The Fuzzification of Systems: The Genesis of Fuzzy Set Theory and its Initial Applications – Developments up to the 1970s

427 Pages·2007·13.919 MB·English

by  Rudolf Seising (auth.)

RudolfSeising
TheFuzzificationofSystems
StudiesinFuzzinessandSoftComputing, Volume 216
Editor-in-chief
Prof.JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
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Rudolf Seising
The Fuzzification
of Systems
The Genesis of Fuzzy Set Theory
and its Initial Applications –
Developments up to the 1970s
With139Figures
Dr.RudolfSeising
MedicalUniversityofVienna
CoreUnitforMedicalStatistics
andInformatics
Spitalgasse23
1090Vienna
Austria
[e-mail]
OriginallypublishedinGerman“DieFuzzifizierungderSysteme.DieEntstehungderFuzzySetTheorie
undihrererstenAnwendungen-IhreEntwicklungbisindie70erJahredes20.Jahrhunderts”,Boethius,
Band54ISBN:9783515087681,SteinerFranzVerlag,2005.
ThetranslationofthisbookwasundertakenbyGavinBruce(Lawrence,Kansas,USA).
TheprintingofthebookwasfundedbyagrantfromtheKurtVogelFoundation(Munich).
MarekReformat(Edmonton,Canada)kindlyprovidedthephotographusedinthefrontmatter.
LibraryofCongressControlNumber:2007926120
ISSNprintedition:1434-9922
ISSNelectronicedition:1860-0808
ISBN 978-3-540-71794-2SpringerBerlinHeidelbergNewYork
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For Julia
Foreword by Rudolf Kruse
Fuzzy Set Theory is turning 40 this year. I am very pleased that the book
which emerged from Rudolf Seising’s Habilitationsschrift (professorial disser-
tation) is being published in time for this anniversary. I have been working in
the field of fuzzy systems for more than 25 years myself, and during this time
I have been able to witness the development of the Fuzzy Set Theory as an
insider.
Oneareainwhichfuzzymethodsaretypicallyappliediscontrolengineer-
ing. Control systems1 are most often described and developed by means of
mathematical models in the form of systems of differential equations, which
normallyadheretophysicallaws.Acommondifficultyinusingmodelstocre-
atedescriptionsisthefactthatsignificantidealizationsareoftennecessaryin
ordertomovefromaconcreteproblemtoasuitablemathematicalmodel.The
reasonsforthiscanbeattributedtothegenerallyhighcomplexityofthemod-
els or to an incomplete understanding of the problem to be described. Lotfi
A. Zadeh’s idea in the mid-1960s was to develop methods that are frequently
more tolerant than traditional exact methods when it comes to phenomena
such as uncertainty, fuzziness, incomplete information and high complexity.
Thegreatpopularityoffuzzyformsofdescriptionwassparkedbytheamazing
success of fuzzy logic in cybernetics.
The subject of the development of the Fuzzy Set Theory is particularly
exciting to a science historian, especially since most of those involved are
still alive and can be consulted. It is for this purpose that Rudolf Seising
has traveled often to the University of California in Berkeley in recent years
to conduct numerous interviews with Lotfi Zadeh. Zadeh’s little office really
is a treasure trove of science history; it can be difficult just to find one’s
way through the mounds of papers to his desk. Seising was granted access to
previously unknown original sources and reports that cast more light on the
developmentofthefieldoffuzzysystems.Moreover,Dr.Seisinghasutilizeda
1 [400] Michels et al. (2006).
X Foreword by Rudolf Kruse
huge base of sources in the form of original articles and literature in the field
of science history.
Therehasbeenawholeslewofpopularscientificpublicationsaboutthede-
velopmentoftheFuzzySetTheoryoraboutZadehhimself.Examplesinclude
the populist book by Daniel McNeill and Paul Freiberger: Fuzzy Logic: The
Discovery of a Revolutionary Computer Technology – and How it is Chang-
ing Our World2 and the adorable book by Lotfi’s wife Fay Zadeh: My Life
and Travels with the Father of Fuzzy Logic3. To my knowledge, however, the
presentworkisthefirsttreatmentofthehistoryoftheFuzzySetTheoryfrom
a science history perspective.
Particularly interesting is Seising’s incorporation of the history of fuzzy
sets into two developments in the history of sciences and technology. He
demonstrates that electrical engineering, with its focus on information and
communication theory, as well as multi-valued logic were very influential in
the development of the Fuzzy Set Theory.
Until1965,mostscientistssubscribedtothenotionexpressed,forexample,
by Gottlob Frege in the book The Basic Laws of Arithmetic 2, published by
Verlag von Hermann Pohle in Jena, 1903: “... A concept that is not sharply
limitedwouldbeakintoanareathatdidnothaveaconsistentlysharpbound-
aryline,butthatinsomeplaceswascompletelyblurredandcrossedoverinto
its surroundings. That would not actually be an area at all; and thus an un-
