FUZZY LOGIC SYSTEMS: ORIGIN, CONCEPTS, AND TRENDS

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FUZZY LOGIC SYSTEMS ORIGIN, CONCEPTS, AND
TRENDS Lotfi A. Zadeh Computer Science
Division Department of EECSUC Berkeley URL
http//www-bisc.cs.berkeley.edu URL
http//zadeh.cs.berkeley.edu/ Email
Zadeh_at_cs.berkeley.edu
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LOTFI A. ZADEH COMPUTER SCIENCE DIVISION,
DEPARTMENT OF EECS UNIVERSITY OF
CALIFORNIA BERKELEY, CA 94720-1776 TEL (510)
642-4959 FAX [phone] SECRETARY (510)
642-8271 HOME FAX [phone] E-MAIL
zadeh_at_cs.berkeley.edu
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BACKDROP
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EVOLUTION OF FUZZY LOGICA PERSONAL PERSPECTIVE
generality
nl-generalization
computing with words and perceptions (CWP)
f.g-generalization
f-generalization
classical bivalent
time
1965
1973
1999
1965 crisp sets fuzzy sets 1973 fuzzy
sets granulated fuzzy sets (linguistic
variable) 1999 measurements perceptions
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WHAT IS FUZZY LOGIC?

• in essence, fuzzy logic (FL) is focused on modes
  of reasoning which are approximate rather than
  exact.
• fuzzy logic is aimed at precisiation of
  approximate reasoning
• in fuzzy logic, everything, including truth, is
  or is allowed to be a matter of degree
• in bivalent logic, everything is either true or
  false
• there is a fundamental conflict between bivalence
  and reality

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WHAT IS FUZZY LOGIC?
fuzzy logic (FL) is aimed at a formalization of
modes of reasoning which are approximate rather
than exact examples exact all men are
mortal Socrates is a man Socrates is
mortal approximate most Swedes are
tall Magnus is a Swede it is likely that
Magnus is tall
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EVOLUTION OF LOGIC

• two-valued (Aristotelian) nothing is a matter of
  degree
• multi-valued truth is a matter of degree
• fuzzy everything is a matter of degree
• principle of the excluded middle every
  proposition is either true or false

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FUZZY LOGIC

• The real world is pervaded with imprecision,
  uncertainty and partiality especially
  partiality of truth, certainty and possibility
• In the real world, almost everything is a matter
  of degree. Absolutes are few and far between
• It is this reality that is the point of departure
  in fuzzy logic
• The cornerstones of fuzzy logic are verity,
  possibility and probability
• The role model for fuzzy logic is the human mind
  and its remarkable capability to operate on
  perception-based information without any
  measurements and any computations

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REASONING WITH WORDS

• Business Week 9-18-95
• a lower deficit leads to a lower dollar and a
  higher deficit pushes the dollar higher. Here is
  the logic behind it
• A growing deficit means the government must
  borrow more, pushing up interest rates. As U.S.
  interest rates rise relative to the rest of the
  worlds, money flows out of foreign assets and
  into U.S. securities, foreigners must dump their
  own currencies and buy dollars.
• Conversely, a decreasing deficit lowers
  government borrowing and thus pushes interest
  rates down. As rates fall, investors seek higher
  returns overseas. They sell dollars and buy
  foreign bonds. The result a depreciating dollar.

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WHAT IS FUZZY LOGIC?

• fuzzy logic has been and still is, though to a
  lesser degree, an object of controversy
• for the most part, the controversies are rooted
  in misperceptions, especially a misperception of
  the relation between fuzzy logic and probability
  theory
• a source of confusion is that the label fuzzy
  logic is used in two different senses
• (a) narrow sense fuzzy logic is a logical system
• (b) wide sense fuzzy logic is coextensive with
  fuzzy set theory
• today, the label fuzzy logic (FL) is used for
  the most part in its wide sense

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SOME COMMENTS ON FUZZY LOGIC

• R.E. Kalman (1972)
• Let me say quite categorically that there is no
  such thing as a fuzzy concept, . We do talk
  about fuzzy things but they are not scientific
  concepts. Some people in the past have discovered
  certain interesting things, formulated their
  findings in a non-fuzzy way, and therefore we
  have progressed in science.

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Professor William Kahan (1975) Fuzzy theory is
wrong, wrong, and pernicious. says William
Kahan, a professor of computer sciences and
mathematics at Cal whose Evans Hall office is a
few doors from Zadehs. I can not think of any
problem that could not be solved better by
ordinary logic. What Zadeh is saying is the
same sort of things Technology got us into this
mess and now it cant get us out. Kahan says.
Well, technology did not get us into this mess.
Greed and weakness and ambivalence got us into
this mess. What we need is more logical thinking,
not less. The danger of fuzzy theory is that it
will encourage the sort of imprecise thinking
that has brought us so much trouble.
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STATISTICS
Count of papers containing the word fuzzy in
title, as cited in INSPEC and MATH.SCI.NET
databases. (data for 2003 are not
complete) Compiled by Camille Wanat, Head,
Engineering Library, UC Berkeley, November 20,
2003
INSPEC/fuzzy
Math.Sci.Net/fuzzy
1970-1979 569 1980-1989 2,404 1990-1999 23,207
2000-present 9,945 1970-present 36,125
443 2,465 5,479 2,865 11,252
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STATISTICS

• Count of books containing the words soft
  computing in title, or published in series on
  soft computing. (source Melvyl catalog)
• Compiled by Camille Wanat, Head,
• Engineering Library, UC Berkeley,
• October 12, 2003
• Count of papers containing soft computing in
  title or published in proceedings of conferences
  on soft computing
• 2494 (1994-2002)

1994 4 1995 2 1996 7 1997 12 1998 15 1999
23 2000 36 2001 43 2002 42 Total 184
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NUMBERS ARE RESPECTEDWORDS ARE NOT

• in science and engineering there is a deep-seated
  tradition of according much more respect to
  numbers than to words. The essence of this
  tradition was stated succinctly by Lord Kelvin in
  1883.

