A New Frontier in ComputationComputation with Information Described in Natural Language

1
A New Frontier in ComputationComputation with
Information Described in Natural Language Lotfi
A. Zadeh Computer Science Division Department
of EECSUC Berkeley SMC06 Taipei,
Taiwan October 9, 2006 URL http//www-bisc.cs.be
rkeley.edu URL http//www.cs.berkeley.edu/zadeh/
Email Zadeh_at_eecs.berkeley.edu
2
PREAMBLE

• Humans have a remarkable capability to reason and
  make decisions in an environment of imprecision,
  uncertainty and partiality of knowledge, truth
  and class membership. One of the principal
  objectives of AI is formalization/mechanization
  of this capability.
• Computation with information described in natural
  language, or NL-Computation for short, is a step
  toward achievement of this objective.

3
A HISTORICAL NOTE

• NL-Computation is a culmination of my
  long-standing interest in exploring what I have
  always believed to be a central issue in fuzzy
  logic, namely, the relationship between fuzzy
  logic and natural languages. My 1971 paper
  entitled Quantitative Fuzzy Semantics, was the
  first in a long series.

4
WHAT IS NL-COMPUTATION?
Here are a few examples

• Trip planning
• I am planning to drive from Berkeley to Santa
  Barbara, with stopover for lunch in Monterey.
  Usually, it takes about two hours to get to
  Monterey. Usually it takes about one hour to have
  lunch. It is likely that it will take about six
  hours to get from Monterey to Santa Barbara. At
  what time should I leave Berkeley to get to Santa
  Barbara, with high probability, before about 6 pm?

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CONTINUED

• Maximization
• f is a function from reals to reals described
  as 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 of f?
• Balls-in-box
• A box contains about twenty balls of various
  sizes. Most are large. What is the number of
  small balls? What is the probability that a ball
  drawn at random is neither small nor large?

6
CONTINUED

• Swedes and Italians
• Most Swedes are much taller than most Italians.
  What is the difference in the average height of
  Swedes and the average height of Italians?
• Flight delay
• Usually most United Airlines flights from San
  Francisco leave on time. What is the probability
  that my flight will be delayed?

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THE BASIC PROBLEMEVALUATION OF A FUNCTION

• ?Zf(X, Y)
• IML(X) information about X expressed in
  mathematical language
• INL(X) information about X expressed in natural
  language
• MLComputation ?Z f(X,Y)
• NLComputation ?Z f(X,Y)

IML
IML
IML
IML
INL
INL
INL
INL
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EQUIVALENCE

• A natural language may be viewed as a system for
  describing perceptions.
• Example Most Swedes are tall
• Information described in natural language
  perception-based information
• INL (X) IPB(X)

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PRECISIATION

• A key concept in NL-Computation is that of
  precisiation of meaning. Precisiation is a
  prerequisite to computation.
• NLComputation MNLComputation
• Precisiand may be viewed as a model of meaning.
• Cointension of precisiand measure of goodness
  of model of meaning

precisiation
precisiend
precisand
precisiation
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BASIC POINTS

• Much of human knowledge is expressed in natural
  language
• A natural language is basically a system for
  describing perceptions
• Perceptions are intrinsically imprecise,
  reflecting the bounded ability of sensory organs,
  and ultimately the brain, to resolve detail and
  store information
• Imprecision of perceptions is passed on to
  natural languages, resulting in semantic
  imprecision
• Semantic imprecision of natural languages places
  the problem of computation with information
  described in natural language beyond the reach of
  existing techniques based on bivalent logic and
  probability theory.

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BASIC STRUCTURE OF NL-COMPUTATION
Basically, NL-Computation is a system of
computation in which the objects of computation
are predicates and propositions drawn from a
natural language
COMPUTATION
PRECISIATION
NL
Pre1(p)
Pren(p)
p q
information
solution
reduction
ans(q/p)
question
Pre1(q)
Pren(q)
final solution
reduction to a standard problem
bridge from NL to MATH
(generalized-constraint-based)
(generalized-constraint-based)
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KEY IDEAS IN NL-COMPUTATION

• FUNDAMENTAL THESIS
• Information closed generalized constraint
• proposition is a carrier of information
• MEANING POSTULATE
• proposition closed generalized constraint
• predicate open generalized constraint
• In our approach, NL-Computation is reduced to
  computation with generalized constraints, that
  is, to generalized-constraint-based computation.
  NL-Computation is based on fuzzy logic.
  NL-Computation is closely related to Computing
  with Words (CW)

13
FUZZY LOGICKEY POINTS

• Humans have a remarkable capability to reason and
  make decisions in an environment of imprecision,
  uncertainty and partiality of knowledge, truth
  and class membership. The principal objective of
  fuzzy logic is formalization/mechanization of
  this capability.

