Re: Abstract

Brian M. Schott (
Fri, 1 Nov 1996 13:38:59 +0100

October 15, 1996
The Key Roles of Fuzzy Information Granulation in Human Reasoning, Fuzzy Logic
and Computing with Words
Lotfi A. Zadeh*


*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: 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

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 logic.

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

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.