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• In physical science the first essential step in
  the direction of learning any subject is to find
  principles of numerical reckoning and practicable
  methods for measuring some quality connected with
  it. I often say that when you can measure what
  you are speaking about and express it in numbers,
  you know something about it but when you cannot
  measure it, when you cannot express it in
  numbers, your knowledge is of a meager and
  unsatisfactory kind it may be the beginning of
  knowledge but you have scarcely, in your
  thoughts, advanced to the state of science,
  whatever the matter may be.

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IN QUEST OF PRECISION

• The risk of a 6.0 quakewhich could be more
  damaging, with one-tenth the destructive power of
  the October 17 quakeis 11 percent during the
  next two months, the surveys scientists say.
• The seismologists in Menlo Park say the
  probability of an aftershock of a magnitude of 5
  or more in the next two months is 45 percent.
• It is very unusual for a quake of this size not
  to come close to the surface. As a result, Dr.
  Holzer said, geologists have begun to doubt their
  ability to make reliable estimates for future
  major earthquakes and to recognize active faults.

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IN QUEST OF PRECISION

• Washington Analysis Corporation
• (The New York Times)
• Bruce Likness, a farm equipment dealer and
  long-time friend of Waletich, estimates that a
  beginner needs 409,780 to 526,487 worth of
  machinery to have a chance of success on a
  1,500-acre farm.

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THE QUEST FOR PRECISION

• Thomas M. Holbrook, a prominent political
  scientist teaching at the University of Milwaukee
  in Wisconsin, at a meeting of the American
  Political Science Association in September, 2000,
  predicted that Gore would win by a landslide vote
  of 60.3 percent (NY Times, 11-7-00)

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THE GAP BETWEEN THEORY AND REALITY

• John Cassidy commenting on the award of Nobel
  Prize to William Vickrey (New Yorker, 12-2-1996)
• Vickrey died just three days after winning the
  prize, but his last words on his subject should
  not be forgotten. Here was a world-renowned
  theorist confirming what many outsiders have long
  suspected-that a good deal of modern economic
  theory, even the kind that wins Nobel prizes,
  simply does not matter much.

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IN QUEST OF PRECISION

• Robert Shuster (Ned Davis Research)
• We classify a bear market as a 30 percent
  decline after 50 days, or a 13 percent decline
  after 145 days.
• Warren Buffet (Fortune 4-4-94)
• It is better to be approximately right than
  precisely wrong.

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WHAT IS FUZZY LOGIC? WHY IS IT NEEDED?

• In the evolution of science a time comes when
  alongside the brilliant successes of a theory, T,
  what become visible are classes of problems which
  fall beyond the reach of T. At that point, the
  stage is set for a progression from T to T--a
  generalization of T
• Among the many historical examples are the
  transitions from Newtonian mechanics to quantum
  mechanics from linear system theory to nonlinear
  system theory and from deterministic models to
  probabilistic models in economics and decision
  analysis

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CONTINUED

• In this perspective, a fundamental point-- a
  point which is not as yet widely recognized-- is
  that there are many classes of problems which
  cannot be addressed by any theory, T, which is
  based on bivalent logic. The problem with
  bivalent logic is that it is in fundamental
  conflict with reality a reality in which almost
  everything is a matter of degree
• To address such problems what is needed is a
  logic for modes of reasoning which are
  approximate rather than exact. This is what fuzzy
  logic is aimed at. In a sense, if bivalent logic
  is the logic of measurements, then fuzzy logic is
  the logic of perceptions.

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THE TRIP-PLANNING PROBLEM

• I have to fly from A to D, and would like to get
  there as soon as possible
• I have two choices (a) fly to D with a
  connection in B or
• (b) fly to D with a connection in C
• if I choose (a), I will arrive in D at time t1
• if I choose (b), I will arrive in D at time t2
• t1 is earlier than t2
• therefore, I should choose (a) ?

B
(a)
A
D
C
(b)
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CONTINUED

• now, let us take a closer look at the problem
• the connection time, cB , in B is short
• should I miss the connecting flight from B to D,
  the next flight will bring me to D at t3
• t3 is later than t2
• what should I do?
• decision f ( t1 , t2 , t3 ,cB ,cC )
• existing methods of decision analysis do not have
  the capability to compute f
• reason nominal values of decision variables ?
  observed values of decision variables

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CONTINUED

• the problem is that we need information about the
  probabilities of missing connections in B and C.
• I do not have, and nobody has, measurement-based
  information about these probabilities
• whatever information I have is perception-based
• with this information, I can compute
  perception-based granular probability
  distributions of arrival times in D for (a) and
  (b)
• the problem is reduced to ranking of granular
  probability distributions

Note subjective probability perception of
likelihood
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THE KERNEL PROBLEM THE SIMPLEST B-HARD DECISION
PROBLEM
time of arrival
missed connection
0
alternatives
a
b

• decision is a function of and perceived
  probability of missing connection
• strength of decision

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THE CONCEPT OF A PROTOFORM AND ITS BASIC ROLE IN
KNOWLEDGE REPRESENTATION, DEDUCTION AND SEARCH

• Informally, a protoformabbreviation of
  prototypical formis an abstracted summary. More
  specifically, a protoform is a symbolic
  expression which defines the deep semantic
  structure of a construct such as a proposition,
  command, question, scenario, or a system of such
  constructs
• Example
• Eva is young A(B) is C

abstraction
young
C
instantiation
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PF-EQUIVALENCE

• Scenario A
• Alan has severe back pain. He goes to see a
  doctor. The doctor tells him that there are two
  options (1) do nothing and (2) do surgery. In
  the case of surgery, there are two possibilities
  (a) surgery is successful, in which case Alan
  will be pain free and (b) surgery is not
  successful, in which case Alan will be paralyzed
  from the neck down. Question Should Alan elect
  surgery?