14
WHAT IS FUZZY LOGIC?

• Fuzzy logic is not fuzzy logic
• Fuzzy logic is a precise logic of reasoning and
  decision making based on information which is
  imprecise, uncertain, incomplete and partially
  true.
• The principal distinguishing features of fuzzy
  logic are
• In fuzzy logic everything is, or is allowed to be
  graduated, that is, be a matter of degree or,
  equivalently fuzzy
• In fuzzy logic everything is, or is allowed to be
  granulated, with a granule being a clump of
  points drawn together by indistinguishability,
  similarity or proximity
• Informal versions of Graduation and granulation
  have a position of centrality in human cognition

15
ANALOGY

• In bivalent logic, one writes and draws with a
  ballpoint pen
• In fuzzy logic, one writes and draws with a spray
  pen which has an adjustable and precisely defined
  spray pattern
• This simple analogy suggests many mathematical
  problems
• What is the maximum value of f?
• Precisiation/imprecisiation principle

Y
X
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A BASIC CONCEPT IN NL-COMPUTATION GRANULAR VALUE
A
granular value of X
a
singular value of X
universe of discourse

• singular X is a singleton
• granular X isr A granule
• a granule is defined by a generalized constraint
• example
• X unemployment
• a 7.3
• A high

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KEY POINTS

• A granule may be viewed as a representation of
  the state of imprecise, uncertain, incomplete
  and/or partially true knowledge about the
  singular value of X
• Information described in natural language
  granular information
• Computation with information described in natural
  language Granular Computing (Zadeh 1979, 1997,
  1998 Lin 1998 Bargiela and Pedrycz 2005).

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GRANULATION OF A VARIABLE

• continuous quantized granulated
• Example Age

middle-aged
µ
µ
old
young
1
1
0
0
Age
quantized
Age
granulated
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GRANULATION OF A FUNCTION
Y
f
0
Y
medium x large
f (fuzzy graph)
perception
f f
summary
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
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COMPUTATION WITH GRANULAR VALUESGRANULAR
COMPUTING

• Problem
• If I leave the hotel at about 10am and usually
  it takes about an hour to get to the airport,
  then at what time will I arrive at the airport?
• Protoformal formulation
• ? Z X Y
• precisiation
• granular computing

usually(b)
a
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PRECISIATION OF approximately a, a
?
1
singleton
s-precisiation
0
x
a
?
1
cg-precisiation
interval
0
a
x
fuzzy interval
g-precisiation
?
type 2 fuzzy interval
0
a
x
fuzzy graph
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CONTINUED
probability distribution
g-precisiation
p
bimodal distribution
g-precisiation
0
x
23
CONTINUED
?A
A
a is A
u
?B
B
b is B
u
?usually
usually is C
u
0
1
usually (b) p(v)
is usually
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CONTINUED
p(v)
p(v)p(v-u)
v
subject to
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EXTENSION PRINCIPLE (Zadeh 1965, 1975)
Yf(X) singular values
granulation
Yf(X) granular values
example f(X) is A g(X) is B
Bsupu(?A(f(u))
subject to
vg(u)
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MAMDANI

• Yf(X)
• granular f
• f if X is Ai then Y is Bi, i1, , n
• f is Ii Ai?Bi
• X is a
• Y is ?i?Ai(a)?Bi

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NL-CAPABILITY

• NL-capability capability to compute with
  information described in natural language
• Existing scientific theories do not have
  NL-capability
• In particular, probability theory does not have
  NL-capability

28
THE CONCEPTS OF PRECISIATION AND
COINTENSIVE PRECISIATION
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UNDERSTANDING VS. PRECISIATION

• Understanding precedes precisiation
• I understand what you said, but can you be more
  precise
• Beyond reasonable doubt
• Use with adequate ventilation
• Unemployment is high unemployment is
  over 5
• Where do you draw the line? Paraphrase The US
  Constitution is an invitation to argue over where
  to draw the line
• Where to draw the line is a key issue in legal
  arguments

precisiation
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WHAT IS PRECISE?
PRECISE
v-precise
m-precise

• precise value
• p X is a Gaussian random variable with mean m
  and variance ?2. m and ?2 are precisely defined
  real numbers
• p is v-imprecise and m-precise
• p X is in the interval a, b. a and b are
  precisely defined real numbers
• p is v-imprecise and m-precise

precise meaning
m-precise mathematically well-defined
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PRECISIATION AND IMPRECISIATION
v-imprecisiation
1
1
0
0
a
v-precisiation
a
x
x
m-precise
m-precise
v-precise
v-imprecise
1
If X is small then Y is small If X is medium then
Y is large If X is large then Y is small
v-imprecisiation
0
x
v-imprecise
v-precise
m-imprecise
m-precise
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v-IMPRECISIATION
v-imprecisiation
forced
deliberate

• forced V is not known precisely
• deliberate V need not be known precisely
• v-imprecisiation principle Precision carries a
  cost. If there is a tolerance for imprecision,
  exploit it by employing v-imprecisiation to
  achieve lower cost, robustness, tractability,
  decision-relevance and higher level of confidence

