PROBABILITY THEORY

1
PROBABILITY THEORY FUZZY LOGIC 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 November 2,
2003 University of Siena, Italy
2
BACKDROP
3
ORGANIZATION

• Part A Historical perspective
• probability theory (PT) vs. fuzzy logic (FL)
• limitations of PT
• Part B Generalization of PT
• from bivalent-logic-based probability theory (PT)
• to fuzzy-logic-based probability theory (PTp)
• Part C Fuzzy-logic-based probability theory (PTp)

4
PART A
5
PROBABILITY THEORY AND FUZZY LOGIC

• How does fuzzy logic relate to probability
  theory?
• This is the question that was raised by Loginov
  in 1966, shortly after the publication of my
  first paper on fuzzy sets (1965).
• Relationship between probability theory and fuzzy
  logic has been, and continues to be, an object of
  controversy.

6
PRINCIPAL VIEWS

• Inevitability of probability
• Fuzzy logic is probability theory in disguise
• The tools provided by fuzzy logic are not of
  importance
• Probability theory and fuzzy logic are
  complementary rather than competitive

7
INEVITABILITY OF PROBABILITY

• The only satisfactory description of uncertainty
  is probability. By this I mean that every
  uncertainty statement must be in the form of a
  probability that several uncertainties must be
  combined using the rules of probability and that
  the calculus of probabilities is adequate to
  handle all situations involving
  uncertaintyprobability is the only sensible
  description of uncertainty and is adequate for
  all problems involving uncertainty. All other
  methods are inadequate anything that can be done
  with fuzzy logic, belief functions, upper and
  lower probabilities, or any other alternative to
  probability can better be done with probability
  Lindley (1987)

8
CONTINUED

• The numerous schemes for representing and
  reasoning about uncertainty that have appeared in
  the AI literature are unnecessary probability is
  all that is needed Cheesman (1985)

9
CONTINUED

• My current view is
• Standard, bivalent-logic-based, probability
  theory, PT, has fundamental limitations
• To remove these limitations, it is necessary to
  restructure probability theory by shifting its
  foundation from bivalent logic to fuzzy logic

10
RELATED PAPER

• Lotfi A. Zadeh, Toward a perception-based
  theory of probabilistic reasoning with imprecise
  probabilities, special issue on imprecise
  probabilities, Journal of Statistical Planning
  and Inference, Vol. 105, pp.233-264, 2002.
• Downloadable from
• http//www-bisc.cs.berkeley.edu/BISCProgram/Projec
  ts.htm

11
THERE IS A FUNDAMENTAL CONFLICT BETWEEN BIVALENCE
AND REALITY

• in bivalent-logic-based probability theory, PT,
  only certainty is a matter of degree
• in perception-based probability theory, PTp,
  everything is, or is allowed to be, a matter of
  degree

12
BIVALENT-LOGIC-BASED THEORIES ARE INTOLERANT OF
IMPRECISION AND PARTIAL TRUTH
UNGRACEFUL DEGRADATION

• most bivalent-logic-based theories, principles,
  algorithms, artifices and definitions, break down
  when X is replaced by approximately X, where X
  is a precisely defined entity, e.g., a number, a
  relation, a quantifier.
• examples
• maximum entropy principle
• definition of symmetry
• definition of maximum
• definition of stationarity
• definition of independence

13
CONTINUED

• transitivity of subsethood
• A ? B and B ? C implies A ? C
• A ?B and B ?C implies I (indeterminate)

14
ANYTHING YOU CAN DO WITH FUZZY LOGIC, I CAN DO
WITH PROBABILITY THEORY

• Challenge Can you solve the following problems?
• The Robert Example
• Usually Robert returns from work at about 6pm.
  What is the probability that Robert is home at
  about 615pm?
• The tall Swedes problem
• Most Swedes are tall. What is the average height
  of Swedes?
• The balls-in-box problem?
• A box contains approximately 20 balls. Most are
  black. There are several times as many black
  balls as white balls. How many are white? What is
  the probability that a ball drawn at random is
  white?

15
CONTINUED

• A function, Y f(X), is defined as follows If X
  is small then Y is small
• If X is medium then Y is large If X is large
  then Y is small. What is the maximum value of f?
• Usually X is not very small. Usually X is not
  very large. What is the probability that X is
  neither small nor large?

16
NEW PROBLEMS AND NEW QUESTIONS

• interpolation with usuality-qualified rules
• If X is Ai then Y isu Bi , I1, , n
• If X is A then Y isr B
• definitions of
• Expected value (not average value)
• Relevance
• Cluster

17
BASIC PROBLEMS WITH PT
PT
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IT IS A FUNDAMENTAL LIMITATION TO BASE
PROBABILITY THEORY ON BIVALENT LOGIC

• A major shortcoming of bivalent-logic-based
  probability theory, PT, relates to the inability
  of PT to operate on perception-based information
• In addition, PT has serious problems with
• (a) brittleness of basic concepts
• (b) the it is possible but not probable dilemma

19
PARADOX

• question Does Robert have a Ph.D. degree?
• evidence Robert is a professor
• 99 of professors have a Ph.D.
• degree
• question What is the probability, p, that Robert
  has a Ph.D. degree
• answer all that can be said is that p is between
  0 and 1
• p is indeterminate