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PF-EQUIVALENCE

• Scenario B
• Alan needs to fly from San Francisco to St.
  Louis and has to get there as soon as possible.
  One option is fly to St. Louis via Chicago and
  the other through Denver. The flight via Denver
  is scheduled to arrive in St. Louis at time a.
  The flight via Chicago is scheduled to arrive in
  St. Louis at time b, with altb. However, the
  connection time in Denver is short. If the flight
  is missed, then the time of arrival in St. Louis
  will be c, with cgtb. Question Which option is
  best?

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PROTOFORM EQUIVALENCE
gain
c
1
2
0
options
a
b
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MEASUREMENTS VS. PERCEPTIONS

• what we are beginning to appreciateand what Lord
  Kelvin did notis the fundamental importance of
  the remarkable human capability to perform a wide
  variety of physical and mental tasks without any
  measurements and any computations.
• in performing such tasks, exemplified by driving
  a car in city traffic, we employ perceptions of
  distance, speed, time, position, shape,
  likelihood, intent, similarity and other
  attributes of physical and mental objects.

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COMPUTATION WITH PERCEPTIONS
Dana is young Tandy is a few years older than
Dana Tandy is ?A
Y is several times larger than X Y is large X is
?A
small X small Y medium medium X large
Y large X is ?A, Y is ?B
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REASONING WITH PERCEPTIONS
simple examples
Dana is young Tandy is a few years older than
Dana Tandy is (young few)
most Swedes are tall most Swedes are
blond (2most-1) Swedes are tall and blond
most Swedes are tall most2 Swedes are very tall
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WHAT IS FUZZY LOGIC (FL) ?
fuzzy logic (FL) has four principal facets
logical (narrow sense FL)
FL/L
F
F.G
FL/E
FL/S
set-theoretic
epistemic
G
FL/R
relational
F fuzziness/ fuzzification G granularity/
granulation F.G F and G
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• The logical facet, FL/L, is focused on logical
  systems in which truth is a matter of degree a
  degree which is allowed to be a fuzzy set
• The set-theoretic facet, FL/S, is concerned, in
  the main, with the theory of fuzzy sets. Most of
  the mathematical literature on fuzzy logic
  relates to FL/S
• The relational facet, FL/R, is focused on fuzzy
  dependencies, granulation, linguistic variables
  and fuzzy rule sets. Most practical applications
  of fuzzy logic relate to FL/R

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• The epistemic facet, FL/E, is concerned, in the
  main, with knowledge representation, natural
  languages, semantics and expert systems.
  Probabilistic and possibilistic modes of
  reasoning are a part of this facet as well as
  FL/L and FL/R

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FROM NUMBERS TO WORDS

• There is a deep-seated tradition in science of
  striving for the ultimate in rigor and precision
• Words are less precise than numbers
• Why and where, then, should words be used?
• When precise information is not available
• When precise information is not needed
• When there is a tolerance for imprecision which
  can be exploited to achieve tractability,
  simplicity, robustness and low solution cost
• When the expressive power of words is greater
  than the expressive power of numbers

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VARIABLES AND LINGUISTIC VARIABLES

• one of the most basic concepts in science is that
  of a variable
• variable -numerical (X5 X(3, 2) )
• -linguistic (X is small (X, Y) is much
  larger)
• a linguistic variable is a variable whose values
  are words or sentences in a natural or synthetic
  language (Zadeh 1973)
• the concept of a linguistic variable plays a
  central role in fuzzy logic and underlies most of
  its applications

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LINGUISTIC VARIABLES AND F-GRANULATION (1973)
example Age primary terms young, middle-aged,
old modifiers not, very, quite, rather,
linguistic values young, very young, not very
young and not very old,
µ
young
old
middle-aged
1
very old
0
Age
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EXAMPLES OF F-GRANULATION (LINGUISTIC VARIABLES)
color red, blue, green, yellow, age young,
middle-aged, old, very old size small, big, very
big, distance near, far, very, not very far,
young
middle-aged
old
1
0
age
100

• humans have a remarkable capability to perform a
  wide variety of physical and mental tasks, e.g.,
  driving a car in city traffic, without any
  measurements and any computations
• one of the principal aims of CTP is to develop a
  better understanding of how this capability can
  be added to machines

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A NEGATIVE VIEW
R.E. Kalman (1972) I would like to comment
briefly on Professor Zadehs presentation. His
proposals could be severely, ferociously, even
brutally criticized from a technical point view.
This would be out of place here. But a blunt
question remains Is Professor Zadeh presenting
important ideas or is he indulging in wishful
thinking? The most serious objection of
fuzzification of system analysis is that lack
of methods of systems analysis is not the
principal scientific problem in the systems
field. That problem is one of developing basic
concepts and deep insight into the nature of
systems, perhaps trying to find something akin
to the laws of Newton. In my opinion, Professor
Zadehs suggestions have no chance to contribute
to the solution of this basic problem.
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GRANULATION OF AGE
Age
1
1