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IMPRECISIATION/ SUMMARIZATION OF FUNCTIONS
L
If X is small then Y is small If X is medium then
Y is large If X is large then Y is small
v-imprecisiation
M
summarization
S
0
S
M
L
If X is small then Y is small If X is medium then
Y is large If X is large then Y is small
(X, Y) is small ? small medium ? large large
? small
mm-precisiation
fuzzy graph
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SUMMARIZATION OF T-NORMS
S
M
L
S
M
L

• To facilitate the chore of an appropriate t-norm,
  each t-norm should be associated with a summary

35
APPROXIMATION VS. SUMMARIZATION

• summarization may be viewed as a form of
  imprecisiation

y
y
approximation
0
0
x
x
y
summarization
0
x
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V-PRECISIATION
A
X variable
g-precisiation
a
X
s-precisiation
s-precisiation
7.3
unemployment
high
g-precisiation

• s-precisiation is used routinely in scientific
  theories and especially in probability theory
• defuzzification may be viewed as an instance of
  s-precisiation

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PRECISIATION/IMPRECISIATION PRINCIPLE
(Zadeh 2005)

• a approximately a
• simple version

f(a) f(a)
Y
Y
X
X
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PRECISE SOLUTION
level set
undominated
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MODALITIES OF m-PRECISIATION
m-precisiation
mh-precisiation
mm-precisiation
machine-oriented
human-oriented
m-precisiation
l precisiand of l (Pre(l))
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PRECISIATION AND DISAMBIGUATION

• Examples
• Overeating causes obesity most of those who
  overeat become obese Count(become.obese/ov
  ereat) is most
• Obesity is caused by overeating most of
  those who are obese were overeating
  Count(were.overeating/obese) is most

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PRECISIATION/ DISAMBIGUATION

• P most tall Swedes P(A) is?

POPULATION (Swedes)
tall Swedes
A
P1 most of tall Swedes P2 mostly tall
Swedes
mh-precisiation
P
mm-precisiation
P1 Count(A/tall.Swedes) is most P2 Count(tall.
Swedes/A) is most
mm-precisiation
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BASIC STRUCTURE OF DEFINITIONS
definiens
definiendum (idea/perception)
concept
mh-precisiand
mm-precisiand
mh-precisiation
mm-precisiation
cointension
cointension
cointension wellness of fit of meaning
Declining market with expectation of further
decline We classify a bear market as a 30
percent decline after 50 days, or a 13 percent
decline after 145 days. (Robert Shuster)
mh-precisiation
bear market
mm-precisiation
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EXAMPLES MOUNTAIN, CLUSTER, STABILITY
mh-precisiation
A natural raised part of the earths surface,
usually rising more or less abruptly, and larger
than a hill
mountain
mm-precisiation
?
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CONTINUED
mh-precisiation
A number of things of the same sort gathered
together or growing together bunch
cluster
mm-precisiation
?

• the concepts of mountain and cluster are
  PF-equivalent, that is, have the same deep
  structure

mh-precisiation
The capacity of an object to return to
equilibrium after having been displaced
stability
mm-precisiation
Lyapuonov definition
mm-precisiation
fuzzy stability definition
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GRANULATION REVISITED

• Granulation plays a key role in human cognition
• In human cognition, v-imprecisiation is followed
  by mh-precisiation. Granulation is
  mh-precisiation-based
• In fuzzy logic, v-imprecisiation is followed by
  mm-precisiation. Granulation is
  mm-precisiation-based

• mm-precisiation-based granulation is a major
• contribution of fuzzy logic. No other logical
  
• system offers this capability

46
DIGRESSIONEXTENSION VS. INTENSION

• extension and intension are concepts drawn from
  logic and linguistics
• basic idea
• object (name (attribute1, value1), ,
  (attribute n, value n))
• more compactly
• object (name, (attribute, value))
• n-ary n-ary

name
attribute name
attribute value
object
attribute name
attribute value
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OPERATIONS ON OBJECTS
function
name-based
extensional
definition
object
intensional
definition
attribute-based
(algorithmic)
object (Michael, (gender, male), , (age,
25)) son (Michael) Ron
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PREDICATE (PROPERTY, CONCEPT, SET, MEMBERSHIP
FUNCTION, INPUT-OUTPUT RELATION)

• A predicate, P, is a truth-valued function
• Denotation of P D(P) XP(X)
• Extension of P Ext(P) D(P) if P(X) is
  name-based
• Intension of P Int(P) D(P) if P(X) is
  attribute-based
• P(X) open predicate (X is a free variable)
• P(a) closed predicate (X is a bound variable(P
  is grounded))

U universe of discourse
generic object
X
P
D(P)
49
EXAMPLE
U population
X
P bachelor
D(bachelor)
Ext (bachelor) Xbachelor (X) Int(bachelor)
50
PRINCIPAL MODES OF DEFINITION

