Fuzzy Implications

310 Pages·2008·8.901 MB·English

by  Michał Baczyński, Balasubramaniam Jayaram (auth.)

MichałBaczyn´skiandBalasubramaniamJayaram
FuzzyImplications
Studiesin Fuzzinessand Soft Computing,Volume231
Editor-in-Chief
Prof.JanuszKacprzyk
SystemsResearchInstitute
PolishAcademyofSciences
ul.Newelska6
01-447Warsaw
Poland
E-mail:[email protected]
Furthervolumesofthisseriescanbefoundonourhomepage:springer.com
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Vol.217.MasoudNikravesh,JanuszKacprzyk, ISBN978-3-540-76972-9
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ForgingNewFrontiers:Fuzzy
JohnN.Mordeson,JoshuaD.Potter,
PioneersI,2007
MarkJ.Wierman
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ForgingNewFrontiers:Fuzzy Vol.226.BhanuPrasad(Ed.)
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Vol.219.RolandR.Yager,LipingLiu(Eds.) Vol.227.EugeneRoventa,TiberiuSpircu
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ofBeliefFunctions,2007 BuildingIntelligentSystems,2008
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Vol.228.AdamKasperski
Vol.220.HumbertoBustince,
DiscreteOptimizationwithIntervalData,2008
FranciscoHerrera,JavierMontero(Eds.)
ISBN978-3-540-78483-8
FuzzySetsandTheirExtensions:
Representation,AggregationandModels,2007 Vol.229.SadaakiMiyamoto,
ISBN978-3-540-73722-3 HidetomoIchihashi,KatsuhiroHonda
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Vol.222.JamesJ.Buckley,
Vol.231.MichałBaczyn´ski,
LeonardJ.Jowers
BalasubramaniamJayaram
MonteCarloMethodsinFuzzy
FuzzyImplications,2008
Optimization,2008
ISBN978-3-540-69080-1
ISBN978-3-540-76289-8
Michał Baczyn´ski and
Balasubramaniam Jayaram
Fuzzy Implications
ABC
Authors
Dr.MichałBaczyn´ski
UniversityofSilesia
InstituteofMathematics
ul.Bankowa14
40-007Katowice
Poland
E-mail:[email protected]
Dr.BalasubramaniamJayaram
DepartmentofMathematicsandComputerSciences
SriSathyaSaiUniversity
PrasanthiNilayam
AnantpurDistrict
AndhraPradesh-515134
India
E-mail:[email protected]
ISBN978-3-540-69080-1 e-ISBN978-3-540-69082-5
DOI10.1007/978-3-540-69082-5
StudiesinFuzzinessandSoftComputing ISSN1434-9922
LibraryofCongressControlNumber:2008928273
(cid:2)c 2008Springer-VerlagBerlinHeidelberg
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To my wife Agnieszka and
our daughters Dominika and Justyna
with love.
Micha(cid:2)l
To Bhagwan Sri Sathya Sai Baba
and His University.
Balasubramaniam
Preface
Nothing new had been done in Logic since Aristotle!
– Kurt Go¨del (1906-1978)
Fuzzyimplicationsareoneofthemainoperationsinfuzzylogic.Theygeneralize
the classical implication, which takes values in {0,1}, to fuzzy logic, where the
truth values belong to the unit interval [0,1]. In classical logic the implication
canbedefinedindifferentways.Threeofthesehavecometoassumegreaterthe-
oreticalimportance,viz.theusualmaterialimplicationfromtheKleenealgebra,
the implication obtained as the residuum of the conjunction in Heyting algebra
(also called pseudo-Boolean algebra) in the intuitionistic logic framework and
the implication (also called as ‘Sasaki arrow’) in the setting of quantum logic.
Interestingly, despite their differing definitions, their truth tables are identical
inclassicalcase.However,the naturalgeneralizationsofthe abovedefinitions in
the fuzzy logic framework are not identical. This diversity is more a boon than
a bane and has led to some intensive researchon fuzzy implications for close to
three decades.It will be our endeavorto coverthe variousworks churnedout in
this period to sufficient depth and allowable breadth in this treatise.
