Fuzzy Discrete Structures

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Springer Science & Business Media, Oct 4, 2000 - Business & Economics - 263 pages
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This ambitious exposition by Malik and Mordeson on the fuzzification of discrete structures not only supplies a solid basic text on this key topic, but also serves as a viable tool for learning basic fuzzy set concepts "from the ground up" due to its unusual lucidity of exposition. While the entire presentation of this book is in a completely traditional setting, with all propositions and theorems provided totally rigorous proofs, the readability of the presentation is not compromised in any way; in fact, the many ex cellently chosen examples illustrate the often tricky concepts the authors address. The book's specific topics - including fuzzy versions of decision trees, networks, graphs, automata, etc. - are so well presented, that it is clear that even those researchers not primarily interested in these topics will, after a cursory reading, choose to return to a more in-depth viewing of its pages. Naturally, when I come across such a well-written book, I not only think of how much better I could have written my co-authored monographs, but naturally, how this work, as distant as it seems to be from my own area of interest, could nevertheless connect with such. Before presenting the briefest of some ideas in this direction, let me state that my interest in fuzzy set theory (FST) has been, since about 1975, in connecting aspects of FST directly with corresponding probability concepts. One chief vehicle in carrying this out involves the concept of random sets.
 

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Contents

Fuzzy Logic Functions
1
12 Relations
2
13 Functions
5
14 Fuzzy Sets
7
15 Semigroups
8
16 Fuzzy Logic
12
17 Fuzzy Functions and Decomposition
16
18 Solution of Fuzzy Logic Inequalities
33
44 References
134
Fuzzy Machines Languages and Grammars
139
52 Irreducibility and Minimality
145
53 On Reductions of Maximin Machines
148
54 ContextFree MaxProduct Grammars
165
55 ContextFree Fuzzy Languages
168
56 Deterministic Acceptors of Regular Fuzzy Languages
173
57 Fuzzy Languages on a Free Monoid
179

19 References
40
Decision Trees
41
22 Fuzzy Decision Tree Algorithms
44
23 Analysis of the BBB Algorithm
53
24 References
55
Networks
57
32 A Maximum Flow Algorithm
59
33 The Max Flow Min Cut Theorem
65
34 Maximum Flow in a Network with Fuzzy Arc Capacities
67
35 The Maximum Flow with Integer Values
74
36 Integer Flows in Network with TwoSided Fuzzy Capacity Constraints
76
37 RealValued Flows in a Network with Fuzzy Arc Capacities
83
38 Petri Nets
91
39 Fuzzy Petri Nets for RuleBased Decisionmaking
95
310 References
103
Fuzzy Graphs and Shortest Paths
107
42 Analysis of the Fuzzy Path Models
116
43 On Valuation and Optimization Problems
120
58 Algebraic Character and Properties of FRegular Languages
182
59 References
195
Algebraic Fuzzy Automata
197
62 Homomorphisms
201
63 Admissible Relations
203
64 Fuzzy Transformation Semigroups
205
65 Submachines
209
66 Retrievability Separability and Connectivity
213
67 Decomposition of Fuzzy Finite State Machines
216
68 Admissible Partitions
219
69 On Fuzzy Recognizers
227
610 Minimal Fuzzy Recognizers
239
611 References
245
Appendix
247
Index
255
List of Symbols
261
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About the author (2000)

D.S. Malik is a Professor of Mathematics and the first recipient and current holder of The Frederick H. and Anna K. Scheerer Endowed Chair in Mathematics at Creighton University. Dr. Malik received his Ph.D. from Ohio University in 1985 and has published more than 50 papers and 18 books on abstract algebra, applied mathematics, graph theory, fuzzy automata theory and languages, fuzzy logic and its applications, programming, data structures, and discrete mathematics.