• Extension name-based meaning
• Intension attribute-based meaning
• Extensional Pu1, , un e-meaning of P
• Ostensive Pu, uk, ul o-meaning of P
• Intensional PuP(u), i-meaning of P

exemplars
51
PROPOSITION (TENTATIVE)

• A proposition, p, is a sentence which may be
  expressed as P(object). Equivalently, p, is a
  closed predicate. Equivalently, p object is P
• very simple example
• p Valentina is young
• example
• p most Swedes are tall
• P most
• object Count(tall.Swedes/Swedes)
• p most(Count(tall.Swedes/Swedes))
• i-meaning of p is associated with i-meaning of P
  and i-meaning of object
• Question D(most.tall.Swedes)?

young(Valentina), e-meaning
extensional
young(Age(Valentina), i-meaning
intensional
52
SUMMARY

• p proposition or predicate
• Extension of p name-based meaning of p
• Intension of p attribute-based meaning of p

53
THE CONCEPT OF COINTENSION

• p, q are predicates or propositions
• CI(p,q) cointension of p and q degree of match
  between the i-meanings of p and q
• q is cointensive w/n to p if GI(p, q) is high
• A definition is cointensive if CI(definiendum,
  definiens) is high
• In practice, CI(p,q) is frequently associated
  with o-meaning of p and i-meaning of q
• The o-meaning of the definiendum is
  perception-based

54
THE CONCEPTS OF COINTENSION AND RESTRICTIVE
COINTENSION
U universe of discourse
q
D(q)
p
D(p)
restriction
R

• CI(p,q)degree of proximity of D(p) and D(q)
• Cointension of q relative to pdegree of
  subsethood of D(q) in D(p)
• Restricted cointension U is restricted to R

55
THE CONCEPT OF COINTENSIVE PRECISIATION

• Precisiation of a concept or proposition, p, is
  cointensive if Pre(p) is cointensive with p.
• Example bear market
• We classify a bear market as a 30 percent
  decline after 50 days, or a 13 percent decline
  after 145 days. (Robert Shuster)
• This definition is clearly not cointensive

56
KEY POINTS

• Precisiandmodel of meaning
• In general, p, may be precisiated in many
  different ways, resulting in precisiands Pre1(p),
  , Pren(p), each of which is associated with the
  degree, CIi, of cointension of Prei(p), i 1, ,
  n. In general, CIi is context-dependent.
• Precisiation is necessary but not sufficient
• To serve its pupose, precisiation must be
  cointensive
• Cointensive precisiation is a key to
  mechanization of natural language understanding

precisiation1
Pre1(p) C1
precisiation2
p
Pre2(p) C2
precisiationn
Pren(p) Cn
57
AN IMPORTANT IMPLICATION FOR SCIENCE

• It is a deep-seated tradition in science to
  employ the conceptual structure of bivalent logic
  and probability theory as a basis for formulation
  of definitions of concepts. What is widely
  unrecognized is that, in reality, most concepts
  are fuzzy rather than bivalent, and that, in
  general, it is not possible to formulate a
  cointensive definition of a fuzzy concept within
  the conceptual structure of bivalent logic and
  probability theory.

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EXAMPLES OF FUZZY CONCEPTS WHOSE STANDARD,
BIVALENT-LOGIC-BASED DEFINITIONS ARE NOT
COINTENSIVE

• stability
• causality
• relevance
• bear market
• recession
• mountain

• independence
• stationarity
• cluster
• grammar
• risk
• linearity

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A GLIMPSE INTO THE FUTURE

• To formulate a cointesive definition of a fuzzy
  concept it is necessary to employ fuzzy logic
• Replacement of bivalent-logic-based definitions
  with fuzzy-logic-based definitions is certain to
  take place but it will be a slow process
• Fuzzy-logic-based definitions will be targeted
  (customized)

60
ANALOGY
S
M(S)
system
model
modelization
l
Pre(l)
lexeme
precisiand
precisiation

• input-output relation intension (test-score
  function)
• system analysis semantical analysis
  (Freges Principle
• of Compositionality)
• degree of match between M(S) and S cointension
• In general, it is not possible to construct a
  cointensive
• model of a nonlinear system from linear
  components

61
CHOICE OF PRECISIAND

• Cointension and tractability are contravariant
• To be tractable, precisiation should not be
  complex
• An optimal choice is one which achieves a
  compromise between tractability and cointension

cointension
tractability
complexity
62
THE KEY IDEA MEANING POSTULATE

• In NL-computation, a proposition, p, is
  precisiated by expressing its meaning as a
  generalized constraint. In this sense, the
  concept of a generalized constraint serves as a
  bridge from natural languages to mathematics.

