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FUZZY LOGIC SYSTEMS: ORIGIN, CONCEPTS, AND TRENDS

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Title: FUZZY LOGIC SYSTEMS: ORIGIN, CONCEPTS, AND TRENDS


1
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
2
LOTFI A. ZADEH COMPUTER SCIENCE DIVISION,
DEPARTMENT OF EECS UNIVERSITY OF
CALIFORNIA BERKELEY, CA 94720-1776 TEL (510)
642-4959 FAX (510) 642-1712 SECRETARY (510)
642-8271 HOME FAX (510) 526-2433 E-MAIL
zadeh_at_cs.berkeley.edu
3
BACKDROP
4
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
5
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

6
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
7
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

8
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

9
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.

10
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

11
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.

12
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.
13
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
14
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
15
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.

16
  • 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.

17
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.

18
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.

19
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)

20
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.

21
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.

22
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

23
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.

24
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)
25
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

26
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
27
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

28
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
29
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?

30
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?

31
PROTOFORM EQUIVALENCE
gain
c
1
2
0
options
a
b
32
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.

33
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
34
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
35
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
36
  • 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

37
  • 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

38
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

39
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

40
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
41
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

42
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.
43
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
44
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

45
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
46
EMERGING APPLICATIONS OF FUZZY LOGIC
  • computational theory of perceptions
  • natural language processing
  • financial engineering
  • biomedicine
  • legal reasoning
  • forecasting

47
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

48
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

49
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

50
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)

51
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

52
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)
53
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

54
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
54
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)

55
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)
56
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

58
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

59
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
60
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

62
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.

68
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?
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