20
SIMPLE EXAMPLES OF QUESTIONS WHICH CANNOT BE
ANSWERED THROUGH THE USE OF PT

• I am driving to the airport. How long will it
  take me to get there?
• Hotel clerk About 20-25 minutes
• PT Cant tell
• I live in Berkeley. I have access to police
  department and insurance company files. What is
  the probability that my car may be stolen?
• PT Cant tell
• I live in the United States. Last year, one
  percent of tax returns were audited. What is the
  probability that my tax return will be audited?
• PT Cant tell

21
BRITTLENESS (DISCONTINUITY)

• Almost all concepts in PT are bivalent in the
  sense that a concept, C, is either true or false,
  with no partiality of truth allowed. For example,
  events A and B are either independent or not
  independent. A process, P, is either stationary
  or nonstationary, and so on. An example of
  brittleness is If all As are Bs and all Bs
  are Cs, then all As are Cs but if almost all
  As are Bs and almost all Bs are Cs, then all
  that can be said is that proportion of As in Cs
  is between 0 and 1.

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BRITTLENESS OF BIVALENT-LOGIC-BASED DEFINITIONS

• when a concept which is in reality a matter of
  degree is defined as one which is not, the
  sorites paradox points to a need for redefinition
• stability
• statistical independence
• stationarity
• linearity

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BRITTLENESS OF DEFINITIONS

• statistical independence
• P (A, B) P(A) P(B)
• stationarity
• P (X1,, Xn) P (X1-a,, Xn-a) for all a
• randomness
• Kolmogorov, Chaitin,
• in PTp, statistical independence, stationarity,
  etc are a matter of degree

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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

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THE DILEMMA OF IT IS POSSIBLE BUT NOT PROBABLE

• A simple version of this dilemma is the
  following. Assume that A is a proper subset of B
  and that the Lebesgue measure of A is arbitrarily
  close to the Lebesgue measure of B. Now, what can
  be said about the probability measure, P(A),
  given the probability measure P(B)? The only
  assertion that can be made is that P(A) lies
  between 0 and P(B). The uniformativeness of this
  assessment of P(A) leads to counterintuitive
  conclusions. For example, suppose that with
  probability 0.99 Robert returns from work within
  one minute of 6pm. What is the probability that
  he is home at 6pm?

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CONTINUED
U
U
B
B
A
A
A ? B proper subset of A
A proper subset of B
0 ? PB(A) ? 1
0 ? PA(B) ? 1

• Counterintuitive conclusion Lebesgue measure of
  B is
• Arbitrarily close to that of A, and yet PB(A)
  is indeterminate

27
CONTINUED

• Using PT, with no additional information or the
  use of the maximum entropy principle, the answer
  is between 0 and 1. This simple example is an
  instance of a basic problem of what to do when we
  know what is possible but cannot assess the
  associated probabilities or probability
  distributions. A case in point relates to
  assessment of the probability of a worst case
  scenario.

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CONSTRAINTS ON PROBABILITIES

• E1, , E2 events in U
• E(r) subsequence of (E1, , En)
• P (E(s)/ E(r)) conditional probability of E(s)
  given E(r)
• problem compute P (E(s)/ E(r)) given
• P (E(q)/ E(p)), , P (E(u)/ E(t))

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ANALYSIS
E2 ? E3
E1
E2
m4
m2
m3
m5
m1
m7
E3
m6
m8
m9
m11
prime subevent
m10
(prime implicant)
E4
mj mass (measure) of jth subevent, m (m1, ,
mk), m scenario mass assignment (support logic
programming)
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SOLUTION

• P (E3 E1, E2) ? given P (E3 E2), P (E2E1)
• given
• maximize/minimize x given y and z
• solution f (m1, , mk) ? x ? g (m1, , mk)
• worst-case scenario/best-case scenario

31
INFORMATION ORTHOGONALITY

• scenario, m, is a variable
• in the absence of knowledge of m, the bounds on
  the desired probability are 0 and 1
• the desired probability is orthogonal to the
  given probabilities

32
EXAMPLEINFORMATION ORTHOGONALITY
B
?
A
?
C
1-2?

• A, B, C are crisp events
• what is the probability of pA(C) given pA(B) and
  pB(C)

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EXAMPLEINFORMATION ORTHOGONALITY

• A, B, C are crisp events
• what is the probability of pA, B(C) given pA(C)
  and pB(C)

C
0.5-?
A
2?
0.5-?
B
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PROBABILITY THEORY AND PERCEPTIONS
35
PREAMBLE

• It is a deep-seated tradition in science to
  strive for the ultimate in rigor and precision.
  But as we enter into the age of machine
  intelligence and automated reasoning, other
  important goals come into view.

36
CONTINUED

• We begin to realize that humans have a remarkable
  capabilitya capability which machines do not
  haveto perform a wide variety of physical and
  mental tasks without any measurements and any
  computations. In performing such tasks, humans
  employ perceptions of distance, speed, direction,
  size, likelihood, intent and other attributes of
  physical and mental objects.

37
CONTINUED

• To endow machines with this capability, what is
  needed is a theory in which the objects of
  computation are, or are allowed to be,
  perceptions. The aim of the computational theory
  of perceptions is to serve this purposepurpose
  which is not served by existing theories.