0
0
years
young
old
middle-aged
130
2
1
refinement
attribute value modifiers very, not very, quite
1
1
2
12

0
12
2
1
months
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F-GRANULARITY AND F-GRANULATION

• perceptions are f-granular (fuzzy and granular)
• fuzzy unsharp class boundaries
• gradual transition from membership to non-
• membership
• granular class elements are grouped into
  granules, with a granule being a clump of
  elements drawn together by indistinguishability,
  similarity, proximity or functionality
• f-granular is a manifestation of a fundamental
  limitation on the cognitive ability of humans to
  resolve detail and store information
• f-granulation serves two major purposes
• (a) Data compression
• (a') Suppression of decision-irrelevant detail
• (b) Divide and conquer

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PRINCIPAL APPLICATIONS OF FUZZY LOGIC
FL

• control
• consumer products
• industrial systems
• automotive
• decision analysis
• medicine
• geology
• pattern recognition
• robotics

CFR
CFR calculus of fuzzy rules
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EMERGING APPLICATIONS OF FUZZY LOGIC

• computational theory of perceptions
• natural language processing
• financial engineering
• biomedicine
• legal reasoning
• forecasting

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CALCULUS OF FUZZY RULES (CFR)

• syntax legal forms of rules
• if X is A then Y is B
• if X is A then Y is B unless Z is C
• taxonomy classification of rules
• categorical
• if X is then Y is B
• qualified
• if X is A then usually (Y is B)
• semantics meaning of rules
• single rule
• collection of rules

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FUZZY IF-THEN RULES

• examples (free form)
• simple If pressure is high then volume is low
• compound if inflation is very low and
  unemployment is very high then a substantial
  reduction in the interest rate is called for
• dynamic if goal is right_turn and light is red
  then stop then if intersection is clear make
  right turn
• fact pressure is low
• command reduce speed if road is slippery
• dispositional usually it is foggy in San
  Francisco in July and August
• gradual the more a tomato is ripe the more it is
  red
• exceptional a tomato is red unless it is unripe

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DEPENDENCY AND COMMAND

• Dependency
• Y is large if X is small
• Y is medium if X is medium
• Y is small if X is large
• Command
• reduce Y slightly if X is small
• reduce Y substantially if X is not small

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TAXONOMY OF RULES IN FDCL

• categorical (examples)
• X is A (fact)
• if X is A then Y is B or equivalently Y is B if X
  is A
• if X is A and Y is B then U is C and W is D
• if X is A then Y is f(A)
• if X is A then Action is B (command)
• if X is A and Context is B then replace X is A
  with X is C (replacement)
• if X is A then delete (if X is B then Y is
  C) (metarule)
• if X is A then add (if X is B then Y is
  C) (metarule)
• the more X is A the more Y is B (gradual)

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TAXONOMY OF RULES IN FDCL

• qualified (examples)
• if X is A then Y is B unless Z is E (exception)
• if X is A then usually (Y is B) (usuality
  qualified)
• usually (if X is A then Y is B)
• if X is A and Prob Y is BX is A is C then
  Action is D
• if X is A then possibly (Y is B) (possibility
  qualified)
• (if X is A then Y is B) is possible
  ? (possibilistic)
• (if X is A then Y is B) is true ? (truth
  qualified)
• hybrid (examples)
• usually (the more X is A the more Y is B)
• If X is A then very likely (Y is B) unless Z is
  E

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SEMANTICS OF SINGLE RULES

• categorical
• If X1 is A1 and Xn is An then Y is B1 and Yn is
  Bn
• If X1 is A1 and Xn is An then Y is (b0 bi
  Xi)
• qualified
• exception if X is A then Y is B unless Z is E
• truth qualified if X is A then Y is B is very
  true
• probability-qualified if X is A then Y is B is
  likely
• possibility-qualified if X is A then Y is B is
  quite possible

(sugeno)
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FUZZY IF-THEN RULES

• increase interest rates slightly if unemployment
  is low and inflation is moderate
• increase interest rates sharply if unemployment
  is low and inflation is moderate but rising
  sharply
• decrease interest rates slightly if unemployment
  is low but increasing and inflation rate is low
  and stable

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HONDA FUZZY LOGIC TRANSMISSION
Fuzzy Set
Not Very Low
High
Close
1
1
1
Low
High
High
Grade
Grade
Grade
Low
Not Low
0
0
0
5
30
130
180
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Throttle
Shift
Speed

• Control Rules
• If (speed is low) and (shift is high) then (-3)
• If (speed is high) and (shift is low) then (3)
• If (throt is low) and (speed is high) then (3)
• If (throt is low) and (speed is low) then (1)
• If (throt is high) and (speed is high) then (-1)
• If (throt is high) and (speed is low) then (-3)

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INTERPOLATION
Y is B1 if X is A1 Y is B2 if X is A2 .. Y is
Bn if X is An Y is ?B if X is A
A?A1, , An
Conjuctive approach (Zadeh 1973) Disjunctive
approach (Zadeh 1971, Zadeh 1973,
Mamdani 1974)
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THE IT IS POSSIBLE BUT NOT PROBABLE DILEMMATHE
ROCK ON WHICH MANY CRISP THEORIES FOUNDER

