . . . . . . . . . . . . . . 114
5.1.3 Types of Fuzzy Relations . . . . . . . . . . . . . . . . . . 115
5.1.4 Similarity Measures and Similarity Classes . . . . . . . . 116
5.1.5 Cardinality-Based Similarity Measures for Sets . . . . . . 119
5.1.6 Inclusion and Equality Measures for Fuzzy Sets . . . . . . 121
5.2 Approximate Reasoning ! Basic Issues . . . . . . . . . . . . . . 125
Contents xi
5.3 Fuzzy Control and Fuzzy Rule-Based Systems . . . . . . . . . . 128
5.3.1 Computational Approach to Fuzzy Rules . . . . . . . . . . 128
5.3.2 Fuzzy Controller . . . . . . . . . . . . . . . . . . . . . . 130
5.3.3 How Does It Work? . . . . . . . . . . . . . . . . . . . . . 133
6 Modeling Incompletely Known Fuzzy Sets . . . . . . . . . . . . . . 139
6.1 Incompletely Known Sets and Their Modeling . . . . . . . . . . 139
6.2 Interval-Valued Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . 142
6.2.1 The Concept of an Interval-Valued Fuzzy Set . . . . . . . 143
6.2.2 General Properties of Uncertainty Degrees . . . . . . . . . 144
6.2.3 Operations on IVFSs . . . . . . . . . . . . . . . . . . . . 145
6.3 I-Fuzzy Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
6.3.1 Basic Notions . . . . . . . . . . . . . . . . . . . . . . . . 148
6.3.2 General Properties of Hesitation Degrees . . . . . . . . . 153
6.3.3 Operations on IFSs . . . . . . . . . . . . . . . . . . . . . 154
6.3.4 Model Examples . . . . . . . . . . . . . . . . . . . . . . 156
Part II Methods of Intelligent Counting under
Information Imprecision
7 General Remarks and Motivations . . . . . . . . . . . . . . . . . . 163
8 Scalar Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
8.1 Sigma f-Counts and Counting in Fuzzy Sets . . . . . . . . . . . 167
8.1.1 Sigma f-Counts . . . . . . . . . . . . . . . . . . . . . . . 167
8.1.2 Main Weighting Functions and Cases of Sigmaf-Counts . 170
8.1.3 The Eight-Bottle Example Continued . . . . . . . . . . . 173
8.2 Arithmetic Aspects . . . . . . . . . . . . . . . . . . . . . . . . . 175
8.2.1 Addition of Sigma f-Counts . . . . . . . . . . . . . . . . 176
8.2.2 Multiplication . . . . . . . . . . . . . . . . . . . . . . . . 178
8.3 Relative Cardinality . . . . . . . . . . . . . . . . . . . . . . . . 179
8.4 Counting in IVFSs and IFSs . . . . . . . . . . . . . . . . . . . . 180
8.4.1 Main Properties . . . . . . . . . . . . . . . . . . . . . . . 181
8.4.2 Relativization . . . . . . . . . . . . . . . . . . . . . . . . 183
8.4.3 The Eight-Bottle Example Once Again . . . . . . . . . . . 185
9 Fuzzy Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.1 Related Methods of Counting in Fuzzy Sets . . . . . . . . . . . . 187
9.1.1 MCAC and the Basic Fuzzy Count . . . . . . . . . . . . . 188
9.1.2 FGCount and FLCount . . . . . . . . . . . . . . . . . . . 193
9.1.3 FECount . . . . . . . . . . . . . . . . . . . . . . . . . . 198
9.1.4 FECount and Classification . . . . . . . . . . . . . . . . . 201