Excerpt from Computer Design
April 1992
The Seven Noble Truths Of Fuzzy Logic
by Earl Cox
TRUTH ONE
There Is Nothing Fuzzy About Fuzzy Logic
The idea that fuzzy logic is fuzzy or intrinsically imprecise is
one of the most commonly expressed fables in the fuzzy logic
mythos. This wide-spread belief comes in two flavors, the first
holds that fuzzy logic violates common sense and the well proven
laws of logic, and the second, perhaps inspired by its name,
holds that fuzzy systems produce answers that are somehow
ad-hoc, fuzzy, or vague. The feeling persists that fuzzy logic
systems somehow, through their handling of imprecise and
approximate concepts, produce results that are approximations of
the answer we would get if we had access to a model that worked
on hard facts and crisp information. Nothing could be further
from fact.
There is nothing fuzzy about fuzzy logic, Fuzzy Sets differ from
classical or crisp sets in that they allow partial or gradual
degrees of membership. We can see the difference easily by
looking at the difference between a conventional (or "crisp")
set and a fuzzy set. Thus someone 34 years, eleven months, and
twenty eight days old is not middle aged. In the Fuzzy
representation, however, we see that as a person grows older he
or she acquires a partial membership in the set of Middle Aged
people, with total membership at forty years old.
But there is nothing ambiguous about the fuzzy set itself. If
we know a value from the domain, say an age of 35 years old,
then we can find its exact and unambiguous membership In the
set, say 82%. This precision at the set level allows us to write
fuzzy rules at a rather high level of abstraction. Thus we can
say, if age is middle-aged, then weight is usually quite heavy;
and means that, to the degree that the individual's age is
considered middle aged, their weight should be considered
somewhat heavy. A weight estimating function, following this
(very simple) rule might infer a weight from age through the
following fuzzy implication process.
Much of the discomfort with fuzzy logic stems from the implicit
assumption that a single ``right'' logical system exists and to
the degree that another system deviates from this right and
correct logic it is in error. This ``correct'' logic, of
course, is Aristotelian or Boolean logic. But as a logic of
continuous and partial memberships, Fuzzy Logic has a deep and
impressive pedigree. Using the metaphor of the river, Heraclitus
aptly points out that a continuous reasoning system more
correctly maps nature's logical ambiguities. From his dictum
that all is flux, nothing is stationary, he devcloped a
rudimentary multi-valued logic two hundred years before
Aristotle. Recently, Bart Kosko, one of the most profound
thinkers in fuzzy logic, has shown that Boolean logic is, in
fact, a special case of fuzzy logic.
TRUTH TWO
Fuzzy Logic Is Different from Probability
The difference between probability and fuzzy logic is clear when
we consider the underlying concept that each attempts to model.
Probability is concerned with the undecidability in the outcome
of clearly defined and randomly occurring events, while fuzzy
logic is concerned with the ambiguity or undecidability inherent
in the description of the event itself. Fuzziness is often
expressed as ambiguity rather than imprecision or uncertainty
and remains a characteristic of perception as well as concept.
TRUTH THREE
Designing the Fuzzy Sets is very asy
Not only are fuzzy sets easy to conceptualize and represent, but
they reflect, in a general "one-to-one" mapping, the way experts
actually think about a problem. Experts can quickly sketch out
the approximate shape of a fuzzy set. Later, after we have run
the model or examined the process, the precise characteristics
of the fuzzy vocabulary can be adjusted if necessary.
TRUTH FOUR
Fuzzy Systems are Stable, Easily Tuned,
and can be conventionally Validated
Creating fuzzy sets and building a fuzzy system is faster and
quicker than conventional knowledge-based systems using "crisp"
constructs. These fuzzy systems routinely show a one or two
order of magnitude reduction in rules since fuzzy logic
simultaneously handles all the interlocking degrees of freedom.
Fuzzy systems are very robust since the over-lapping of the
fuzzy regions, representing the continuous domain of each
control and solution variable, contributes to a well-behaved and
predictable system operation. These systems are validated in
the same manner as conventional system. The tuning of fuzzy
systems, however, is usually much simpler since there are fewer
rules; representation if visually centered around fuzzy sets,
and operations act simultaneously on the output areas.
TRUTH FIVE
Fuzzy Systems are Different From
and Complementary to Neural Networks
There is a close relationship between fuzzy logic and neural
systems. A fuzzy system attempts to find a region that
represents the space defined by the intersection, union, or
complement of the fuzzy control variables. This has analogies
to both neural network classifiers and linear programming
models. Yet fuzzy systems approach the problem differently with
a deeper and more robust epistemology. In a fuzzy system, the
classification and bounding process is much more open to the
developer and user with capabilities for explanations, rule and
fuzzy set calibration, performance measurements, and controls
over the way the solution is ultimately derived.
TRUTH SIX
Fuzzy logic "ain't just process control anymore"
Historically we have come to view fuzzy logic as a process
control and signal analysis technique, but fuzzy logic is really
a way of logically representing and analyzing information,
independent of particular applications. The information
management field in particular has, until recently, ignored
fuzzy logic, delaying its introduction into expert system and
decision support technology. Recently, however, new types of
knowledge base construction tools have emerged. Such tools will
make it easier for experts who are not computer experts to
intuitively represent and manipulate information.
TRUTH SEVEN
Fuzzy Logic is a Representation and Reasoning Process
Not the "Magic Bullet" for all AI's current problems - Fuzzy
Logic is a tool for representing imprecise, ambiguous, and vague
information. Its power lies in its ability to perform meaningful
and reasonable operations on concepts that are outside the
definitions available in conventional Boolean logic. We have
used fuzzy logic in such applications as project management,
product pricing models, health care provider fraud detection,
sales forecasting, market share demographic analysis, criminal
identification, capital budgeting, and company acquisition
analysis. Although fuzzy logic is a powerful and versatile tool,
it is not a solution to all problems. Nevertheless, it opens the
door for the modeling of problems that have generally been
extremely difficult or intractable.
Earl Cox, CEO
Metus Systems
White Plains, NY
(914) 238-0647
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