Definition 4.7.6 ([242]) A vertex v in a fuzzy graph G is a complete vertex if the fuzzy subgraph induced by its strong neighbors form a complete fuzzy graph. Theorem 4.7.7 Let G = (V, o, pu) be a complete fuzzy graph.

Author: John N. Mordeson

Publisher: Springer

ISBN: 9783030042158

Category: Technology & Engineering

Page: 209

View: 267

This book builds on two recently published books by the same authors on fuzzy graph theory. Continuing in their tradition, it provides readers with an extensive set of tools for applying fuzzy mathematics and graph theory to social problems such as human trafficking and illegal immigration. Further, it especially focuses on advanced concepts such as connectivity and Wiener indices in fuzzy graphs, distance, operations on fuzzy graphs involving t-norms, and the application of dialectic synthesis in fuzzy graph theory. Each chapter also discusses a number of key, representative applications. Given its approach, the book provides readers with an authoritative, self-contained guide to – and at the same time an inspiring read on – the theory and modern applications of fuzzy graphs. For newcomers, the book also includes a brief introduction to fuzzy sets, fuzzy relations and fuzzy graphs.

Section 3.1 deals with defuzzificaton of three types of fuzzy pre-orders by means of ≽ R. More precisely, we use previous results on pos and negative transitivities of a strict component to prove that a strongly complete fuzzy ...

Author: Janusz Kacprzyk

Publisher: Springer

ISBN: 9783319668246

Category: Technology & Engineering

Page: 628

View: 173

This volume constitutes the proceedings of two collocated international conferences: EUSFLAT-2017 – the 10th edition of the flagship Conference of the European Society for Fuzzy Logic and Technology held in Warsaw, Poland, on September 11–15, 2017, and IWIFSGN’2017 – The Sixteenth International Workshop on Intuitionistic Fuzzy Sets and Generalized Nets, held in Warsaw on September 13–15, 2017. The conferences were organized by the Systems Research Institute, Polish Academy of Sciences, Department IV of Engineering Sciences, Polish Academy of Sciences, and the Polish Operational and Systems Research Society in collaboration with the European Society for Fuzzy Logic and Technology (EUSFLAT), the Bulgarian Academy of Sciences and various European universities. The aim of the EUSFLAT-2017 was to bring together theoreticians and practitioners working on fuzzy logic, fuzzy systems, soft computing and related areas and to provide a platform for exchanging ideas and discussing the l atest trends and ideas, while the aim of IWIFSGN’2017 was to discuss new developments in extensions of the concept of a fuzzy set, such as an intuitionistic fuzzy set, as well as other concepts, like that of a generalized net. The papers included, written by leading international experts, as well as the special sessions and panel discussions contribute to the development the field, strengthen collaborations and intensify networking.

(3) A fuzzy relation u of A into B is called a partial fuzzy function, if Va e A, b, b' e B, u(a,b) = u(a,b) > 0 implies that b = b'. (4) A fuzzy relation u of A into B is called a fuzzy function if it is a complete partial fuzzy ...

Author: John N. Mordeson

Publisher: Springer

ISBN: 9783540371250

Category: Mathematics

Page: 319

View: 515

Lotfi Zadeh introduced the notion of a fuzzy subset of a set in 1965. Ris seminal paper has opened up new insights and applications in a wide range of scientific fields. Azriel Rosenfeld used the notion of a fuzzy subset to put forth cornerstone papers in several areas of mathematics, among other discplines. Rosenfeld is the father of fuzzy abstract algebra. Kuroki is re sponsible for much of fuzzy ideal theory of semigroups. Others who worked on fuzzy semigroup theory, such as Xie, are mentioned in the bibliogra phy. The purpose of this book is to present an up to date account of fuzzy subsemigroups and fuzzy ideals of a semigroup. We concentrate mainly on theoretical aspects, but we do include applications. The applications are in the areas of fuzzy coding theory, fuzzy finite state machines, and fuzzy languages. An extensive account of fuzzy automata and fuzzy languages is given in [100]. Consequently, we only consider results in these areas that have not appeared in [100] and that pertain to semigroups. In Chapter 1, we review some basic results on fuzzy subsets, semigroups, codes, finite state machines, and languages. The purpose of this chapter is to present basic results that are needed in the remainder of the book. In Chapter 2, we introduce certain fuzzy ideals of a semigroup, namely, fuzzy two-sided ideals, fuzzy bi-ideals, fuzzy interior ideals, fuzzy quasi ideals, and fuzzy generalized bi-ideals.