NL
Mathematics
p
p (GC(p))
precisiation
generalized constraint

• The concept of a generalized constraint is the
  
• centerpiece of NL-computation

63
TEST-SCORE SEMANTICS (ZADEH 1982)
Prinicipal Concepts and Ideas

• Test-score semantics has the same conceptual
  structure as systems analysis
• In test-score semantics, a lexeme, p, is viewed
  as a composite constraint
• Each constraint is associated with a test-score
  function which defines the degree to which the
  constraint is satisfied given the values of
  constraint variables
• Semantic analysis involves computation of the
  test-score function associated with p in terms of
  the test-score functions associated with f
  components of p
• the operation of composition and the resulting
  test-score function constitute the meaning of p

64
CONTINUED

• Constraints are represented as relations
• The system of relations associated with p
  constitutes an explanatory database ED
• ED may be viewed as a description of a possible
  world
• Test-score semantics has a much higher expressive
  power than possible-world semantics

65
EXAMPLE

• p young men like young women
• p most young men like most young women
• ED POPULATION Name Gender Age
• LIKES Name1 Name2 ?
• YOUNG Age ?
• MOST Proportion ?

POPULATION
ED (explanatory database)
men
women
possible world
Namei
Namej
young
66
CONTINUED

• P likes mostly young women
• P(Namei) Count ((POPULATION Name Gender is F
  Age is young)/ LIKES Name is Namei Name2
  Gender is F Age) is most
• ts(p) Count (POPULATIONName is P/ POPULATION
  Name Gender is M Age is young) is most

women liked by Namei
young women
Namei
67
THE CONCEPT OF A GENERALIZED CONSTRAINT
68
PREAMBLE

• The centerpiece of fuzzy logic is the concept of
  a generalized constraint. Constraints are
  ubiquitous. In scientific theories,
  representation of constraints is generally over
  simplified. Over simplification of constraints is
  a necessity because existing constrained
  definition languages have a very limited
  expressive power. The concept of a generalized
  constraint is intended to provide a basis for
  construction of a maximally expressive constraint
  definition language which can also serve as a
  meaning representation/precisiation language for
  natural languages.

69
GENERALIZED CONSTRAINT (Zadeh 1986)

• Bivalent constraint (hard, inelastic,
  categorical)

X ? C
constraining bivalent relation

• Generalized constraint on X GC(X)

GC(X) X isr R
constraining non-bivalent (fuzzy) relation
index of modality (defines semantics)
constrained variable
r ? ? ? ? blank p v u rs
fg ps
bivalent
primary

• open GC(X) X is free (GC(X) is a predicate)
• closed GC(X) X is instantiated (GC(X) is a
  proposition)

70
CONTINUED

• constrained variable
• X is an n-ary variable, X (X1, , Xn)
• X is a proposition, e.g., Leslie is tall
• X is a function of another variable Xf(Y)
• X is conditioned on another variable, X/Y
• X has a structure, e.g., X Location
  (Residence(Carol))
• X is a generalized constraint, X Y isr R
• X is a group variable. In this case, there is a
  group, G (Name1, , Namen), with each member of
  the group, Namei, i 1, , n, associated with an
  attribute-value, hi, of attribute H. hi may be
  vector-valued. Symbolically

71
CONTINUED

• G (Name1, , Namen)
• GH (Name1/h1, , Namen/hn)
• GH is A (µA(hi)/Name1, , µA(hn)/Namen)
• Basically, GH is a relation and GH is A is a
  fuzzy restriction of GH
• Example
• tall Swedes SwedesHeight is tall

72
SIMPLE EXAMPLES

• Check-out time is 1 pm, is an instance of a
  generalized constraint on check-out time
• Speed limit is 100km/h is an instance of a
  generalized constraint on speed
• Vera is a divorcee with two young children, is
  an instance of a generalized constraint on Veras
  age

73
GENERALIZED CONSTRAINTMODALITY 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
Standard constraints bivalent possibilistic,
bivalent veristic and probabilistic
74
CONTINUED
r bm bimodal constraint X is a random
variable R is a bimodal distribution 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 g
group constraint X isg R means that R constrains
the attribute-values of the group
75
PRIMARY GENERALIZED CONSTRAINTS

• Possibilistic X is R
• Probabilistic X isp R
• Veristic X isv R
• Primary constraints are formalizations of three
  basic perceptions (a) perception of possibility
  (b) perception of likelihood and (c) perception
  of truth
• In this perspective, probability may be viewed as
  an attribute of perception of likelihood

76
EXAMPLES POSSIBILISTIC

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

X
R
X
R
R
X
77
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)

78
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

79
STANDARD CONSTRAINTS

• Bivalent possibilistic X ? C (crisp set)
• Bivalent veristic Ver(p) is true or false
• Probabilistic X isp R
• Standard constraints are instances of generalized
  constraints which underlie methods based on
  bivalent logic and probability theory

80
GENERALIZED CONSTRAINTSEMANTICS
A generalized constraint, GC, is associated with
a test-score function, ts(u), which associates
with each object, u, to which the constraint is
applicable, the degree to which u satisfies the
constraint. Usually, ts(u) is a point in the unit
interval. However, if necessary, it may be an
element of a semi-ring, a lattice, or more
generally, a partially ordered set, or a bimodal
distribution. example possibilistic constraint,
X is R X is R Poss(Xu) µR(u) ts(u) µR(u)
81
TEST-SCORE FUNCTION

