To: BISC Group
From: L. A. Zadeh <zadeh@cs.berkeley.edu>
As could be expected, my message "Probability Theory Needs an
Infusion of Fuzzy Logic to Enhance its Ability to Deal with Real-World
Problems" generated many comments and responses. Please feel free to
circulate your views to the BISC Group, with cc to me.
There were several typos in the version which you got.
Following is a corrected version, identified as Part 1.
In Part 2, which follows Part 1, I add to what I said in Part
1 and arrive at what I believe to be a novel conclusion, namely, that
what might be called Computational Semantics of Natural Languages
(CSNL) should be assigned a key role in enhanced versions of
probability theory. Such a role is played by CSNL in PT++.
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Probability Theory Needs an Infusion of Fuzzy Logic to Enhance its
Ability to Deal with Real-World Problems (Part1)
Discussions and debates centering on the relationship between
fuzzy set theory and fuzzy logic, on one side, and probability theory,
on the other, have a long history starting shortly after the
publication of my first paper on fuzzy sets (1965). There is an
extensive literature on this sub ject, including several papers of
mine of which the latest [Zadeh 1995] appears in a special issue of
Technometrics -- an issue that focuses on the relationship in
question.
The following is an updated expression of my views. Your
comments would be appreciated. Please do not hesitate to disagree
with me and feel free to convey your views to the BISC Group.
First, a point of clarification. By probability theory in the
heading of this message is meant standard probability theory (PT) of
the kind found in textbooks and taught in courses. By infusion is
meant generalization. Thus, the crux of my argument is that PT is in
need of generalization, first by f-generalization (fuzzification) and
second by g-generalization (granulation). In combination,
f-generalization and g-generalization give rise to f.g-generalization
(fuzzy granulation). F-generalization of PT leads to what might be
denoted as PT+. Then, g-generalization of PT+ leads to PT++, which
may be viewed as f.g-generalization of PT. Basically, fuzzy
granulation reflects the finite cognitive ability of humans to resolve
detail and store information. In summary, my contention is that to
enhance its effectiveness PT should be generalized to PT++. This has
already been done to a significant extent by a number of contributors,
but much more remains to be done.
What is not in dispute is that standard probability theory
(PT) provides a vast array of concepts and techniques which are
effective in dealing with a wide variety of problems in which the
available information is lacking in certainty. But what may come as a
surprise to some is that along with such problems stand many very
simple problems for which PT offers no solutions. Here is an sample.
It should be noted that the underlying difficulties in these problems
are well-known to probability theory professionals.
DB 1. What is the probability that my tax return will be
audited?
DB 2. What is the probability that my car may be stolen?
DB 3. How long does it take to get from the hotel to the
airport by taxi?
DB 4. What is the probability that Mary is telling the truth?
DB 5. A and B played 10 times. A won 7 times. The last 3 times
B won. What is the probability that A will win?
Questions of this kind are routinely faced and answered by
humans. The answers, however, are not numbers; they are linguistic
descriptions of fuzzy perceptions of probabilities, e.g., not very
high, quite unlikely, about 0.8, etc. Such answers cannot be arrived
at through the use of PT. What is needed for this purpose is PT++.
I discussed Problems 1-5 with Professor David Blackwell (UC
Berkeley), who is one of the world's leading authorities on
probability theory. The initials DB signify that he agrees with my
assessment.
What are the sources of difficulty in using PT? In Problem 1,
the difficulty comes from the basic property of conditional
probabilities, namely, given p(X), all that can be said is that the
value of p(X|Y) is between 0 and 1. Thus, if I start with the
knowledge that 1% of tax returns are audited, it tells me nothing
about the probability that my tax return will be audited. The same
holds true when I add more detailed information about myself, e.g., my
profession, income, age, place of residence, etc.. IRS may be able to
tell me what fraction of returns in a particular category are audited,
but all that can be said about the probability that my return will be
audited is that it is between 0 and 1.