We recall that the notion of fuzzy lattice (more general, fuzzy (sub)algebra) is cutworthy: every cut set of a fuzzy sublattice of M is a crisp sublattice of M. We also use the notion of a complete fuzzy lattice, which we define to be a ...

Author: Aijun An

Publisher: Springer

ISBN: 9783540725305

Category: Computers

Page: 588

View: 627

This book constitutes the refereed proceedings of the 11th International Conference on Rough Sets, Fuzzy Sets, Data Mining, and Granular Computing, RSFDGrC 2007, held in Toronto, Canada in May 2007 in conjunction with the Second International Conference on Rough Sets and Knowledge Technology, RSKT 2007, both as part of the Joint Rough Set Symposium, JRS 2007.

Nair and Cheng for first time [98] defined clique for FG, but, as per their definition fuzzy subgraph induced by a fuzzy clique is not complete [141]. Also, their definition of fuzzy clique fails to draw any connection between fuzzy ...

Author: Madhumangal Pal

Publisher: Springer Nature

ISBN: 9789811588037

Category: Mathematics

Page: 311

View: 771

This book provides an extensive set of tools for applying fuzzy mathematics and graph theory to real-life problems. Balancing the basics and latest developments in fuzzy graph theory, this book starts with existing fundamental theories such as connectivity, isomorphism, products of fuzzy graphs, and different types of paths and arcs in fuzzy graphs to focus on advanced concepts such as planarity in fuzzy graphs, fuzzy competition graphs, fuzzy threshold graphs, fuzzy tolerance graphs, fuzzy trees, coloring in fuzzy graphs, bipolar fuzzy graphs, intuitionistic fuzzy graphs, m-polar fuzzy graphs, applications of fuzzy graphs, and more. Each chapter includes a number of key representative applications of the discussed concept. An authoritative, self-contained, and inspiring read on the theory and modern applications of fuzzy graphs, this book is of value to advanced undergraduate and graduate students of mathematics, engineering, and computer science, as well as researchers interested in new developments in fuzzy logic and applied mathematics.

Thus (A1 UA2, E1 UE2) is a partial fuzzy subgraph of G1 UG2. ... Let A1, A2, E1, E2 be fuzzy subsets of V1,V2, X1, X2, respectively. ... We use the notation Cn(A, E) for a complete fuzzy graph where V = m. Definition 2.15 (A, E) is ...

Author: John N. Mordeson

Publisher: Physica

ISBN: 9783790818086

Category: Mathematics

Page: 314

View: 681

In the mid-1960's I had the pleasure of attending a talk by Lotfi Zadeh at which he presented some of his basic (and at the time, recent) work on fuzzy sets. Lotfi's algebra of fuzzy subsets of a set struck me as very nice; in fact, as a graduate student in the mid-1950's, I had suggested similar ideas about continuous-truth-valued propositional calculus (inffor "and", sup for "or") to my advisor, but he didn't go for it (and in fact, confused it with the foundations of probability theory), so I ended up writing a thesis in a more conventional area of mathematics (differential algebra). I especially enjoyed Lotfi's discussion of fuzzy convexity; I remember talking to him about possible ways of extending this work, but I didn't pursue this at the time. I have elsewhere told the story of how, when I saw C. L. Chang's 1968 paper on fuzzy topological spaces, I was impelled to try my hand at fuzzi fying algebra. This led to my 1971 paper "Fuzzy groups", which became the starting point of an entire literature on fuzzy algebraic structures. In 1974 King-Sun Fu invited me to speak at a U. S. -Japan seminar on Fuzzy Sets and their Applications, which was to be held that summer in Berkeley.

The extension of such a notion to the fuzzy subsets is obvious. Definition 7.4. We say that a fuzzy subset k of S is 1-1-complete if: - k is recursively enumerable - every recursively enumerable fuzzy subset is one-one reducible to k.