38
KEY IDEA

• In the computational theory of perceptions,
  perceptions are dealt with through their
  descriptions in a natural language

39
COMPUTATIONAL THEORY OF PERCEPTIONS (CTP) BASIC
POSTULATES

• perceptions are intrinsically imprecise
• imprecision of perceptions is a concomitant of
  the bounded ability of sensory organsand
  ultimately the brainto resolve detail and store
  information

40
KEY POINTS

• a natural language is, above all, a system for
  describing and reasoning with perceptions
• in large measure, human decisions are
  perception-based
• one of the principal purposes of CWP (Computing
  with Words and Perceptions) is that of making it
  possible to construct machines that are capable
  of operating on perception-based information
  expressed in a natural language
• existing bivalent-logic-based machines do not
  have this capability

41
MEASUREMENT-BASED VS. PERCEPTION-BASED INFORMATION
INFORMATION
measurement-based numerical
perception-based linguistic

• it is 35 C
• Eva is 28
• probability is 0.8

• It is very warm
• Eva is young
• probability is high
• 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

42
MEASUREMENT-BASED VS. PERCEPTION-BASED CONCEPTS
measurement-based perception-based expected
value usual value stationarity regularity con
tinuous smooth Example of a regular
process T (t0 , t1 , t2 ) ti travel time from
home to office on day i.
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WHAT IS CWP?
THE BALLS-IN-BOX PROBLEM

• Version 1. Measurement-based
• a box contains 20 black and white balls
• over 70 are black
• there are three times as many black balls as
  white balls
• what is the number of white balls?
• what is the probability that a ball drawn at
  random is white?
• I draw a ball at random. If it is white, I win
  20 if it is black, I lose 5. Should I play the
  game?

44
CONTINUED

• Version 2. Perception-based
• a box contains about 20 black and white balls
• most are black
• there are several times as many black balls as
  white balls
• what is the number of white balls?
• what is the probability that a ball drawn at
  random is white?
• I draw a ball at random. If it is white, I win
  20 if it is black, I lose 5. Should I play the
  game?

45
CONTINUED

• Version 3. Perception-based
• a box contains about 20 black balls of various
  sizes
• most are large
• there are several times as many large balls as
  small balls
• what is the number of small balls?
• what is the probability that a ball drawn at
  random is small?

box
46
COMPUTATION (version 1)

• measurement-based
• X number of black balls
• Y2 number of white balls
• X ? 0.7 20 14
• X Y 20
• X 3Y
• X 15 Y 5
• p 5/20 .25
• (integer programming)

• perception-based
• X number of black balls
• Y number of white balls
• X most 20
• X several Y
• X Y 20
• P Y/N
• (fuzzy integer programming)

47
FUZZY INTEGER PROGRAMMING
Y
X most 20
XY 20
X several y
x
1
48
PART B
49
PROBLEMS WITH PT

• Bivalent-logic-based PT is capable of solving
  complex problems
• But, what is not widely recognized is that PT
  cannot answer simple questions drawn from
  everyday experiences
• To deal with such questions, PT must undergo
  three stages of generalization, leading to
  perception-based probability theory, PTp

50
BASIC STRUCTURE OF PROBABILITY THEORY
PROBABILITY THEORY
measurement- based
perception- based
frequestist objective
bivalent-logic- based
fuzzy-logic- based
Bayesian subjective
PTp
PT
generalization

• In PTp everything is or is allowed to be
  perception-based

51
THE NEED FOR A RESTRUCTURING OF PROBABILITY THEORY

• to circumvent the limitations of PT three stages
  of generalization are required
• f-generalization
• f.g-generalization
• nl-generalization

PT
PT
PT
PTp
f-generalization
f.g-generalization
nl-generalization
52
FUNDAMENTAL POINTS

• the point of departure in perception-based
  probability theory (PTp) is the postulate
• subjective probabilityperception of likelihood
• perception of likelihood is similar to
  perceptions of time, distance, speed, weight,
  age, taste, mood, resemblance and other
  attributes of physical and mental objects
• perceptions are intrinsically imprecise,
  reflecting the bounded ability of sensory organs
  and, ultimately, the brain, to resolve detail and
  store information
• perceptions and subjective probabilities are
  f-granular

53
F-GENERALIZATION

• f-generalization of a theory, T, involves an
  introduction into T of the concept of a fuzzy set
• f-generalization of PT, PT , adds to PT the
  capability to deal with fuzzy probabilities,
  fuzzy probability distributions, fuzzy events,
  fuzzy functions and fuzzy relations

?
?
A
A
1
X
X
0
0
54
F.G-GENERALIZATION

• f.g-generalization of T, T, involves an
  introduction into T of the concept of a
  granulated fuzzy set
• f.g-generalization of PT, PT , adds to PT
  the capability to deal with f-granular
  probabilities, f-granular probability
  distributions, f-granular events, f-granular
  functions and f-granular relations

?
?
A
A
1
1
X
0
X
0
55
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

56
NL-GENERALIZATION
nl-generalization
A
AP
PNL-defined set
crisp set
nl-generalization
PT
PTP
crisp probability PNL-defined probability crisp
relation PNL-defined relation crisp
independence PNL-defined independence
57
NL-GENERALIZATION

• Nl-generalization of T. Tnl , involves an
  addition to T of a capability to operate on
  propositions expressed in a natural language
• nl-generalization of T adds to T a capability
  to operate on perceptions described in a natural
  language
• nl-generalization of PT, PTnl , adds to PT a
  capability to operate on perceptions described in
  a natural language
• nl-generalization of PT is perception-based
  probability theory, PTp
• a key concept in PTp is PNL (Precisiated Natural
  Language)