• GC(X) generalized constraint on X
• X takes values in U u
• test-score function ts(u) degree to which u
  satisfies GC
• ts(u) may be defined (a) directly (extensionally)
  as a function of u or indirectly (intensionally)
  as a function of attributes of u
• intensional definitionattribute-based
  definition
• example (a) Andrea is tall 0.9
• (b) Andreas height is 175cm µtall(175)0.9
  Andrea is 0.9 tall

82
CONSTRAINT QUALIFICATION

• p isr R means r-value of p is R
• in particular
• p isp R Prob(p) is R (probability
  qualification)
• p isv R Tr(p) is R (truth (verity)
  qualification)
• p is R Poss(p) is R (possibility
  qualification)
• examples
• (X is small) isp likely ProbX is small
  is likely
• (X is small) isv very true VerX is small
  is very true
• (X isu R) ProbX is R is usually

83
GENERALIZED CONSTRAINT LANGUAGE (GCL)

• GCL is an abstract language
• GCL is generated by combination, qualification,
  propagation and counterpropagation of generalized
  constraints
• examples of elements of GCL
• X/Age(Monika) is R/young (annotated element)
• (X isp R) and (X,Y) is S)
• (X isr R) is unlikely) and (X iss S) is likely
• If X is A then Y is B
• the language of fuzzy if-then rules is a
  sublanguage of GCL
• deduction generalized constraint propagation and
  counterpropagation

84
CONSTRAINTS
generalized constraints
primary constraints
standard constraints

• generalized X isr R , r possibilistic,
  probabilistic, veristic, random
  set, usuality, group,
• primary possibilistic, probabilistic, veristic
• standard bivalent possibilistic, probabilistic,
  bivalent veristic
• existing scientific theories are based on primary
  constraints

85
PRECISIATION TRANSLATION INTO GCLBASIC
STRUCTURE
NL
GCL
p
p
precisiation
precisiand of p GC(p)
translation
generalized constraint

• annotation
• p X/A isr R/B GC-form of p
• example
• p Carol lives in a small city near San
  Francisco
• X/Location(Residence(Carol)) is R/NEARCity ?
  SMALLCity

86
v-PRECISIATION
s-precisiation
g-precisiation

• conventional (degranulation)
• a a
• approximately a

GCL-based (granulation)
precisiation
a
precisiation
X isr R
p
proposition
GC-form
common practice in probability theory

• cg-precisiation crisp granular precisiation

87
PRECISIATION OF approximately a, a
?
1
singleton
s-precisiation
0
x
a
?
1
cg-precisiation
interval
0
a
x
fuzzy interval
g-precisiation
?
type 2 fuzzy interval
0
a
x
fuzzy graph
88
CONTINUED
probability distribution
g-precisiation
p
bimodal distribution
g-precisiation
0
x
89
EXAMPLE

• p Speed limit is 100 km/h

poss
cg-precisiation r blank (possibilistic)
p
speed
100
110
poss
g-precisiation r blank (possibilistic)
p
100
110
prob
g-precisiation r p (probabilistic)
p
speed
100
110
90
CONTINUED
prob
g-precisiation r bm (bimodal)
p
100
110
120
speed
If Speed is less than 110, Prob(Ticket) is
low If Speed is between 110 and 120,
Prob(Ticket) is medium If Speed is greater than
120, Prob(Ticket) is high
91
GC-BASED DEFINITION OF GRANULAR VALUE

• X is a singular value
• X is A granular value
• A is defined as a generalized constraint
• example
• X is small granular value

singleton
granule
fuzzy set
92
GRANULAR COMPUTING (GrC) REVISITED

• The objects of computation in granular computing
  are granular values of variables and parameters
• Granular computing has a position of centrality
  in fuzzy logic
• Granular computing plays a key role in
  precisiation and deduction
• Informally
• granular computingballpark computing

93
GRANULAR COMPUTING AND DEDUCTION

• The principal rule of deduction in fuzzy logic is
  the Extension Principle (Zadeh 1965, 1975).

f(X) is A g(X) is B
subject to
94
CONTINUED

• Generalized extension principle
• Zf(X,Y) singular values
• Zf(X,Y) granular values

extension
95
EXAMPLE

• p most Swedes are tall
• p ?Count(tall.Swedes/Swedes) is most
• further precisiation
• X(h) height density function (not known)
• X(h)du fraction of Swedes whose height is in h,
  hdu, a ? h ? b

96
PRECISIATION AND CALIBRATION

• µtall(h) membership function of tall (known)
• µmost(u) membership function of most (known)

?height
?most
1
1
0
0
height
fraction
0.5
1
1
X(h)
height density function
0
h (height)
b
a
97
CONTINUED