IntheforewordtoKlirandYuan’sbook[147],ProfessorLotfiA.Zadehstates
the following:
“The problem is that the term ‘fuzzy logic’ has two different meanings.
More specifically, in a narrow sense, fuzzy logic, FL , is a logical sys-
n
temwhichmaybe viewedasanextensionandgeneralizationofclassical
multivalued logics. But in a wider sense, fuzzy logic, FL , is almost
w
synonymous with the theory of Fuzzy Sets. In this context, what is im-
portant to recognize is that:
a) FL is much broader than FL and subsumes FL as one of its
w n n
branches;
b) the agenda of FL is very different from the agendas of classical
n
multivalued logics; and
c) atthisjuncture,thetermfuzzylogicisusuallyusedinitswiderather
than the narrowsense, effectively equating Fuzzy Logic with FL .”
w
Since a fuzzy implication is a generalization of the classical (Boolean) implica-
tion to the unit interval [0,1], its study predominantly belongs to the domain
of FL . Keeping the above concern and the subsequent distinction, we have
n
VIII Preface
made asincereattempt todiscuss fuzzy implicationsinthe contextofFL too,
w
vis-´a-vis the role of fuzzy implications in potential applications. Moreover, the
organization of the book is made to reflect the above philosophy. This treatise
has been divided into the following 3 parts:
Part I Analytical Study of Fuzzy Implications;
Part II Algebraic Study of Fuzzy Implications;
Part III Applicational Study of Fuzzy Implications.
Chap. 1 introduces the readers to fuzzy implications. After defining a fuzzy
implication,wepresentsomeofthemostdesirablepropertiesthatafuzzyimpli-
cation can possess and discuss their interrelationships. Based on this study we
list out, what we consider, as 9 basic fuzzy implications.
Part I, consisting of Chaps. 2-5, deals with the analytical aspects of fuzzy
implications, viz., different approaches to generating fuzzy implications from
other fuzzy logic operations,the families obtained throughsuch processes,their
properties,representations,characterizationsand the overlapsthat exist among
these families.
PartII,consistingofChaps.6and7,dealswiththealgebraicaspectsoffuzzy
implications,viz.,differentapproachestogeneratingfuzzyimplicationsfromex-
isting fuzzy implications, the closure of such processes, the mathematical struc-
tures that can be imposed on different classes or families of fuzzy implications
and the functional equations/tautologies they satisfy.
While Parts I and II can be seen as dealing with fuzzy implications in the
setting of FL , Part III, consisting of Chap. 8, is an ‘applicational’ study of
n
fuzzy implications, wherein we deal with the effecting role that fuzzy implica-
tions play in approximate reasoning(AR) and fuzzy control(FC) which occupy
the center stage in the framework of FL . Towards helping the readers better
w
appreciatethisfacetoffuzzyimplications,abasicoverwiewofARisgiven,with
particular emphasis on two of the most important inference schemes therein,
namely, Compositional Rule of Inference and Similarity Based Reasoning. Fol-
lowing this, we investigate how different families of fuzzy implications with the
myriad properties they possess and tautologies they satisfy influence both the
performance and functioning of the above schemes of inference.
This is probably the first comprehensive treatise on fuzzy implications. The
terms first and comprehensive may seem contradictory and mutually exclusive.
We should confess that it was for the most part during the writing of the book.
Thechallengeofachievingthistwin,butseeminglyconflictinggoals,hasspurred
us to put inour sincereefforts inmaking this up-to-date withallthe latestrele-
vant results, complete with proofs, examples and counter-examples to highlight
the concepts and plots of different operations for better illustration. Eachchap-
terisprecededbyashortsummaryandendswithsomebibliographicalremarks
on the contents of the chapter, viz., historical details, the context in which the
operations were proposed and their subsequent line of analysis.