58
PERCEPTION OF A FUNCTION
Y
f
0
Y
medium x large
f (fuzzy graph)
perception
f f
if X is small then Y is small if X is
medium then Y is large if X is large then Y
is small
0
X
59
BIMODAL DISTRIBUTION (PERCEPTION-BASED
PROBABILITY DISTRIBUTION)
probability
P3
P2
P1
X
0
A2
A1
A3
P(X) Pi(1)\A1 Pi(2)\A2 Pi(3)\A3 Prob X
is Ai is Pj(i)
P(X) low\small high\medium low\large
60
CONTINUED

• function if X is small then Y is large
• (X is small, Y is large)
• probability distribution low \ small low \
  medium high \ large
• Count \ attribute value distribution 5 \ small
  8 \ large
• PRINCIPAL RATIONALES FOR F-GRANULATION
• detail not known
• detail not needed
• detail not wanted

61
BIMODAL PROBABILITY DISTRIBUTIONS (LAZ 1981)
(a) possibility\probability
(b) probability\\possibility
P
U
g
P3
P2
P1
X
A3
A2
A1
62
BIMODAL PROBABILITY DISTRIBUTION
X a random variable taking values in U g
probability density function of X
g
P3
g
f-granulation
P2
P1
X
X
A3
A2
A1
63
CONTINUED
P defines a possibility distribution of g
problems a) what is the probability of a
perception-based event A in U b) what is the
perception-based expected value of X
64
PROBABILITY OF A PERCEPTION-BASED EVENT
knowing ?(g)
problem Prob X is A is ?B
Extension Principle
subject to
65
CONTINUED
subject to
66
EXPECTED VALUE OF A BIMODAL PD
Extension Principle
subject to
67
PERCEPTION-BASED DECISION ANALYSIS
ranking of f-granular probability distributions
PA
0
X
PB
0
X
maximization of expected utility ranking of
fuzzy numbers
68
USUALITY CONSTRAINT PROPAGATION RULE
X random variable taking values in U g
probability density of X
X isu A Prob X is B is C
X isu A
Prob X is A is usually
subject to
69
New Tools
70
NEW TOOLS
computing with numbers
computing with words and perceptions


CWP
CN
PNL
IA
precisiated natural language
computing with intervals
PTp
CTP
THD
PFT
PT
CTP computational theory of
perceptions PFT protoform theory PTp
perception-based probability theory THD
theory of hierarchical definability
probability theory
71
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
72
PRECISIATED NATURAL LANGUAGE
PNL
73
CWP AND PNL

• a concept which plays a central role in CWP is
  that of PNL (Precisiated Natural Language)
• basically, a natural language, NL, is a system
  for describing perceptions
• perceptions are intrinsically imprecise
• imprecision of natural languages is a reflection
  of the imprecision of perceptions
• the primary function of PNL is that of serving as
  a part of NL which admits precisiation
• PNL has a much higher expressive power than any
  language that is based on bivalent logic

74
PRINCIPAL FUNCTIONS OF PNL

• knowledgeand especially world knowledgedescripti
  on language
• Robert is tall
• heavy smoking causes lung cancer
• definition language
• smooth function
• stability
• deduction language

A is near B B is near C C is not far from A
75
PNL
KEY POINTS

• PNL is a subset of precisiable propositions/comman
  ds/questions in NL
• PNL is equipped with two dictionaries
• (1) from NL to GCL and (2) from GCL to PFL and
  (3) a modular multiagent deduction database (DDB)
  of rules of deduction (rules of generalized
  constrained propagation) expressed in PFL
• the deduction database includes a collection of
  modules and submodules, among them the WORLD
  KNOWLEDGE module

76
THE CONCEPT OF PRECISIATION
NL (natural language)
PL (precisiable language)
p
p
translation
precisiation
translate of p precisiation of p
proposition

• p is precisiable w/r to PL p is translatable
  into PL
• criterion of precisiability p is an object of
  computation
• PL propositional logic
• predicate logic
• modal logic
• Prolog
• LISP
• SQL
• Generalized Constraint Language (GCL) p
  GC-form

77
PRECISIATION

• precisiation is not coextensive with meaning
  representation
• precisiation of p precisiation of meaning of p
• example
• p usually Robert returns from work at about
  6pm.
• I understand what you mean but can you be more
  precise?
• yes
• p Prob (Time (Robert.returns.from.work) is
  6) is usually

µ
µ
6
usually
1
1
0
0
6
0.5
1
78
THE CONCEPT OF A GENERALIZED CONSTRAINT (1985)
GC-form
X isr R
granular value of X
constraining relation
modal variable (defines modality)
constrained variable

• principal modalities
• possibilistic (r blank) X is R ,
  Rpossibility distribution of X
• probabilistic (r p) X isp R
  Rprobability distribution of X
• veristic (r v) X isv R
  Rverity (truth) distribution of X
• usuality (ru) X isu R Rusual value
  of X
• random set (rrs) X isrs R
  Rfuzzy-set-valued distribution of X
• fuzzy graph (rfg) X isfg Rfuzzy
  graph of X
• bimodal (rbm) X isbm R Rbimodal
  distribution of X
• Pawlak set (rps) X isps R upper and
  lower approximation to X

79
GENERALIZED CONSTRAINT

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

X isr R
copula
GC-form (generalized constraint form of type r)
type (modality) 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)

