Fuzzy Discrete Structures

278 Pages·2000·19.416 MB·English

by  Prof. Davender S. Malik, Prof. John N. Mordeson (auth.)

Fuzzy Discrete Structures
Studies in Fuzziness and Soft Computing
Editor-in-chief
Prof. Janusz Kacprzyk
Systems Research Institute
Polish Academy of Sciences
ul. Newelska 6
01-447 Warsaw, Poland
E-mail: [email protected]
http://www.springer.de/cgi-binlsearch_book.pl?series=2941
Vol. 3. A. Geyer-Schulz Vol. 14. E. Hisdal
Fuzzy Rule-Based Expert Systems Logical Structures for Representation of Knowledge
and Genetic Machine Learning, 2nd ed. 1996 and Uncertainty. 1998
ISBN 3-7908-0964-0 ISBN 3-7908-I 056-8
Vol. 4. T. Onisawa and J. Kaeprzyk (Eds.) Vol. 15. G.J. Klir and M.J. Wierman
Uncertainty-Based Information, 2nd ed.. 1999
Reliability and Safety Analyses under Fuzziness.
1995 ISBN 3-7908-1242-0
ISBN 3-7908-0837-7 Vol. 16. D. Driankov and R. Palm (Eds.)
Advances in Fuzzy Control, 1998
Vol. 5. P. Bose and J. Kacprzyk (Eds.) ISBN 3-7908-1090-8
Fuzziness in Database Management Systems. 1995
ISBN 3-7908-0858-X Vol. 17. L. Reznik. V. Dirnitrov and J. Kacprzyk
(Eds.)
Vol. 6. E. S. Lee and Q. Zhu Fuzzy Systems Design. 1998
Fuzzy and Evidence Reasoning. 1995 ISBN 3-7908-1118-1
ISBN 3-7908-0880-6
Vol. 18. L. Polkowski and A. Skowron (Eds.)
Rough Sets in Knowledge Discovery 1. 1998
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Tracing Chains-of-Thought. 1996
ISBN 3-7908-0922-5 Vol. 19. L. Polkowski and A. Skowron (Eds.)
Rough Sets in Knowledge Discovery 2. 1998
Vol. 8. F. Herrera and J. L. Verdegay (Eds.) ISBN 3-7908-1120-3
Genetic Algorithms and Soft Computing. 1996
ISBN 3-7908-0956-X Vol. 20. J.N. Mordeson and P.S. Nair
Fuzzy Mathematics. 1998
Vol. 9. M. Sato et al. ISBN 3-7908-1121-1
Fuzzy Clustering Models and Applications. 1997
Vol. 21. L. C. Jain and T. Fukuda (Eds.)
ISBN 3-7908-1026-6
Soft Computing for Inrelligenr Robotic Systems.
1998
Vol. 10. L. C. Jain (Ed.)
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Soft Computing Techniques in Knowledge-based
Intelligenr Engineering Systems. 1997 Vol. 22. J. Cardoso and H. Camargo (Eds.)
ISBN 3-7908-1035-5 Fuzziness in Petri Nets. 1999
ISBN 3-7908-1158-0
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Fuzzy Logic Techniques in Power Systems. 1998, Vol. 23. P. S. Szczepaniak (Ed.)
ISBN 3-7908-10444 Computational Intelligence and Applications, 1999
ISBN 3-7908-1161-0
Vol. 12. B. Bouchon-Meunier (Ed.) Vol. 24. E. Orlowska (Ed.)
Aggregation and Fusion of Impeifect Information,
Logic at Work, 1999
1998 ISBN 3-7908-1164-5
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Incomplete Information: Rough Set Analysis. 1998 1999
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continued on page 264
Davender S. Malik
John N. Mordeson
Fuzzy
Discrete Structures
With 37 Figures
and 7 Tables
Springer-Verlag Berlin Heidelberg GmbH
Prof. Davender S. Malik
Prof. John N. Mordeson
Creighton University
Omaha, Nebraska 68178
USA
E-mail: [email protected]
[email protected]
ISBN 978-3-7908-2477-3 ISBN 978-3-7908-1838-3 (eBook)
DOI 10.1007/978-3-7908-1838-3
Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Malik, Davender S.: Fuzzy discrete structures: with 7 tables I Davender S. Malik;
John N. Mordeson. - Heidelberg; New York: Physica-Verl., 2000
(Studies in fuzziness and soft computing; Vol. 58)
1bis work is subject to copyright. All rights are reserved, whether the whole or
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© Springer-Verlag Berlin Heidelberg 2000
Originally published by Physica-Verlag Heidelberg New York in 2000
Softcover reprint of the hardcover 1st edition 2000
The use of general descriptive names. registered names. trademarks. etc. in this
publication does not imply. even in the absence of a specilic statement, that such
names are exempt from the relevant protective laws and regulations and therefore
free for general use.
Hardcover Design: Erich Kirchner. Heidelberg
To
Lotfi A. Zadeh
Foreword
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. At the outset, a ba
sic interesting point (no proverbial pun intended!) connected with random
sets - or rather subsets of a given domain - is that while all of the ordinary
laws of probability are obeyed and formally all one is doing is replacing in
"ordinary" probability evaluations "points" (or vectors) by sets of points
Vlll Foreword
(or vectors) in the randomization process, something very profound has
taken place at even the simple finite domain cases: While the probabilities
of the distinct outcomes of the random set in question must add up to
unity, unlike our normal image of a probability function over that domain
representing an ordinary random variable over same domain, the distinct
outcomes, i.e., subsets of the domain, need not be disjoint, unlike our image
of points. Of course, if we were to interpret such point as really representing
possibly overlapping concepts as random sets do in general, without vio
lating probability constraints, the same idea would unfold. But, the human
side of this traditionally has been that when one sees a probability function
- and equivalently, a discrete-valued random variable (or random vector)
- one almost invariably thinks of complete disjointness. (Parenthetically,
it would be interesting to see if any associated psychological studies have
ever been initiated in this direction concerning our intuitive concepts of
probability for random sets.) In turn, the very overlapping of the outcomes
of a random set allow for a richer expression of probability concepts than
just a point-limited random variable over the same domain.
In a related vein, certain types of possible overlap of random sets with
fixed (but arbitrary) given sets from the same domain serves as a basis