As I have alluded to earlier, when I prepare my tax return, I
do have a fuzzy perception of the chance that I may be audited. To
arrive at this fuzzy perception what is needed is fuzzy logic, or,
more specifically, PT++. I will discuss in a later installment how
this can be done through the use of PT++.
In Problems 3 and 5, the difficulty is that we are dealing
with a time series drawn from a nonstationary process. In such cases,
probabilities do not exist. Unfortunately, this is frequently the
case when PT is used.
In Problem 3, when I pose the question to a hotel clerk, he
may tell me that it would take approximately 20-25 minutes. The point
is that the answer could not be deduced in a rigorous way by applying
PT. Again, to arrive at the clerk's answer what is needed is PT++.
In Problem 4, the difficulty is that truth is a matter of
degree. For example, if I ask Mary how old she is, and she tells me
that she is 30 but in fact is 31, the degree of truth might be 0.9.
However, if she tells me that she is 25, the degree of truth might be
0.5. The point is that the event "telling the truth" is, in general,
a fuzzy event, PT does not support fuzzy events.
Another class of simple problems which PT cannot handle relate
to commonsense reasoning exemplified by:
6. Most young men are healthy; Robert is young.
What can be said about Robert's health?
7. Most young men are healthy; it is likely that Robert is young.
What can be said about Robert's health?
8. Slimness is attractive; Cindy is slim.
What can be said about Cindy's attractiveness?
In what ways do PT+ and PT++ enhance the ability of
probability theory to deal with real-world problems? In relation to
PT, PT+ has the capability to deal with:
1. fuzzy events, e.g., warm day
2. fuzzy numbers, quantifiers and probabilities, e.g., about 0.7,
most, not very likely
3. fuzzy relations, e.g., much larger than
4. fuzzy truths and fuzzy possibilities, e.g., very true, quite
possible
In addition, PT+ has the potential -- as yet largely unrealized -- to
fuzzify such basic concepts as independence, stationarity and
normality.
PT++ adds to PT+ further capabilities which derive from the
use of granulation. They are, mainly:
1. linguistic (granular) variables
2. fuzzy rule sets and fuzzy graphs
3. granular goals and constraints
At this juncture there exists an extensive and growing
literature centering on the calculi of imprecise and fuzzy
probabilities. The concepts of PT+ and PT++ serve to clarify some of
the basic issues in these calculi and suggest directions in which the
ability of probability theory to deal with real-world problems may be
enhanced.
Reference:
L. A. Zadeh, "Probability Theory and Fuzzy Logic are Complementary
Rather Than Competitive," Technometrics, vol. 37, pp. 271-276, 1995.
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Probability Theory Needs an Infusion of Fuzzy Logic to Enhance its
Ability to Deal with Real-World Problems (Part2)
Standard probability theory, PT, may be likened to a factory
equipped with precision machinery. The raw materials consist of the
knowledge of real-world probabilities, utilities, dependencies, goals
and constraints.
The problem with PT is that precision of its machinery is in
large measure incompatible with the pervasive imprecision of much of
the raw materials which the machinery must process. Viewed in this
perspective, PT++ adds to the machinery of PT a variety of devices,
tools and constructs which are capable of processing the imprecise raw
materials which are drawn from the real world. The tradeoff is that
conclusions which are arrived at through the use of PT++ are, in
general, less precise but more realistic than those yielded by PT.
In my view, a point of crucial importance is that much of our
knowledge of real-world probabilities, utilities, dependencies, goals
and constraints is based on perceptions rather than measurements --
perceptions of distance, time, force, weight, numbers, proportions,
likelihood and truth, among others. In particular, our estimates of
subjective probabilities are based for the most part on perceptions.
A basic difference between measurements and perceptions is that, in
general, measuremetns are crisp whereas perceptions are fuzzy.
In my recently initiated computational theory of perceptions
(CTP) [Zadeh 1999], perceptions are dealt with via descriptions in a
natural language. For example, "most balls are black," is a
description of my perception of the proportion of black balls in a
box. Similarly, "it is very unlikely that there will be a significant
increase in the price of oil in the near future," may be viewed as my
perception of the likelihood of the event "a significant increase in
the price of oil in the near future."