80
CONSTRAINT QUALIFICATION

• constraint qualification (X isr R) is q
• q
• example (X is small) is unlikely

qualifier
possibility
probability
verity (truth)
81
INFORMATION PRINCIPAL MODALITIES

• possibilistic r blank
• X is R (R possibility distribution of X)
• probabilistic r p
• X isp R (R probability distribution of X)
• veristic r v
• X isv R (R verity (truth) distribution of X)
  
• if r is not specified, default mode is
  possibilistic

82
EXAMPLES (POSSIBILISTIC)

• Eva is young Age (Eva) is young
• Eva is much younger than Maria
• (Age (Eva), Age (Maria)) is much younger
• most Swedes are tall
• ?Count (tall.Swedes/Swedes) is most

X
R
R
X
R
X
83
EXAMPLES (PROBABILISITIC)

• X is a normally distributed random variable with
  mean m and variance ?2
• X isp N(m, ?2)
• X is a random variable taking the values u1, u2,
  u3 with probabilities p1, p2 and p3, respectively
• X isp (p1\u1p2\u2p3\u3)

84
EXAMPLES (VERISTIC)

• Robert is half German, quarter French and quarter
  Italian
• Ethnicity (Robert) isv (0.5German
  0.25French 0.25Italian)
• Robert resided in London from 1985 to 1990
• Reside (Robert, London) isv 1985, 1990

85
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

86
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
RANDOM SET MODULE
FUZZY LOGIC MODULE
EXTENSION PRINCIPLE MODULE
87
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

88
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
89
WHAT IS A PROTOFORM?

• protoform abbreviation of prototypical form
• informally, a protoform, A, of an object, B,
  written as APF(B), is an abstracted summary of B
• usually, B is lexical entity such as proposition,
  question, command, scenario, decision problem,
  etc
• more generally, B may be a relation, system,
  geometrical form or an object of arbitrary
  complexity
• usually, A is a symbolic expression, but, like B,
  it may be a complex object
• the primary function of PF(B) is to place in
  evidence the deep semantic structure of B

90
TRANSLATION FROM NL TO PFL
examples Eva is young A (B) is C 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
Age
Eva
young
Age
Eva
Age
much younger
Pat
usually
about 6 pm
Time (Robert returns from work)
91
PNL AS A DEFINITION LANGUAGE
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
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
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) 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
PROTOFORM LANGUAGE
PFL
104
DEDUCTION (COMPUTING) WITH PERCEPTIONS
deduction
p1 p2 pn
pn1
example
Dana is young Tandy is a few years older than
Dana Tandy is (youngfew)
deduction with perceptions involves the use of
protoformal rules of generalized constraint
propagation
105
MULTILEVEL STRUCTURES

• An object has a multiplicity of protoforms
• Protoforms have a multilevel structure
• There are three principal multilevel structures
• Level of abstraction (?)
• Level of summarization (?)
• Level of detail (?)
• For simplicity, levels are implicit
• A terminal protoform has maximum level of
  abstraction
• A multilevel structure may be represented as a
  lattice

106
ABSTRACTION LATTICE
example
most Swedes are tall
Q Swedes are tall
most As are tall
most Swedes are B
Q Swedes are B
Q As are tall
most As are Bs
Q Swedes are B
Q As are Bs
Count(B/A) is Q
107
LEVELS OF SUMMARIZATION

• example
• p it is very unlikely that there will be a
  significant increase in the price of oil in the
  near future
• PF(p) Prob(E) is A

very.unlikely
significant increase in the price of oil in the
near future
108
CONTINUED
semantic network representation of E
E
E
modifier
variation
attribute
mod
var
attr
significant
increase
price
oil
epoch
future
mod
near
109
CONTINUED

• E significant increase in the price of oil in
  the near future
• f function of time
• PF(E) (B(f) is C, D(f) is E)

significant.increase
variation
near.future
epoch
110
CONTINUED
Precisiation (f.b.-concept) E Epoch
(Variation (Price (oil)) is significant.increase)
is near.future
Price
significant increase
Price
current
present
Time
near.future
111
CONTINUED
precisiation of very unlikely
µ
1
likely
unlikely ant(likely)
very unlikely 2ant(likely)
V
1
0
µvery.unlikely (v) (µlikely (1-v))2
112
PROTOFORM OF A QUERY

• largest port in Canada?
• second tallest building in San Francisco

B
A
X
?X is selector (attribute (A/B))
San Francisco
buildings
height
2nd tallest
113
TEST QUERY (GOOGLE)

• population of largest city in Spain failure
• largest city in Spain Madrid, success
• population of Madrid success

114
PROTOFORM OF A DECISION PROBLEM

• buying a house
• decision attributes
• measurement-based price, taxes, area, no. of
  rooms,
• perception-based appearance, quality of
  construction, security
• normalization of attributes
• ranking of importance of attributes
• importance function w(attribute)
• importance function is granulated L(low), M
  (medium), H (high)

115
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
Count(A/B) is Q
116
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

117
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

118
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

119
ORGANIZATION OF WORLD KNOWLEDGEEPISTEMIC
(KNOWLEDGE-DIRECTED) LEXICON (EL)
(ONTOLOGY-RELATED)
j
rij
wij granular strength of association between i
and j
wij
i
K(i)
network of nodes and links
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