Author: G. Gerla

Publisher: Springer Science & Business Media

ISBN: 9789401596602

Category: Mathematics

Page: 271

View: 304

Fuzzy logic in narrow sense is a promising new chapter of formal logic whose basic ideas were formulated by Lotfi Zadeh (see Zadeh [1975]a). The aim of this theory is to formalize the "approximate reasoning" we use in everyday life, the object of investigation being the human aptitude to manage vague properties (as, for example, "beautiful", "small", "plausible", "believable", etc. ) that by their own nature can be satisfied to a degree different from 0 (false) and I (true). It is worth noting that the traditional deductive framework in many-valued logic is different from the one adopted in this book for fuzzy logic: in the former logics one always uses a "crisp" deduction apparatus, producing crisp sets of formulas, the formulas that are considered logically valid. By contrast, fuzzy logical deductive machinery is devised to produce a fuzzy set of formulas (the theorems) from a fuzzy set of formulas (the hypotheses). Approximate reasoning has generated a very interesting literature in recent years. However, in spite of several basic results, in our opinion, we are still far from a satisfactory setting of this very hard and mysterious subject. The aim of this book is to furnish some theoretical devices and to sketch a general framework for fuzzy logic. This is also in accordance with the non Fregean attitude of the book.

Author: Vladimir G. IvancevicPublish On: 2007-04-11

For example, Motorola's 68HC12 MCU features a complete fuzzy–logic function set in assembly language. A slightly finer 2 −→ 1 nonlinear dynamical system would have a generic structure as depicted in Figure 4.6.

Author: Vladimir G. Ivancevic

Publisher: Springer

ISBN: 9783540483960

Category: Computers

Page: 730

View: 535

This book represents a comprehensive introduction into both conceptual and rigorous brain and cognition modelling. It is devoted to understanding, prediction and control of the fundamental mechanisms of brain functioning. The reader will be provided with a scientific tool enabling him or her to perform a competitive research in brain and cognition modelling. This is a graduate–level monographic textbook.

M is a coaccessible fuzzy recognizer & Q = {p e Q|T + (X")"(p) > 0} + Vp e Q, Evo e X", and go e Q such that T(qo). A pl"(po, wo, qo) > 0 <= Vp € Q, ... 7.8 Complete Fuzzy Machines Recall that a fuzzy finite state machine M = (Q, X, p.) ...

Author: John N. Mordeson

Publisher: CRC Press

ISBN: 9781420035643

Category: Computers

Page: 576

View: 276

The huge number and broad range of the existing and potential applications of fuzzy logic have precipitated a veritable avalanche of books published on the subject. Most, however, focus on particular areas of application. Many do no more than scratch the surface of the theory that holds the power and promise of fuzzy logic. Fuzzy Automata and Languages: Theory and Applications offers the first in-depth treatment of the theory and mathematics of fuzzy automata and fuzzy languages. After introducing background material, the authors study max-min machines and max-product machines, developing their respective algebras and exploring properties such as equivalences, homomorphisms, irreducibility, and minimality. The focus then turns to fuzzy context-free grammars and languages, with special attention to trees, fuzzy dendrolanguage generating systems, and normal forms. A treatment of algebraic fuzzy automata theory follows, along with additional results on fuzzy languages, minimization of fuzzy automata, and recognition of fuzzy languages. Although the book is theoretical in nature, the authors also discuss applications in a variety of fields, including databases, medicine, learning systems, and pattern recognition. Much of the information on fuzzy languages is new and never before presented in book form. Fuzzy Automata and Languages incorporates virtually all of the important material published thus far. It stands alone as a complete reference on the subject and belongs on the shelves of anyone interested in fuzzy mathematics or its applications.

LEMMA 4.28 Let T be a Henkin fuzzy theory and r a special constant for (Va.).A. Then T Ha (Va.) A iff T. H. A.[r]. 4.3.6 Complete fuzzy theories Recall that classical theory is complete if every formula or its negation is provable (a ...

Author: Vilém Novák

Publisher: Springer Science & Business Media

ISBN: 9781461552178

Category: Mathematics

Page: 320

View: 701

Mathematical Principles of Fuzzy Logic provides a systematic study of the formal theory of fuzzy logic. The book is based on logical formalism demonstrating that fuzzy logic is a well-developed logical theory. It includes the theory of functional systems in fuzzy logic, providing an explanation of what can be represented, and how, by formulas of fuzzy logic calculi. It also presents a more general interpretation of fuzzy logic within the environment of other proper categories of fuzzy sets stemming either from the topos theory, or even generalizing the latter. This book presents fuzzy logic as the mathematical theory of vagueness as well as the theory of commonsense human reasoning, based on the use of natural language, the distinguishing feature of which is the vagueness of its semantics.