• decision is based on information
• in most real-world settings, decision-relevant
  information is incomplete, uncertain and
  imprecise
• to assess the consequences of a decision when
  decision-relevant information is not complete,
  requires consideration of all possible scenarios
• among such scenarios, a scenario that plays a
  pivotal role is the worst-case scenario

57
THE DILEMMA

• worst-case scenario is possible
• what is the probability of the worst-case
  scenario?
• the problem is that, in general, the probability
  of worst-case scenario does not lend itself to
  crisp assessment
• this problem is a rock on which many crisp
  theories founder

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NEW TOOLS
computing with words and perceptions
computing with numbers



CWP
CN
IA
GrC
PNL
precisiated natural language
computing with granules
computing with intervals
PTp
CTP
THD
CTP computational theory of
perceptions PTp perception-based
probability theory THD theory of hierarchical
definability

• a granule is defined
• by a generalized
• constraint

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GRANULAR COMPUTINGGENERALIZED
VALUATIONvaluation assignment of a value to a
variable

• X 5 0 X 5 X is small X
  isr R
• point interval fuzzy interval
  generalized

singular value measurement-based
granular values perception-based
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COMPUTATIONAL THEORY OF PERCEPTIONS

• the point of departure in the computational
  theory of perceptions is the assumption that
  perceptions are described by propositions
  expressed in a natural language
• examples
• economy is improving
• Robert is very honest
• it is not likely to rain tomorrow
• it is very warm
• traffic is heavy

• in general, perceptions are summaries
• perceptions are intrinsically imprecise

61
MEASUREMENT-BASED VS. PERCEPTION-BASED INFORMATION
INFORMATION
measurement-based numerical
perception-based linguistic

• it is 35 C
• Eva is 28

• It is very warm
• Eva is young
• it is cloudy
• traffic is heavy
• it is hard to find parking near the campus

• measurement-based information may be viewed as
  special case of perception-based information

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CONTINUED

• imprecision of perceptions is a manifestation of
  the bounded ability of sensory organs and,
  ultimately, the brain, to resolve detail and
  store information
• perceptions are f-granular in the sense that (a)
  the boundaries of perceived classes are fuzzy
  and (b) the values of perceived attributes are
  granular, with a granule being a clump of values
  drawn together by indistinguishability,
  similarity, proximity or functionality
• it is not possible to construct a computational
  theory of perceptions within the conceptual
  structure of bivalent logic and probability theory

63
KEY POINT

• words are less precise than numbers
• computing with words and perceptions(CWP) is less
  precise than computing with numbers (CN)
• CWP serves two major purposes
• provides a machinery for dealing with problems in
  which precise information is not available
• provides a machinery for dealing with problems in
  which precise information is available, but there
  is a tolerance for imprecision which can be
  exploited to achieve tractability, robustness,
  simplicity and low solution cost

64
EXAMPLE

• I am driving to the airport. How long will it
  take me to get there?
• Hotel clerks perception-based answer about
  20-25 minutes
• about 20-25 minutes cannot be defined in the
  language of bivalent logic and probability theory
• To define about 20-25 minutes what is needed is
  PNL

65
PRECISIATED NATURAL LANGUAGE
PNL
66
WHAT IS PRECISIATED NATURAL LANGUAGE (PNL)?
PRELIMINARIES

• a proposition, p, in a natural language, NL, is
  precisiable if it translatable into a
  precisiation language
• in the case of PNL, the precisiation language is
  the Generalized Constraint Language, GCL
• precisiation of p, p, is an element of GCL
  (GC-form)

67
WHAT IS PNL?

• PNL is a sublanguage of precisiable propositions
  in NL which is equipped with two dictionaries
  (1) NL to GCL (2) GCL to PFL (Protoform
  Language) and (3) a modular multiagent database
  of rules of deduction (rules of generalized
  constrained propagation) expressed in PFL.

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THE BASIC IDEA
P
GCL
NL
precisiation
description
p
NL(p)
GC(p)
description of perception
precisiation of perception
perception
PFL
GCL
abstraction
GC(p)
PF(p)
precisiation of perception
GCL (Generalized Constrain Language) is maximally
expressive
69
GENERALIZED CONSTRAINT

• standard constraint X ? C
• generalized constraint X isr R

X isr R
copula
GC-form (generalized constraint form of type r)
type identifier
constraining relation
constrained variable

• X (X1 , , Xn )
• X may have a structure XLocation
  (Residence(Carol))
• X may be a function of another variable Xf(Y)
• X may be conditioned (X/Y)

70
GC-FORM (GENERALIZED CONSTRAINT FORM OF TYPE r)
X isr R
r equality constraint XR is abbreviation of
X isR r inequality constraint X
R r? subsethood constraint X ? R r
blank possibilistic constraint X is R R is the
possibility distribution of X r v veristic
constraint X isv R R is the verity distributio
n of X r p probabilistic constraint X isp R R
is the probability distribution of X
71
CONTINUED
r rs random set constraint X isrs R R is the
set- valued probability distribution of X r
fg fuzzy graph constraint X isfg R X is a
function and R is its fuzzy graph r u usuality
constraint X isu R means usually (X is R) r
ps Pawlak set constraint X isps ( X, X) means
that X is a set and X and X are the lower and
upper approximations to X
72
GENERALIZED CONSTRAINT LANGUAGE (GCL)

• GCL is generated by combination, qualification
  and propagation of generalized constraints
• in GCL, rules of deduction are the rules
  governing generalized constraint propagation
• examples of elements of GCL
• (X isp R) and (X,Y) is S)
• (X isr R) is unlikely) and (X iss S) is likely
• if X is small then Y is large
• the language of fuzzy if-then rules is a
  sublanguage of PNL