120
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
121
EPISTEMIC LEXICON
FORMAT OF RELATIONS
perception-based relation
lexine
attributes
granular values
example
car
G 20 \ ? 15k 40 \ 15k, 25k
granular count
122
PROTOFORM-BASED DEDUCTION
123
THE CONCEPT OF i.PROTOFORM

• i.protoform idealized protoform
• the key idea is to equate the grade of
  membership, µA(u), of an object, u, in a fuzzy
  set, A, to the distance of u from an i.protoform
• this idea is inspired by E. Roschs work (ca
  1972) on the theory of prototypes

fuzzy set
U
A
u
object
distance of u from i.protoform
d
i.protoform

• d is defined via PNL

124
PROTOFORM-CENTERED CONCEPTS EXAMPLE EXPECTED
VALUE (f.f-concept)

• X real-valued random variable with probability
  density g
• standard definition of expected value of X
• the label expected value is misleading

E( X ) average value of X
125
i.PROTOFORM-BASED DEFINITION OF EXPECTED VALUE
g
g
U
0
µ
normalized g
1
i.protoform of expected value
U
0
126
CONTINUED
gn
normalized probability density of X
i.protoform E(X)
U
0

• E(X) is a fuzzy set
• grade of membership of a particular function,
  E(X), in the fuzzy set of expected value of X is
  the distance of E(X) form best-fitting
  i.protoform

127
PROTOFORM AND PF-EQUIVALENCE
knowledge base (KB)
PF-equivalence class (P)
P
protoform (p) Q As are Bs
p
most Swedes are tall
q
few professors are rich

• P is the class of PF-equivalent propositions
• P does not have a prototype
• P has an abstracted prototype Q As are Bs
• P is the set of all propositions whose protoform
  is Q As are Bs

128
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?

129
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 aconnection time in Denver is short. If the flight
  is missed, then the time of arrival in St. Louis
  will be c, with cb. Question Which option is
  best?

130
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)
131
PROTOFORM EQUIVALENCE
gain
c
1
2
0
options
a
b
132
PROTOFORM-CENTERED KNOWLEDGE ORGANIZATION
knowledge base
PF-module
PF-module
PF-submodule
133
EXAMPLE
module
submodule
set of cars and their prices
134
TEST QUERY (GOOGLE)

• distance between largest city in Spain and
  largest city in Portugal failure
• largest city in Spain Madrid (success)
• largest city in Portugal Lisbon (success)
• distance between Madrid and Lisbon (success)

135
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))

136
PROTOFORMAL (PROTOFORM-BASED) DEDUCTION
precisiation
abstraction
antecedent
GC(p)
PF(p)
p
proposition
Deduction Database
instantiation
retranslation
consequent
q
PF(q)
proposition
137
FORMAT OF PROTOFORMAL DEDUCTION RULES
protoformal rule
symbolic part
computational part
138
PROTOFORM DEDUCTION RULE GENERALIZED MODUS PONENS
fuzzy logic
classical
X is A If X is B then Y is C Y is D
A A B B
symbolic
D A(BC)
(fuzzy graph Mamdani)
computational 1
D A(B?C)
(implication conditional relation)
computational 2
139
PROTOFORMAL RULES OF DEDUCTION
examples
X is A (X, Y) is B Y is A?B
symbolic part
computational part
Prob (X is A) is B Prob (X is C) is D
subject to
140
PROTOFORM-BASED (PROTOFORMAL) DEDUCTION

• Rules of deduction in the Deduction Database
  (DDB) are protoformal
• examples (a) compositional rule of inference

X is A (X, Y) is B Y is AB
symbolic
computational
(b) extension principle
X is A Y f(X) Y f(A)
Subject to
symbolic
computational
141
THE TALL SWEDES PROBLEM

• p most Swedes are tall
• Q What is the average height of Swedes?
• Try
• p p Count (B/A) is Q
• q q F(C/A) is ?R
• answer to q cannot be inferred from p
• level of summarization of p has to be reduced

Swedes
height attribute
functional of height attribute
142
CONTINUED
precisiation

• p p Prop(tall.Swedes/Swedes) is most
• q q Ave.height is ?R
• p p Prob F(B/A) is ?Q
• q q Ave F(B/A) is ?R
• protoformal deduction rule
• symbolic Prop (F(B/A)) is Q
• Ave F(B/A) is R
• computational
• subject to

precisiation
abs
abs
143
CONTINUED

• example
• IDS p Most Swedes are tall
• TDS q What is the average height of Swedes?
• g(u) count density g(u)du number of Swedes
  whose height is between u and udu

g
g(u)
u
250cm M
height
144
PARTICULARIZATION (LAZ 1975)

• P population of objects
• R relation describing P
• example
• R population of Swedes
• R Height weight age
• R particularized R
• R Height is tall population of tall Swedes

145
CONTINUED

• p p Count(SwedesHeight is tall/Swedes) is
  most
• p Count(RA is B/R) is Q
• q q ? Ave (RA is B A)

rule
Count(RA is B/R) is Q Ave(RA is B is ?C
146
CONTINUED
g
g
gdg
g
0
u
height
udu
g(u) height distribution
is most
is ?C
147
CONTINUED
subject to
148
RULES OF DEDUCTION

• Rules of deduction are basically rules governing
  generalized constraint propagation
• The principal rule of deduction is the extension
  principle