However, it should be noted that in this treatise we have dealt only with
the classical approaches to fuzzy implications, owing mainly to keep the size of
this monographto amanageableextent.Forexample,studieson interval-valued
Preface IX
and intuitionistic fuzzy implications (see Bustince et al. [44], Cornelis et al.
[61, 64], Deschrijver and Kerre [74, 75]) and those that deal with fuzzy
implications where the underlying set is any discrete and/or finite (see Mas et
al.[172,173,177],MayorandTorrens[178])havenotbeenincludedindetail
inthistreatise.Fromthe applicationalaspectsoffuzzyimplications,inPartIII,
we have covered only their influence in approximate reasoning, while there are
many other areas wherein they wield a dominant influence - see, for example,
the excellent survey of Mas et al. [176]. Another notable omission is the role
of fuzzy implications in fuzzy logics, in the sense of FL , viz., the Basic-Logic
n
(BL) of Ha´jek [119] which was shown to be the logic of continuous t-norms
and their residua by Cignoli et al. [53], the Monoidal T-norm Logic (MTL)
of Esteva and Godo [97] - which was shown to be the logic of left-continuous
t-norms and their residua - and later on extended to Involutive MTL, WNML
and NML in Esteva et al. [98, 96], etc. In this case, the predominant reason
being the availability of many excellent treatises on such topics, e.g., Cignoli
et al. [55], Gottwald [113], Ha´jek [120], Turunen [243].
This book is intended for any researcher in fuzzy logic operations and it can
serveasanauxiliarytextbookfordifferentcoursesonfuzzylogicorfuzzycontrol.
It does not assume any special background in fuzzy set theory, but the reader
shouldhavesome knowledgein analysis,algebraandclassicallogic(onthe level
of a graduate student).
Parts of this book may also be of interest to practitioners in fuzzy logic ap-
plications, especially in approximate reasoning or fuzzy control, where fuzzy
implications play a central role.
Acknowledgments
Itissaidthat“Nobookiswritteninisolation”.Thisbookisanexceptiontothis
rule,inthat,mostpartsofthisbookwaswrittenwhenoneoftheauthorswasin
PolandwhiletheotherwasinIndia-surelyisolationcannotgetanyfartherthan
this.Itisimmediatelyobviousthattosurmountthispart-challenge-part-obstacle
of geographicaldistance the help and benevolence of many were involved.
Firstonourlistareourrespectivesupervisors,Prof.Jo´zefDrewniakandProf.
C. Jagan Mohan Rao, who magnanimously gave their time, energy and knowl-
edge that have opened many vistas, plugged many gaps in the various topics,
shapedthesubsequentorganizationandcontentsofthistreatise.Followingthem
areourcolleaguesandfellowresearchersDr.Pawe(cid:4)lDryga´s,Dr.UrszulaDudziak,
MgrAnnaKro´landDr.MartinS˘t˘epni˘ckawhohavepatientlyreadmanypartsof
the variousversionsofour draftandwhose suggestionshavemade aremarkable
difference in the final outcome. Thank you, all.
We thank the constant support extended to us by our host universities, Uni-
versity of Silesia, Poland and Sri Sathya Sai University, India, respectively, and
the colleagues at our respective departments for the excellent ambience they
provided, without which a venture of this sort would have been difficult to un-
dertake.
While the gaps in the academic content were narrowed with the help of the
above, the geographical distance was bridged by Google. The authors would
X Preface
like to gratefully acknowledgethe excellent voice-chattool ‘Google Talk’, which
broughtustohearing-distancewitheachother.Thecountlesshoursofdiscussion
we have had online with the help of GTalk is a testimony to the power of the
Internet in conquering distances.
This geographical gap finally vanished in the final stages of this venture,
thanks to the SAIA scholarship under the National Scholarship Program of
the Slovak Republic, which made it possible for Balasubramaniam Jayaram to
traveltoBratislava,Slovakiawheremanyimportantdecisionsweremadeandthe
book finalized inthis currentform.BalasubramaniamJayaramsincerely thanks
SAIAandProf.RadkoMesiarofDMDG,FCE,SlovakUniversityofTechnology,
Bratislava for graciously hosting him during this period.