73
DICTIONARIES
1
precisiation
proposition in NL
p
p (GC-form)
? Count (tall.Swedes/Swedes) is most
most Swedes are tall
2
protoform
precisiation
PF(p)
p (GC-form)
? Count (tall.Swedes/Swedes) is most
Q As are Bs
74
THE CONCEPT OF A PROTOFORM AND ITS BASIC ROLE IN
KNOWLEDGE REPRESENTATION, DEDUCTION AND SEARCH

• Informally, a protoformabbreviation of
  prototypical formis an abstracted summary. More
  specifically, a protoform is a symbolic
  expression which defines the deep semantic
  structure of a construct such as a proposition,
  command, question, scenario, or a system of such
  constructs
• Example
• Eva is young A(B) is C

abstraction
young
C
instantiation
75
TRANSLATION FROM NL TO PFL
examples Most Swedes are tall Count
(A/B) is Q Eva is much younger than Pat
(A (B), A (C)) is R usually Robert returns
from work at about 6pm Prob A is B is C
much younger
Pat
Age
Eva
Age
usually
about 6 pm
Time (Robert returns from work)
76
BASIC POINTS

• annotation specification of class or type
• Eva is young A(B) is C
• A/attribute of B, B/name, C/value of A
• abstraction has levels, just as summarization
  does
• most Swedes are tall most As are tall
• most As are B QAs are Bs
• P and q are PF-equivalent (at level ?) iff they
  have identical protoforms (at level ?)
• most Swedes are tallfew professors are rich

77
BASIC STRUCTURE OF PNL
NL
PFL
GCL
p


p
p
precisiation
GC(p)
PF(p)
precisiation (a)
abstraction (b)
DDB
WKDB
world knowledge database
deduction database

• In PNL, deductiongeneralized constraint
  propagation
• DDB deduction databasecollection of
  protoformal rules governing generalized
  constraint propagation
• WKDB PNL-based

78
WORLD KNOWLEDGE

• examples
• icy roads are slippery
• big cars are safer than small cars
• usually it is hard to find parking near the
  campus on weekdays between 9 and 5
• most Swedes are tall
• overeating causes obesity
• Ph.D. is the highest academic degree
• an academic degree is associated with a field of
  study
• Princeton employees are well paid

79
WORLD KNOWLEDGE
KEY POINTS

• world knowledgeand especially knowledge about
  the underlying probabilitiesplays an essential
  role in disambiguation, planning, search and
  decision processes
• what is not recognized to the extent that it
  should, is that world knowledge is for the most
  part perception-based

80
WORLD KNOWLEDGE EXAMPLES

• specific
• if Robert works in Berkeley then it is likely
  that Robert lives in or near Berkeley
• if Robert lives in Berkeley then it is likely
  that Robert works in or near Berkeley
• generalized
• if A/Person works in B/City then it is likely
  that A lives in or near B
• precisiated
• Distance (Location (Residence (A/Person),
  Location (Work (A/Person) isu near
• protoform F (A (B (C)), A (D (C))) isu R

81
ORGANIZATION OF WORLD KNOWLEDGEEPISTEMIC
(KNOWLEDGE-DIRECTED) LEXICON (EL)
network of nodes and links
j
rij
wij granular strength of association between i
and j
wij
i
K(i)
lexine

• i (lexine) object, construct, concept
  (e.g., car, Ph.D. degree)
• K(i) world knowledge about i (mostly
  perception-based)
• K(i) is organized into n(i) relations Rii, ,
  Rin
• entries in Rij are bimodal-distribution-valued
  attributes of i
• values of attributes are, in general, granular
  and context-dependent

82
EPISTEMIC LEXICON
lexinej
rij
lexinei
rij i is an instance of j (is or isu) i is a
subset of j (is or isu) i is a superset of
j (is or isu) j is an attribute of i i causes
j (or usually) i and j are related
83
EPISTEMIC LEXICON
FORMAT OF RELATIONS
perception-based relation
lexine
attributes
granular values
example
car
G 20 \ ? 15k 40 \ 15k, 25k
granular count
84
BASIC STRUCTURE OF PNL
DICTIONARY 1
DICTIONARY 2
GCL
PFL
NL
GCL
p
GC(p)
GC(p)
PF(p)
MODULAR DEDUCTION DATABASE
POSSIBILITY MODULE
PROBABILITY MODULE
FUZZY ARITHMETIC MODULE
agent
SEARCH MODULE
FUZZY LOGIC MODULE
EXTENSION PRINCIPLE MODULE
85
PROTOFORMAL SEARCH RULES

• example
• query What is the distance between the largest
  city in Spain and the largest city in Portugal?
• protoform of query ?Attr (Desc(A), Desc(B))
• procedure
• query ?Name (A)Desc (A)
• query Name (B)Desc (B)
• query ?Attr (Name (A), Name (B))

86
PROTOFORMAL (PROTOFORM-BASED) DEDUCTION
precisiation
abstraction
antecedent
GC(p)
PF(p)
p
proposition
Deduction Database
instantiation
retranslation
consequent
q
PF(q)
proposition
87
PNL AS A DEFINITION LANGUAGE
88
BRITTLENESS OF DEFINITIONS (THE SORITES PARADOX)

• statistical independence
• A and B are independent PA(B) P(B)
• suppose that (a) PA(B) and P(B) differ by an
  epsilon (b) epsilon increases
• at which point will A and B cease to be
  independent?
• statistical independence is a matter of degree
• degree of independence is context-dependent
• brittleness is a consequence of bivalence