X is A f(X,) is B
Subject to
computational
symbolic
149
GENERALIZATIONS OF THE EXTENSION PRINCIPLE
information constraint on a variable
f(X) is A g(X) is B
given information about X
inferred information about X
Subject to
150
CONTINUED
f(X1, , Xn) is A g(X1, , Xn) is B
Subject to
(X1, , Xn) is A gj(X1, , Xn) is Yj , j1,
, n (Y1, , Yn) is B
Subject to
151
PROBLEM
X real-valued random variable
f(X) isp P g(X) isr ?Q
g(X) X f(X)
q1
p1
q2
p2
q3
p3
q4
q1 ? p1 q2 ? p1 q1 q2 p1
152
COUNT-AND MEASURE-RELATED RULES
?
Q
crisp
1
ant (Q)
Q As are Bs ant (Q) As are not Bs
r
0
1
?
Q As are Bs Q1/2 As are 2Bs
1
Q
Q1/2
r
0
1
most Swedes are tall ave (height) Swedes is ?h
Q As are Bs ave (BA) is ?C
,
153
CONTINUED
not(QAs are Bs) (not Q) As are Bs
Q1 As are Bs Q2 (AB)s are Cs Q1 Q2
As are (BC)s
Q1 As are Bs Q2 As are Cs (Q1 Q2 -1)
As are (BC)s
154
DEDUCTION MODULE

• rules of deduction are rules governing
  generalized constraint propagation
• rules of deduction are protoformal
• examples
• generalized modus ponens

X is A if X is B then Y is
C Y is A (B C)
Prob (A) is B Prob (C) is D
subject to
155
REASONING WITH PERCEPTIONS DEDUCTION MODULE
initial data set
initial generalized constraint set
IDS
IGCS
perceptions p
GC-forms GC(p)
translation
explicitation precisiation
IGCS
IPS
initial protoform set
GC-form GC(p)
protoforms PF(p)
abstraction
deinstantiation
TPS
TDS
IPS
terminal data set
terminal protoform set
initial protoform set
goal-directed
deinstantiation
deduction
156
PROTOFORMAL CONSTRAINT PROPAGATION
p
GC(p)
PF(p)
Age (Dana) is young
Dana is young
X is A
Age (Tandy) is (Age (Dana))
Tandy is a few years older than Dana
Y is (XB)
few
X is A Y is (XB) Y is AB
Age (Tandy) is (youngfew)
157
EXAMPLE OF DEDUCTION
most Swedes are tall ? R Swedes are very tall
s/a-transformation
most Swedes are tall
Q As are Bs
Q As are Bs Q½ As are 2Bs
1
most
most
most½ Swedes are very tall
r
0
1
0.25
0.5
158
INTERPOLATION
is ?A
subject to
159
CONTINUED

• (g) possibility distribution of g
• ?(g)

extension principle
?(g) ?(f(g))
?(v) supg(?(g))
subject to
v f(g)
160
EXPECTED VALUE
is ?A
subject to
161
CONTINUED

• Prob X is Ai is Pj(i), i1, , m , j1, ,
  n
• g(u)du1
• G is small ?u(g(u) is small)

Prob X is A is ?v
g(u)?Ai(u)du
Prob X is Ai
construct
162
PROBABILITY MODULE
163
INTERPOLATION OF BIMODAL DISTRIBUTIONS
P
g(u) probability density of X
p2
p
p1
pn
X
0
A1
A2
A
An
pi is Pi granular value of pi , i1, , n (Pi ,
Ai) , i1, , n are given A is given (?P, A)
164
INTERPOLATION MODULE AND PROBABILITY MODULE
Prob X is Ai is Pi , i 1, , n Prob X is
A is Q
subject to
165
PROBABILISTIC CONSTRAINT PROPAGATION RULE (a
special version of the generalized extension
principle)
is R
is ?S
subject to
166
USUALITY SUBMODULE
167
CONJUNCTION
X is A X is B X is A B
X isu A X isu B X isr A B

• determination of r involves interpolation of a
  bimodal distribution

168
USUALITY QUALIFIED RULES
X isu A X isun (not A)
X isu A Yf(X) Y isu f(A)
169
USUALITY QUALIFIED RULES
X isu A Y isu B Z f(X,Y) Z isu f(A, B)
170
EXTENSION PRINCIPLE MODULE
171
PRINCIPAL COMPUTATIONAL RULE IS THE EXTENSION
PRINCIPLE (EP)
point of departure function evaluation
Y
f
f(a)
X
0
a
Xa Yf(X) Yf(a)
172
EXTENSION PRINCIPLE HIERARCHY
EP(0,0)
argument
function
EP(0,1)
EP(0,1b)
EP(1,0)
Extension Principle
EP(0,2)
EP(1,1)
EP(1,1b)
EP(2,0)
Dempster-Shafer
Mamdani (fuzzy graph)
173
VERSION EP(0,1) (1965 1975)
Y
f(A)
f
X
0
A
X is A Yf(X) Yf(A)
subject to
174
VERSION EP(1,1) (COMPOSITIONAL RULE OF INFERENCE)
(1965)
Y
R
f(A)
X
0
A
X is A (X,Y) is R Y is A R
175
EXTENSION PRINCIPLE EP(2,0) (Mamdani)
Y
fuzzy graph (f)
X
0
a
(if X is AI then Y is BI)
176
VERSION EP(2,1)
Y
f (granulated f)
f(A)
X
0
A
X is A (X, Y) is R Y is ?i mi ? Bi
R ?i AiBi
mi supu (µA(u) ? µAi (u)) matching coefficient
177
VERSION EP(1,1b) (DEMPSTER-SHAFER)
X isp (p1\u1 pu\un) (X,Y) is R
Y isp (p1\R(u1) pn\R(un))
Y is a fuzzy-set-valued random variable
µR(ui) (v) µR (ui, v)
178
VERSION GEP(0,0)
f(X) is A g(X) is g(f -1(A))
subject to
179
GENERALIZED EXTENSION PRINCIPLE
f(X) is A g(Y) is B Zh(X,Y)
Z is h (f-1(A), g-1 (B))
subject to
180
U-QUALIFIED EXTENSION PRINCIPLE
Y
Bi
X
0
Ai
If X is Ai then Y isu Bi, i1,, n X isu
A Y isu ?I mi?Bi
m supu (µA(u)?µAi(u)) matching coefficient
181
THE ROBERT EXAMPLE
182
PROTOFORMAL DEDUCTIONTHE ROBERT EXAMPLE