To be able to process information which is presented in the
form of perceptions, it is necessary to have a method of representing
the meaning of propositions expressed in a natural language in a way
that makes meaning amenable to computation. In the computational
theory of perceptions, this is done through the use of the methodology
of computing with words (CW) [Zadeh 1996]. The computational
semantics of natural languages (CSNL) is a key component of CW.
More specifically, in CSNL the meaning of a proposition, p, is
expressed as a generalized constraint on a variable; X isr R, in which
X is the constrained variable; R is the constraining relation; isr is
a variable copula which defines the way in which R constrains X; and r
is a variable whose value identifies the type of constraint that is
involved. For example, when the value of r is blank, the constraint
is possibilistic; when r is v the constraint is veristic; when r is p
the constraint is probabilistic; and when r is rs the constraint is of
random set type. The reason why r is allowed to take a multiplicity
of values is that the meaning of propositions in a natural language is
in general too comples to lend itself to representation in terms of
conventional constraints of the form X is A, where A is a crisp set.
In effect, meaning representation in CSNL involves a
translation of propositions expressed in a natural language into what
is referred to as the generalized constraint laguage, GCL.
Expressions in GCL are generalized constraints and reasoning with
propositions expressed in a natural language involves generalized
constraint propagation from antecedent constraints to consequent
constraints. This process is the core of the methodology of computing
with words (CW).
As a simple example of meaning representation in CSNL,
consider the perception, p, described as "most young men are healthy,"
in which most, young and healthy are assumed to be fuzzy rather than
crisp. The meaning if this simple proposition cannot br expressed in
predicate logic. In fuzzy logic there are basically two ways in which
the meaning of p can be expressed: (a) using the concept of crisp
cardinality of a fuzzy set; and (b) using the concept of fuzzy
cardinality of a fuzzy set. Using (a), which is simpler than (b), the
meaning may be expressed as a possibilistic constraint on the
proportion of healthy men among young men, with the constraining
relation being the fuzzy proportion most.
This very simple example explains why PT cannot lead to an
answer to the question, (6), posed in Part1, namely" Most young men
are healthy; Robert is young. What can be said about Robert's health?
The inability of PT to deal with perceptions is one of its
principal limitations. Incorporation of the machinery of CW in PT++
and the ability of PT++ to deal with perceptions through the use of
CSNL is one of the main reasons why PT++ has a much wider range of
applicability to real world probelms than PT.
It is of interest to observe that the capability to manipulate
perceptions plays an important role not only in enhanced versions of
probability theory but, more generally, in all fields of science. It
is this capability that underlies the remarkable human ability to
perform a wide variety of physical and mental tasks without any
measurements and any computations.
What this suggeests is that adding to an existing theory, T,
the machinery for manipulation of perceptions may enhance the ability
of T to deal with real-world problems -- as it does in the case of PT.
This is an intriguing possibility that may be worthy of exploration.
References
L.A. Zadeh, "Fuzzy Logic = Computing with Words," IEEE Transactions
on Fuzzy Systems, vol. 4, pp. 103-111, 1996.
L.A. Zadeh, "From Computing with Numbers to Computing with Words --
>From Manipulation of Measurements to Manipulation of Perceptions," (in
press), IEEE Transactions on Circuits and Systems, January, 1999.
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P.S. If you cannot access the papers referenced in this message, send
me a message whth your fax number and mailing address.
Warm regards to all,
Lotfi A. Zadeh
------------------------------------------------------
Lotfi A. Zadeh
Professor in the Graduate School and Director,
Berkeley Initiative in Soft Computing (BISC)
CS Division, Department of EECS
University of California
Berkeley, CA 94720-1776
Tel/office: (510) 642-4959 Fax/office: (510) 642-1712
Tel/home: (510) 526-2569 Fax/home: (510) 526-2433
email: zadeh@cs.berkeley.edu
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