89
STABILITY IS A FUZZY CONCEPT

• graduality of progression from stability to
  instability

D

• Lyapounovs definition of stability leads to the
  counterintuitive conclusion that the system is
  stable no matter how large the ball is
• In reality, stability is a matter of degree

90
SIMPLE QUESTIONS THAT ARE HARD TO ANSWER

• WHAT ARE THE DEFINITIONS OF
• length
• volume
• edge
• cluster
• summary
• relevance
• density

91
MAXIMUM ?
Y
Y
maximum (possibilistic)
maximum
interval-valued
0
X
0
X
Y
Pareto maximum
Y
fuzzy-interval-valued
interval-valued
0
X
0
X
Y
fuzzy graph
Bi
Y isfg (?iAiBi)
0
X
92
HIERARCHY OF DEFINITION LANGUAGES
PNL
F.G language
fuzzy-logic-based
F language
B language
bivalent-logic-based
NL
NL natural language B language standard
mathematical bivalent-logic-based language F
language fuzzy logic language without
granulation F.G language fuzzy logic language
with granulation PNL Precisiated Natural Language
Note the language of fuzzy if-then rules is a
sublanguage of PNL
Note a language in the hierarchy subsumes all
lower languages
93
SIMPLIFIED HIERARCHY
PNL
fuzzy-logic-based
B language
bivalent-logic-based
NL
The expressive power of the B language the
standard bivalence-logic-based definition
language is insufficient
Insufficiency of the expressive power of the B
language is rooted in the fundamental conflict
between bivalence and reality
94
EVERYDAY CONCEPTS WHICH CANNOT BE DEFINED
REALISTICALY THROUGH THE USE OF B

• check-out time is 1230 pm
• speed limit is 65 mph
• it is cloudy
• Eva has long hair
• economy is in recession
• I am risk averse

95
DEFINITION OF p ABOUT 20-25 MINUTES
?
1
b-definition
0
20
25
time
?
1
f-definition
0
20
25
time
?
1
f.g-definition
0
20
25
time
P
PNL-definition (bimodal distribution)
Prob (Time is A) is B
B
6
time
A
96
INSUFFICIENCY OF THE B LANGUAGE

• Concepts which cannot be defined
• causality
• relevance
• intelligence
• Concepts whose definitions are problematic
• stability
• optimality
• statistical independence
• stationarity

97
DEFINITION OF OPTIMALITYOPTIMIZATIONMAXIMIZATION
?
gain
gain
yes
unsure
0
0
X
a
a
b
X
gain
gain
no
hard to tell
0
0
a
b
X
a
b
c
X

• definition of optimal X requires use of PNL

98
MAXIMUM ?
Y

• ?x (f (x)? f(a))
• (?x (f (x) gt f(a))

f
m
0
X
a
Y
extension principle
Y
Pareto maximum
f
f
0
X
0
X
b) (?x (f (x) dominates f(a))
99
MAXIMUM ?
Y
f (x) is A
0
X
Y
f
f ?i Ai ? Bi f if X is Ai then Y is Bi, i1,
, n
Bi
0
X
Ai
100
EXAMPLE

• I am driving to the airport. How long will it
  take me to get there?
• Hotel clerks perception-based answer about
  20-25 minutes
• about 20-25 minutes cannot be defined in the
  language of bivalent logic and probability theory
• To define about 20-25 minutes what is needed is
  PNL

101
EXAMPLE
PNL definition of about 20 to 25 minutes
Prob getting to the airport in less than about
25 min is unlikely Prob getting to the airport
in about 20 to 25 min is likely Prob getting
to the airport in more than 25 min is unlikely
P
granular probability distribution
likely
unlikely
Time
20
25
102
PNL-BASED DEFINITION OF STATISTICAL INDEPENDENCE
Y
contingency table
L
?C(M/L)
L/M
L/L
L/S
3
M
?C(S/S)
M/M
M/S
M/L
2
S
X
S/S
S/M
S/L
1
0
1
2
3
S
M
L
?C (M x L)
? (M/L)
?C (L)

• degree of independence of Y from X
• degree to which columns 1, 2, 3 are identical

PNL-based definition
103
LYAPOUNOV STABILITY IS COUNTERINTUITIVE
D
equilibrium state

• the system is stable no matter how large D is

104
PNL-BASED DEFINITION OF STABILITY

• a system is F-stable if it satisfies the fuzzy
  Lipshitz condition

fuzzy number

• interpretation

0
degree of stabilitydegree to which f is in
105
F-STABILITY
0
106
WHY IS EXPRESSIVE POWER AN IMPORTANT FACTOR?

• Definition of a concept, construct or metric may
  be viewed as a precisiation of perception of the
  definiendum
• The language in which a definition is expressed
  is a definition language
• The expressive power of a definition language
  places a limit on the complexity of the
  definiendum and on the degree to which definition
  of the definiendum approximates to its perception

107
EVERYDAY CONCEPTS WHICH CANNOT BE DEFINED
REALISTICALY THROUGH THE USE OF B

• check-out time is 1230 pm
• speed limit is 65 mph
• it is cloudy
• Eva has long hair
• economy is in recession
• I am risk averse