• The Robert example is intended to serve as an
  illustration of protoformal deduction. In
  addition, it is intended to serve as a test of
  ability of standard probability theory, PT, to
  operate on perception-based information
• IDS Usually Robert returns from work at about 6
  pm
• TDS What is the probability that Robert is home
  at about t pm?

183
SOLUTION

• Precisiation
• p usually Robert returns from work at about 6
  pm
• p?p Prob(Return.Robert.from.work is about.6 pm
• is usually)
• What is the probability that Robert is home at
  about t pm?
• q?q Prob(Robert.home.at.about.t pm) is ? D
• Abstraction
• p?p Prob(X is A) is B
• q?q Prob(Y is C) is ?D

X
A
B
Y
C
D
184
CONTINUED

• Search in Deduction Database
• desired rule Prob(X is A) is B
• Prob(Y is C) is ?D
• top-level agent reports that desired rule is not
  in DDB, but that a variant rule,
• Prob(X is A) is B
• Prob(X is C) is ?D ,
• is in DDB
• Can the desired rule be linked to the variant
  rule?

185
CONTINUED

• Computation
• Prob(X is A) is B
• Prob(X is C) is ?D
• computational part (g probability density of X)

subject to
186
CONTINUED

• Search for linkage
• If Robert does not leave his home after returning
  from work, then
• Robert is at home at about.t pm
• Robert returns from work at.or.before t pm
• consequently
• Y is about t pm X is ? about.t pm

187
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
188
CONTINUED

• Answer
• Instantiation D Prob Robert is home at about
  t
• X Time (Robert returns from work)
• A 6
• B usually
• C ? t

subject to
189
SUMMATIONBASIC POINTS

• Among a large number and variety of perceptions
  in human cognition, there are three that stand
  out in importance
• perception of likelihood
• perception of truth (similarity, compatibility,
  correspondence)
• perception of possibility (ease of attainment)
• These perceptions, as most others, are a matter
  of degree
• In bivalent-logic-based probability theory, PT,
  only perception of likelihood is a matter of
  degree
• In perception-based probability theory, PTp, in
  addition to the perception of likelihood,
  perceptions of truth and possibility are, or are
  allowed to be, a matter of degree

190
CONCLUSION

• Conceptually, computationally and mathematically,
  perception-based probability theory is
  significantly more complex than measurement-based
  probability theory.
• Complexity is the price that has to be paid to
  reduce the gap between theory and reality.

191
COMMENTS
from preface to the Special Issue on Imprecise
Probabilities, Journal of Statistical Planning
and Inference, Vol. 105, 2002 There is a wide
range of views concerning the sources and
significance of imprecision. This ranges from de
Finettis view, that imprecision arises merely
from incomplete elicitation of subjective
probabilities, to Zadehs view, that most of the
information relevant to probabilistic analysis is
intrinsically imprecise, and that there is
imprecision and fuzziness not only in
probabilities, but also in events, relations and
properties such as independence. The research
program outlined by Zadeh is a more radical
departure from standard probability theory than
the other approaches in this volume. (Jean-Marc
Bernard)
192
CONTINUED
From Peter Walley (Co-editor of special
issue) "I think that your ideas on
perception-based probability are exciting and I
hope that they will be published in probability
and statistics journals where they will be widely
read. I think that there is an urgent need for a
new, more innovative and more eclectic, journal
in the area. The established journals are just
not receptive to new ideas - their editors are
convinced that all the fundamental ideas of
probability were established by Kolmogorov and
Bayes, and that it only remains to develop them!
"  
193
CONTINUED
From Patrick Suppes (Stanford) I am not
suggesting I fully understand what the final
outcome of this direction of work will be, but I
am confident that the vigor of the debate, and
even more the depth of the new applications of
fuzzy logic, constitute a genuinely new turn in
the long history of concepts and theories for
dealing with uncertainty.
194
STATISTICS
Count of papers containing the word fuzzy in
the title, as cited in INSPEC and MATH.SCI.NET
databases. (data for 2002 are not
complete) Compiled by Camille Wanat, Head,
Engineering Library, UC Berkeley, April 17, 2003
INSPEC/fuzzy
Math.Sci.Net/fuzzy
1970-1979 569 1980-1989 2,404 1990-1999 23,207
2000-present 8,745 1970-present 34,925
443 2,466 5,472 2,319 10