While writing this monograph we have also employed and gained familiarity
with a number of fantastic tools. This text was typeset entirely in LATEX2e and
all the 3-d plots were made in Matlab 7.0. We have still not ceased to wonder
at these magical tools and marvel at how the lifeless characters that we type
suddenlymetamorphoseintosuchbreath-takingsymbolsandplots.Isthiswhere
science meets art?
No acknowledgement section in a book is deemed complete without the au-
thors’part-apologetic,part-warningbut neverthelessproudproclamationofthe
errors that they have managed to retain in their book. We would like to part
with this tradition with the supreme knowledge that errors are endowed with a
life-expectancy thatsurvivesany editorialvettingandhaveanuncannysense of
direction to creep seemingly innocuously, in the least obtrusive way, into that
part of the text where it lies hidden only to leap into the full view of all the
readers, especially the reviewer and except the writer, causing maximum catas-
trophe.To errmay not be the right ofthe authors,but to ignore it is left to the
readers.
Katowice, Puttaparthi Michal(cid:4) Baczyn´ski
June 2008 Balasubramaniam Jayaram
Notations and Some Preliminaries
Mathematics is written for mathematicians.
– Nicholaus Copernicus (1473-1543)
In this book we have tried to employ notations that are generally used in the
mathematical literature. For the classical logical operations like conjunction,
disjunction,negationandimplicationwewrite∧,∨,¬and→,respectively.IfA
is a subset of B, where A = B is possible, we denote this by A⊆ B or B ⊇ A.
If A is a subset of, but not equal to, B, then A is called a proper subset of
B, written A (cid:2) B or B (cid:3) A. The union, the intersection and the difference of
two sets A and B are denoted by A∪B, A∩B and A\B, respectively. The
complement of a subset A⊂X is denoted by A(cid:2). The Cartesian product of two
sets A and B is denoted by A×B. The cardinality of a set A is denoted by
card A. The symbol N denotes the set of positive integers, Z denotes the set of
allintegersandR standsforthe setofallrealnumbers.MoreoverN =N∪{0}.
0
If f: X → Y is a function, then X is called the domain of f denoted by
Dom(f) and the setY is called the codomain. The range of f is given by
Ran(f)={f(x)|x∈X}. If A⊆ Dom(f), then f(A) is the subset of the range
consisting of all images of elements of A, i.e., f(A) = {f(x) ∈ Y |x ∈ A}. The
preimage(orinverseimage)ofasubsetBofthecodomainY underafunctionf is
thesubsetofthedomainX definedbyf−1(B)={x∈X|f(x)∈B}.Thecompo-
sitionoftwofunctionsf: X →Y andg: Y →Z isgivenby(g◦f)(x)=g(f(x))
for all x ∈ X. The identity function idX: X → X is defined by idX(x) = x for
all x ∈ X. If f is a function from X to Y and A is any subset of X, then the
restriction of f to A is denoted by f|A = f ◦idA. Let F: X ×Y → Z be a
function of two variables. Then for each fixed x ∈ X, the vertical section of F
0
is denoted by F(x , ·). Similarly, for each fixed y ∈ Y, the horizontal section
0 0
of F is denoted by F(·,y ).
0
For a closed interval we write [a,b], for an open interval (a,b) and for half-
open intervals [a,b) or (a,b]. The set R∪{−∞,+∞} is denoted by [−∞,+∞]
and the set [a,b]×[a,b] is denoted by [a,b]2. The conventions when arithmetic
operations are done on ∞ and −∞ (e.g., the symbols ∞+(−∞) or ∞·(−∞))
will be explained in the context.
Let A be a subset of R, and let f: A→R be a function. Then we say that
• f is increasing, if x≤y implies that f(x)≤f(y) for all x,y ∈A;
• f is strictly increasing, if x<y implies that f(x)<f(y) for all x,y ∈A;