108
PRECISIATION/DEFINITION OF PERCEPTIONS
?
Perception ABOUT 20-25 MINUTES
1
interval
B definition
0
20
25
time
?
1
fuzzy interval
F definition
0
20
25
time
?
1
fuzzy graph
F.G definition
0
20
25
time
P
f-granular probability distribution
PNL definition
0
time
20
25
109
DEFINITION OF OPTIMALITYOPTIMIZATIONMAXIMIZATION
?
gain
gain
yes
unsure
0
0
X
a
a
b
X
gain
gain
no
hard to tell
0
0
a
b
X
a
b
c
X

• definition of optimal X requires use of PNL

110
EXAMPLE
PNL definition of about 20 to 25 minutes
Prob getting to the airport in less than about
25 min is unlikely Prob getting to the airport
in about 20 to 25 min is likely Prob getting
to the airport in more than 25 min is unlikely
P
granular probability distribution
likely
unlikely
Time
20
25
111
THE ROBERT EXAMPLE
112
THE ROBERT EXAMPLE

• the Robert example relates to everyday
  commonsense reasoning a kind of reasoning which
  is preponderantly perception-based
• the Robert example is intended to serve as a test
  of the deductive capability of a reasoning system
  to operate on perception-based information

113
THE ROBERT EXAMPLE
Version 1. My perception is that Robert
usually returns from work at about
600pm q1 What is the probability that
Robert is home at about t pm? q2 What
is the earliest time at which the probability
that Robert is home is high?
114
THE ROBERT EXAMPLE (VERSION 3)

• IDS Robert leaves office between 515pm and
  545pm. When the time of departure is about
  520pm, the travel time is usually about 20min
  when the time of departure is about 530pm, the
  travel time is usually about 30min when the time
  of departure is about 540pm, the travel time is
  about 20min
• usually Robert leaves office at about 530pm
• What is the probability that Robert is home at
  about t pm?

115
THE ROBERT EXAMPLE
Version 4

• Usually Robert returns from work at about 6 pm
• Usually Ann returns from work about
  half-an-hour later
• What is the probability that both Robert and
  Ann are
• home at about t pm?

Ann
P
Robert
1
0
time
600
t
116
CONTINUED (VERSION 1)
Q what is the probability that Robert is home at
t?
CF(q)
is ?P
?
1
t
time
0
6 pm
t
PF(q) Prob(C) is ? D
117
PROTOFORMAL DEDUCTION
THE ROBERT EXAMPLE

• IDS p usually Robert returns from work at about
  6 pm.
• TDS q what is the probability that Robert is
  home at
• about 615 pm?
• precisiation
• p Prob Time (Robert returns from work is
• about 6 pm is usually
• q Prob Time (Robert is home) is about 615 pm
• is ?D
• calibration µusually , µt , t about t
• abstraction
• p Prob X is A is B
• q Prob Y is C is ?D

118
CONTINUED
4. search in Probability module for applicable
rules
Prob X is A is B Prob Y is C is D
not found
Prob X is A is B Prob X is C is D
Prob X is A is B Prob f(X) is C is D
found
5. back to IDS and TDS event equivalence Robert
is home at t Robert returns from work before t
119
THE ROBERT EXAMPLE
event equivalence
Robert is home at about t pm Robert returns from
work before about t pm
?
before t
1
t (about t pm)
0
time
T
t
time of return
Before about t pm o about t pm
120
CONTINUED
6. back to Probability module
Prob X is A is B Prob X is C is D
7. Instantiation D Prob Robert is home at
about 615 X Time (Robert returns from
work) A 6 B usually C ? 615
121
CONCLUSION

• Existing scientific theories are based on
  bivalent logica logic in which everything is
  black or white, with no shades of gray allowed
• What is not recognized, to the extent that it
  should, is that bivalent logic is in fundamental
  conflict with reality
• Fuzzy logic is not in conflict with bivalent
  logicit is a generalization of bivalent logic in
  which everything is, or is allowed to be, a
  matter of degree
• Fuzzy logic provides a foundation for the
  methodology of computing with words and
  perceptions

122
STATISTICS
Count of papers containing the word fuzzy in
title, as cited in INSPEC and MATH.SCI.NET
databases. (data for 2003 are not
complete) Compiled by Camille Wanat, Head,
Engineering Library, UC Berkeley, November 20,
2003
INSPEC/fuzzy
Math.Sci.Net/fuzzy
1970-1979 569 1980-1989 2,404 1990-1999 23,207
2000-present 9,945 1970-present 36,125
443 2,465 5,479 2,865 11,252
123
STATISTICS

• Count of books containing the words soft
  computing in title, or published in series on
  soft computing. (source Melvyl catalog)
• Compiled by Camille Wanat, Head,
• Engineering Library, UC Berkeley,
• October 12, 2003
• Count of papers containing soft computing in
  title or published in proceedings of conferences
  on soft computing
• 2494 (1994-2002)

1994 4 1995 2 1996 7 1997 12 1998 15 1999
23 2000 36 2001 43 2002 42 Total 184
124
DEFINITION OF p ABOUT 20-25 MINUTES
1
c-definition
0
20
25
time
1
f-definition
0
20
25
time
1
f.g-definition
0
20
25
time
P
PNL-definition
Prob (Time is A) is B
B
6
time
A
125
WHAT IS A RANDOM SAMPLE?

• In most cases, a sample is drawn from a
  population which is a fuzzy set, e.g., middle
  class, young women, adults
• In the case of polls, fuzziness of the population
  which is polled may reflect the degree
  applicability of the question to the person who
  is polled
• example (Atlanta Constitution 5-29-95)
• Is O.J. Simpson guilty?
• Random sample of 1004 adults polled by phone.
• 61 said yes. Margin of error is 3
• to what degree is this question applicable to a
  person who is n years old?