sharply defined concept is incorrectly called a concept. Logic cannot accept
concept-like formations such as this; it is impossible to establish precise laws
from them. After all, the law of the excluded middle is actually just another
formoftherequirementthattheconceptbesharplylimited.Anygivenobject
x either falls under the concept y or it does not: tertium non datur...”.
It is clear that the rejection of this principle by Berkeley’s Zadeh, already
internationallyrenownedasanelectricalengineeratthattime,wouldprovoke
passionate reactions in professional circles. Zadeh’s paper Fuzzy Sets, which
appeared in the journal Information and Control4 in 1965, has accordingly
also become an extremely frequently cited article. In my (subjective) view,
this paper was accepted because it was submitted by a professor of electrical
engineering at the university in Berkeley who also happened to be a member
of the editorial board of the journal Information and Control; additionally,
the title Fuzzy Sets was very unusual. The paper contains many concepts
used in later applications, such as fuzzy operators, fuzzy intersections, fuzzy
relations and the combinations of these, as well as convex fuzzy sets. These
concepts are still discussed today in every lecture on the subject of fuzzy
systems.
In this context, it is interesting to see that earlier related approaches to
model imprecision were not accorded the same level of attention. Let me just
2 [370] McNeil et al. Fuzzy (1993).
3 [574] Zadeh F., Life (1998).
4 [612] Zadeh, Fuzzy (1965).
Foreword by Rudolf Kruse XI
mention the paper by the philosopher Max Black, who wrote his article on
Vagueness5 in 1937, and the paper by Łukasiewics, who studied many-valued
systems of propositional logic in 1930.6 In his book, Seising defines Zadeh’s
discrete perception of the theory of fuzzy systems quite conclusively as a
general system theory by means of which so-called input-output analyses of
systems can be carried out.
During the period Seising describes, the exact mathematical foundations
of the methods used in fuzzy systems (such as the precise semantics of de-
grees of membership) were seldom questioned by users in the industry. This
necessarily led to conflicts with proponents of other schools of science. In
particular, there were fervent discussions with researchers from the fields of
(subjective) probability theory and logic, but even classically trained control
engineerswereextremelyskeptical.Itwasparticularlythemathematicalsim-
plicity and tolerance of suboptimal solutions to real problems, qualities that
are appreciated today, that were a thorn in the side of many researchers at
the time.
Seising reports in detail on these (distribution) struggles and, in his dis-
cussion of fuzzy automata, fuzzy algorithms and fuzzy Turing machines, he
introduces three examples of areas in which research was being conducted
during this early stage. Researchers in this field used the principle of fuzzi-
fication, by which any theory involving the concept of sets can be expanded
into a corresponding theory with fuzzy sets: The extension principle as well
as the representation theorem are applied here, and this sometimes results in
interesting generalizations. In 1983 I used this method myself to generalize
concepts from statistics – although at the time, the publisher at Reidel Ver-
lag urged me, for various reasons, to publish the results in a book entitled
Statistics With Vague Data7 and thus at least avoid using the word “Fuzzy”
inthetitle. AsSeisingdescribes,theconceptofthefirstfuzzylogic controller
(a simple characteristic diagram controller) was developed by Mamdani and
his student Assilian over the course of one weekend with the help of a fuzzi-
fication process. This methodology is still routinely used for many simple
feedback control problems today, and even the AG4 automatic transmission
in the Volkswagen New Beetle was developed using this design methodology.
Today fuzzy techniques, especially those logical combinations of different
methods that have been developed in the last few years (such as neuro-fuzzy
systems), are being applied in numerous other technical and non-technical
fields and have meanwhile also become part of a standard education at uni-
versities.
This account of the history of the Fuzzy Set Theory is very well done, in
my opinion. I consider it technically and factually correct.
5 [83] Black, Vagueness (1937).
6 [331] Łukasiewics, Remarks (1930).
7 [300] Kruse et al., Statistics (1987).
XII Foreword by Rudolf Kruse
I hope the book enjoys broad distribution, and not just among historians
of science and technology but also among all scientists and engineers who use
fuzzy methods and have ever asked themselves how this theory of fuzzy sets
came about.
Magdeburg, Germany Rudolf Kruse
June 2006