for the basic functions of Dempster-Shafer Theory - belief (superset re
lation), plausibility (incidence relation), doubt, and a fourth not usually
named (each representing also the four basic Choquet infinite capacities
- see Nguyen's now classic exposition of 1978, pp. 531-542 in the Jour
nal of Mathematical Analysis fj Applications). In particular, for the fixed
sets restricted to be singletons from the domain, when overlaps occur they
represent one-point coverage events generated by the effect of the ran
dom set covering such given points. Then, somewhat analogous to the way
Dempster-Shafer Theory can be explained via random sets acting upon
fixed sets, one can show that a fuzzy set (in the first order sense with mem
bership function having range only in the unit interval) can be likewise in
terpreted as the action of a random set covering individual specified points
of the fuzzy set's domain. In general, unlike the Dempster-Shafer situation,
there may be many such (even an infinity of truly different) random sets
providing such action resulting in the same fuzzy set representation. On
the other hand, when the fuzzy set is actually a crisp set, the random set is
uniquely determined as the constant crisp set itself. In fact, in general, each
possible truly distinct random set representing a given fuzzy set in the above
one-point coverage way is completely determine by both the given fuzzy set
and a particular (but arbitrary) choice of copula, co-copula pair. Moreover,
while even the algebraic structure of Zadeh's classical min-max-l-(.) logi
cal operators over fuzzy sets is not equivalent isomorphically to that of the
ordinary (boolean set) algebraic structure of sets - and hence random sets
and their associated ordinary events, including one-point coverage events -
certain localized equivalences can be obtained, including "isomorphic-like"
correspondences between a number of fuzzy logic concepts and one-point
Foreword IX
coverage events. For example, Zadeh's extension principle couched in gen
eral co-copula form (extending max) applied to some crisp function g with
arguments described by fuzzy set f can be interpreted as the one-point cov
erage action of the functional image under g of the random set determined
by f and the copula-cocopula pair chosen on singletons of the domain of
f.
For further information on these ideas, see, e.g., the articles by Goodman
& Nguyen, "Applications of conditional and relational event algebra to the
s.
defining of fuzzy logic concepts", Froc. P. I.E. Signal Processing, Sensor
Fusion f3 Target Recognition VIII (vol. 3720), Orlando, FL, April, 1999,
pp. 25-36, or Goodman & Kramer, "Extension of rational and conditional
event algebra to random sets with applications to data fusion", in Random
Sets: Theory f3 Applications (J. Goutsias, R.P. Mahler & H.T. Nguyen,
eds.) Springer-Verlag, New York, 1997, pp. 209-242.)
Finally, with some of the basic ideas outlined above relating fuzzy set
concepts with corresponding random set ones, one may ask whether any
natural connections can be established between the various fuzzified crisp
concepts analyzed in Fuzzy Discrete Structures and naturally correspond
ing random set concepts. For example, section (4) on fuzzy graphs and
shortest path depends obviously on how one carries out the fuzzification
of graphs and subgraphs, as well as the valuation of the arcs of a fuzzy
graph. While each of these concepts has a rote one-point coverage random
set representation, the challenge is to determine whether a nontrivial cor
respondence exists between the theorems and procedures associated with
these fuzzy concepts and natural crisp counterparts applied to random sets,
as in the case of the extension principle. If such correspondences could be
found for either the currently used or suitably modified definitions, then an
even stronger case can be made for justifying the very direction of fuzzifi
cation of the crisp concepts involved. Much work remain in this essentially
unexplored area.
LR. Goodman
Space & Naval Warfare Systems Center
San Diego, CA 92152
Preface
In 1965, L.A. Zadeh introduced the concepts of a fuzzy subset of a set as
a way for representing uncertainty. Zadeh's ideas stirred researchers world
wide. His ideas have been applied to a wide range of scientific areas. Here
we consider the area of mathematics which is normally known as discrete
structures. Since texts on discrete structures cover a wide variety of sub
jects, the topics appearing in such books do not appear in the depth they
would in books specializing in these topics. This book deals with fuzzy
logic, fuzzy switching functions, fuzzy decision trees, fuzzy networks, fuzzy
Petri nets, fuzzy path problems, fuzzy automata, and fuzzy languages. As
in crisp books on discrete structures, the topics in this book do not appear
in the depth they would in books specializing in these topics. The book
should be of interest to research mathematicians, engineers, and computer
scientists interested in applications.
In Chapter 1, we present just enough material on fuzzy logic to study
fuzzy switching functions and their decomposition. We concentrate on the
minimization of completely specified fuzzy functions. We touch briefly on
the minimization of incompletely specified fuzzy functions, leaving the in
terested reader to pursue the issue elsewhere, say in Kandel and Lee, [4].
The chapter concludes with a section on the solution of fuzzy logical in
equalities.
Chapter 2 is concerned with fuzzy decision trees. We begin the chapter
with a review of crisp decision trees. We introduce the idea of a fuzzy
decision tree and present a branch-bound-backtrack algorithm which has
an effective backtracking mechanism leading to the optimal solution while