• fraction of tall Swedes
• constraint on X(h)

is most
granular value
98
DEDUCTION
q How many Swedes are short q is ?
Q deduction is
most given
is ? Q needed

• Frege principle of compositionalityprecisiated
  version
• precisiation of a proposition requires
  precisiations
• (calibrations) of its constituents

99
CONTINUED
deduction
given
is ? Q needed
solution
subject to
100
CONTINUED
q What is the average height of Swedes? q
is ? Q deduction is most
is ? Q
101
LOOKAHEAD--PROTOFORMAL DEDUCTION

• Example
• most Swedes are tall 1/n?Count(GH is R)
  is Q

Height
102
PROTOFORMAL DEDUCTION RULE
1/n?Count(GH is R) is Q
1/n?Count(GH is S) is T
?i µR(hi) is Q
?i µS(hi) is T
µT(v) suph1, , hn(µQ(?i µR(hi))
subject to
v ?i µS(hi)
values of H h1, , hn
103
PROTOFORM LANGUAGE AND PROTOFORMAL DEDUCTION
PFL
104
THE CONCEPT OF A PROTOFORM
PREAMBLE

• As we move further into the age of machine
  intelligence and automated reasoning, a daunting
  problem becomes harder and harder to master. How
  can we cope with the explosive growth in
  knowledge, information and data. How can we
  locateand infer fromdecision-relevant
  information which is embedded in a large
  database.
• Among the many concepts that relate to this
  issue there are four that stand out in
  importance search, precisiation and deduction.
  In relation to these concepts, a basic underlying
  concept is that of a protoforma concept which is
  centered on the confluence of abstraction and
  summarization

105
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

106
CONTINUED
object space
object p
protoform space
summary of p
protoform
summarization
abstraction
S(p)
A(S(p))
PF(p)

• PF(p) abstracted summary of p
• deep structure of p
• protoform equivalence
• protoform similarity

107
PROTOFORMS
object space
protoform space
PF-equivalence class

• at a given level of abstraction and
  summarization, objects p and q are PF-equivalent
  if PF(p)PF(q)
• example
• p Most Swedes are tall Count (A/B) is Q
• q Few professors are rich Count (A/B) is Q

108
EXAMPLES
instantiation

• Monika is young Age(Monika) is young A(B) is C
• Monika is much younger than Robert
• (Age(Monika), Age(Robert) is much.younger
• D(A(B), A(C)) is E
• Usually Robert returns from work at about 615pm
• ProbTime(Return(Robert) is 615 is usually
• ProbA(B) is C is D

abstraction
usually
615
Return(Robert)
Time
109
CONTINUEDEXTENSION VS INTENSION
Q As are Bs
(attribute-free extension)

• most Swedes are tall

Count(GH is A) is Q
(attribute-based intension)
110
EXAMPLES
gain
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?
2
1
0
option 2
option 1
Y
Y
object
i-protoform
X
0
X
0
111
PROTOFORMAL DEDUCTION
NL
GCL
PFL
p q
p q
p q
precisiation
summarization
precisiation
abstraction
WKM
DM
r
World Knowledge Module
a
answer
deduction module
112
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
113
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
114
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
115
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
116
EXAMPLE OF DEDUCTION

• p Most Swedes are much taller than most Italians
• q What is the difference in the average height
  of Swedes and Italians?
• Solution
• Step 1. precisiation translation of p into GCL
• S S1, , Sn population of Swedes
• I I1, , In population of Italians
• gi height of Si , g (g1, , gn)
• hj height of Ij , h (h1, , hn)
• µij µmuch.taller(gi, hj) degree to which Si is
  much taller than Ij

117
CONTINUED

• Relative ?Count of Italians in relation to
  whom Si is much taller
• ti µmost (ri) degree to which Si is much
  taller than most Italians
• v Relative ?Count of Swedes who are
  much taller than most Italians
• ts(g, h) µmost(v)
• p generalized constraint on S and I
• q d

118
CONTINUED

• Step 2. Deduction via Extension Principle

subject to
119
DEDUCTION PRINCIPLE

• Precisiate query
• Precisiate query-relevant information
• Employ constraint propagation (Extension
  Principle) to deduce the answer to query
• example
• q What is the average height of Swedes?
• Assume that P is a population of Swedes,
  P(Name1, , Namen), with hiHeight(Namei), i1,
  , n.

