Toward a Restructuring of the Foundations of Fuzzy Logic (FL)
Speaker: Prof. Lotfi A. Zadeh
Computer Science Division, EECS
University of California, Berkeley
email: zadeh@cs.berkeley.edu
Thursday, October 2nd, 1997
4:00-5:00pm
310 Soda Hall
Abstract:
The proposed restructuring has three objectives. First, to clarify
and solidify the foundations of fuzzy logic; second to make more
transparent the links between fuzzy logic and other methodologies;
and third, to suggest alternative ways for further development
of fuzzy logic and its applications.
The point of departure for the restructuring is the premise that
fuzzy logic has many distinct facets --facets which overlap and have
unsharp boundaries. Among these facets there are four that stand
out in importance. They are: (i) the logical facet, L: (ii) the
set-theoretic facet, S; (iii) the relational face, R: and (iv)
the epistemic facet, E.
The logical facet of FL, L, is a logical system or, more
accurately, a collection of logical systems which include a special
case both two-valued and multiple-valued systems. As in any
logical system, at the core of the logical facet of FL lies a
system of rules of inference. In FL, however, the rules of inference
play the role of rules which govern propagation of various types of fuzzy
constraints. Concomitantly, a proposition, p, is viewed as fuzzy constraint
on an explicitly or implicitly defined variable. Th elogical
facet of FL plays a pivotal role in th eapplications of FL to
knowledge representation and inference from information which is
imprecise, incomplete, uncertain or partially true.
Th eset-theoretic facet of FL, S, is concerned with classes or sets
whose boundaries are not sharply defined. The initail development of FL
was focused on this facet. Most of the applications of FL in
mathematics have been and continue to be related to the set-theoretic facet.
Among the examples of such applications are : fuzzy topology, fuzzy groups,
fuzzy differential equations and fuzzy arithmetic.
The relational facet of FL, R, is concerned in the main with representation
and manipulation of imprecisely defined functions and relations. It is this
facet of FL that plays a pivotal role in its applications to system
analysis and control. The three basic concepts that lie at the core
of this facet of FL are those of a linguistic variable, fuzzy if-then
rule and fuzzy graph. The relatrional facet of FL provides a foundation
for the fuzzy-logic-based methodology of computing with words (CW).
The epistemic facet of FL, E, is linked to its logical facet and is focused
on the applications of FL to knowledge representation, information systems,
fuzzy databases and the theory of possibility and probability. A particularly
important application area for the epistemic facet of FL relates to the
conception and design of information/intelligent systems.
At the core of FL lie two basic concepts: (a) fuzziness/fuzzification;
and (b) granularity/granulation. Fuzziness is a condition which relates
to classes (sets) whose boundaries are unsharply defined, while
fuzzification refers to replacing a crisp set with a set whose boundaries
are fuzzy. For example, the number 5 is fuzzified when it is transformed
into approximately 5.
In a similar spirit granularity relates to clumpiness of structure while
granulation refers to partitioning an object into a collection of
granules, with a granule being a clump of objects (points) drawn
together by indistinguishability, similarity, proximity or functionality.
For example, the granules of a human body might be head, neck, chest,
stomach, legs, etc. Granulation may be crisp or fuzzy; dense or sparse;
and physical or mental.
A concept which plays a pivotal role in fuzzy logic is that of
''fuzzy information granulation'', or fuzzy IG, for short. In crisp IG,
the granules are crisp while in fuzzy IG the granules are fuzzy.
The importance of fuzzy logic --especially in the relam of applications--
derives in large measure from the fact that FL is the only methodology
that provides a machinery for fuzzy information granulation.
A new element in the proposed restructuring relates to linking fuzzy
logic to the concept of what might be called ''generalization group'',
or GG, for short. The elements of GG are various modes of generalization
of concepts, constructs, methods and theories. In relation to fuzzy logic,
the principal modes are : set-to-fuzzy set generalization (fuzzification);
fuzzy-set-to-granulated fuzzy set generalization (fuzzy granulation);
and function-to-fuzzy relation generalization. Various modes of generalization
can be combined through composition. For example, fuzzy granulation may be
viewed
as the composition of granulation and fuzzification.
The rules of inference in fuzzy logic are interpreted as rules of constraint
propagation. The rules are derived by successive application of fuzzification
and fuzzy granulation to what is the ''initial construct''. More specifically,
the initial construct is taken to be the basic argument assignment/function
evaluation rule: x=a, Y=f(x), Y=f(a), where a is a singleton and f is
a crisp function. The generalization group may be interpreted as a language
whose terminal symbols are modes of generalization. Inference rules are
linked to the initial construct by derivation chains.
The concept of a generalization group suggests new directions in the
development and application of fuzzy logic. Of particular importance
in the need for a better understanding of decision-making in an
environment of generalized imprecision, uncertainty and partial truth.
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Please direct questions with regard to the contents of the talk
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Frank Hoffmann UC Berkeley
Computer Science Division Department of EECS
Email: fhoffman@cs.berkeley.edu phone: 1-510-642-8282
URL: http://http.cs.berkeley.edu/~fhoffman fax: 1-510-642-5775
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Yu-Chi Ho
Pierce Hall
Harvard University
Cambridge, MA 02138
617-495-3992(phone) 617-496-6404(fax)
HOME PAGE: hrl.harvard.edu/people/faculty/ho