Hadas Zies (hzies@cs.berkeley.edu)
Thu, 17 Oct 1996 17:52:43 +0100

To: BISC Group
From: L. A. Zadeh

The following abstract summarizes my recent thinking about fuzzy
information granulation. I believe that fuzzy IG will evolve into
an important field in its own right.

With warm regards,

Lotfi Zadeh

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October 15, 1996
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The Key Roles of Fuzzy Information Granulation in Human Reasoning, Fuzzy Logic
and Computing with Words
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Lotfi A. Zadeh*
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*Computer Science Division and the Electronics Research
Laboratory, Department of EECS, University of California, Berkeley, CA
94720-1776; Telephone: 510-642-4959; Fax: 510-642-1712; E-mail:
zadeh@cs.berkeley.edu. Research supported in part by NASA Grant NCC 2-275,
ONR Grant N00014-96-1-0556, LLNL Grant 442427-26449, and the BISC
Program of UC Berkeley.
The concepts of granulation and organization play fundamental roles in human
cognition. In a general setting, granulation involves a decomposition of
whole into parts. Conversely, organization involves an integration of parts
into whole.
In more specific terms, information granulation (IG) relates to partitioning
a class of points (objects) into granules, with a granule being a clump of
points drawn together by indistinguishability, similarity or functionality.
The concept of a granule is more general than that of a cluster.
Modes of information granulation in which granules are crisp play important
roles in a wide variety of methods, approaches and techniques. Among them are:
interval analysis, quantization, rough set theory, diakoptics, divide and
conquer, Dempster-Shafer theory, machine learning from examples, chunking,
qualitative process theory, decision trees, semantic networks, analog-to-digital
conversion, constraint programming, cluster analysis and many others.
Important though it is, crisp information granulation (crisp IG) has a major
blind spot. More specifically, it fails to reflect the fact that in much --
perhaps most -- of human reasoning and concept formation granules are fuzzy
rather than crisp. For example, fuzzy granules of a human head are the nose,
forehead, hair, cheeks, etc. Each of the fuzzy granules is associated with a
set of fuzzy attributes, e.g., in the case of the fuzzy granule hair, the fuzzy
attributes are color, length, texture, etc. In turn, each of the fuzzy attributes
is associated with a set of fuzzy values. Specifically, in the case of the fuzzy
attribute length(hair), the fuzzy values are long, short, not very long, etc.
The fuzziness of granules is characteristic of the ways in which human concepts
are formed, organized and manipulated.
In human cognition, fuzziness of granules is a direct consequence of fuzziness of
the concepts of indistinguishability, similarity and functionality. Furthermore,
it is entailed by the finite capacity of the human mind to store information and
resolve detail. In this perspective, fuzzy information granulation (fuzzy IG)
may be viewed as a form of lossy data compression.
Fuzzy information granulation underlies the remarkable human ability to make
rational decisions in an environment of imprecision, uncertainty and partial
truth. And yet, despite its intrinsic importance, fuzzy information granulation
has received scant attention except in the context of fuzzy logic, in which
fuzzy IG underlies the basic concepts of linguistic variable, fuzzy if-then rule
and fuzzy graph. In fact, the effectiveness and successes of fuzzy logic in
dealing with real-world problems rest in large measure on the use of the
machinery of fuzzy information granulation. This machinery is unique to fuzzy
Recently fuzzy information granulation has come to play a central role in the
methodology of computing with words. More specifically, in a natural language
words play the role of labels of fuzzy granules. In computing with words, a
proposition is viewed as an implicit fuzzy constraint on an implicit variable.
The meaning of a proposition is the constraint which it represents.
In CW, the initial data set (IDS) is assumed to consist of a collection of
propositions expressed in a natural language. The result of computation --
referred to as the terminal data set (TDS) -- is likewise a collection of
propositions expressed in a natural language. To infer TDS from IDS the rules
of inference in fuzzy logic are used for constraint propagation from premises
to conclusions.
There are two main rationales for computing with words. First, computing with
words is a necessity when the available information is not precise enough to
justify the use of numbers. And second, computing with words is advantageous
when there is a tolerance for imprecision, uncertainty and partial truth that
can be exploited to achieve tractability, robustness, low solution cost and
better rapport with reality. In coming years, computing with words is likely to
evolve into an important methodology in its own right with wide-ranging
applications on both basic and applied levels.
Inspired by the ways in which humans granulate human concepts, we can proceed to
granulate conceptual structures in various fields of science. In a sense, this
is what motivates computing with words. An intriguing possibility is to
granulate the conceptual structure of mathematics. This would lead to what may
be called granular mathematics. Eventually, granular mathematics may evolve
into a distinct branch of mathematics having close links to the real world.
In the final analysis, fuzzy information granulation is central to human
reasoning and concept formation. It is this aspect of fuzzy IG that underlies
its essential role in the conception and design of intelligent systems. What is
conclusive is that there are many, many tasks which humans can perform with
ease and that no machine could perform without the use of fuzzy information
granulation, This conclusion has a thought-provoking implication for AI: Without
the methodology of fuzzy IG in its armamentarium, AI cannot achieve its goals.