120
CONTINUED

• q (h1hn)
• (qri) I Most Swedes are tall
• I (µtall(h1)µtall(hn) is most
• GC(h) (µmost( (?iµtall(hi)) , h (hi, ,
  hn)

121
CONTINUED

• constraint propagation
• (µmost( (?iµtall(hi))
• Ave(h) ?ihi
• Extension Principle
• (µAve(h)(v) suph(µmost( ?iµtall(hi)) ,
  (h1hn)
• subject to
• v ?ihi

122
DEDUCTION PRINCIPLEGENERAL FORMULATION

• Point of departure question, q
• Data D (X1/u1, , Xn/un)
• ui is a generic value of Xi
• Ans(q) answer to q
• If we knew the values of the Xi, u1, , un, we
  could express Ans(q) as a function of the ui
• Ans(q)g(u1, ,un) u(u1, , un)
• Our information about the ui, I(u1, , un) is a
  generalized constraint on the ui. The constraint
  is defined by its test-score function
• f(u)f(u1, , un)

123
CONTINUED

• Use the extension principle

subject to
124
MODULAR DEDUCTION DATABASE
POSSIBILITY MODULE
PROBABILITY MODULE
FUZZY ARITHMETIC MODULE
agent
SEARCH MODULE
FUZZY LOGIC MODULE
EXTENSION PRINCIPLE MODULE
125
PROBABILITY MODULE
126
THE CONCEPT OF BIMODAL DISTRIBUTION (ZADEH 1979)
X isbm R
bimodal distribution
random variable

• A bimodal distribution is a collection of ordered
  pairs of the form
• R (P1, A1), , (Pn, An)
• or equivalently
• ?i(Pi \Ai) , i1, , n
• where the Pi are fuzzy probabilities and the Ai
  are fuzzy sets

127
CONTINUED

• Special cases
• The Pi are crisp the Ai are fuzzy
• The Pi are fuzzy the Ai are crisp
• The Pi are crisp the Ai are crisp
• The Demspter-Shafer theory of evidence is
  basically a theory of crisp bimodal distributions

128
EXAMPLE FORD STOCK

• I am considering buying Ford stock. I ask my
  stockbroker, What is your perception of the
  near-term prospects for Ford stock? He tells me,
  A moderate decline is very likely a steep
  decline is unlikely and a moderate gain is not
  likely. My question is What is the probability
  of a large gain?

129
CONTINUED

• Information provided by my stockbroker may be
  represented as a collection of ordered pairs
• Price ((unlikely, steep.decline),
• (very.likely, moderate.decline),
• (not.likely, moderate.gain))
• In this collection, the second element of an
  ordered pair is a fuzzy event or, equivalently, a
  possibility distribution, and the first element
  is a fuzzy probability.
• The importance of the concept of a bimodal
  distribution derives from the fact that in the
  context of human-centric systems, most
  probability distributions are bimodal

130
BIMODAL DISTRIBUTIONS

• Bimodal distributions can assume a variety of
  forms. The principal types are Type 1, Type 2 and
  Type 3. Type 1, 2 and 3 bimodal distributions
  have a common framework but differ in important
  detail

131
BIMODAL DISTRIBUTIONS (Type 1, 2, 3)
U
A2
A1
An
A

• Type 1 (default) X is a random variable taking
  values in U
• A1, , An, A are events (fuzzy sets)
• pi Prob(X is Ai) , i 1, , n
• ?ipi is unconstrained
• pi is Pi (granular probability)
• BMD bimodal distribution ((P1, A1), , (Pn,
  An))
• X isbm (P1\A1
  Pn\An)
• Problem What is the probability, p, of A? In
  general, this probability is
  fuzzy-set-valued, that is, granular

132
CONTINUED

• Type 2 (fuzzy random set) X is a
  fuzzy-set-valued random variable with
  values A1, , An (fuzzy
  sets)
• pi Prob(X Ai), i 1, , n
• BMD X isrs (p1\A1 pn\An)
• ?ipi 1
• Problem What is the probability, p, of A? p is
  not definable. What are definable are (a) the
  expected value of the conditional
  possibility of A given BMD, and (b) the
  expected value of the conditional necessity
  of A given BMD

133
CONTINUED

• Type 3 (augmented random set Dempster-Shafer)
• X is a set-valued random variable taking the
  values X1, , Xn with respective probabilities
  p1, , pn
• Yi is a random variable taking values in Ai, i
  1, , n
• Probability distribution of Yi in Ai, i 1, ,
  n, is not specified
• X isp (p1\X1pn\Xn)
• Problem What is the probability, p, that Y1 or
  Y2 or Yn is in A? Because probability
  distributions of the Yi in the Ai are not
  specified, p is interval-valued. What is
  important to note is that the concepts of upper
  and lower probabilities break down when the Ai
  are fuzzy sets

134
IS DEMPSTER SHAFER COINTENSIVE?

• In applying Dempster Shafer theory, it is
  important to check on whether the data fit Type 3
  model.
• Caveat In many cases the cointensive
  (well-fitting) precisiand (model) of a problem
  statement is bimodal distribution of Type 1
  rather than Type 3 (Demspter-Shafer)

Bimodal Type 1
precisiation
NL description of problem
precisiation
Bimodal Type 2
precisiation
Dempster-Shafer
Bimodal Type 3
135
BASIC BIMODAL DISTRIBUTION (BMD) (Type
1)(PERCEPTION-BASED PROBABILITY DISTRIBUTION)
X is a real-valued random variable
probability
P3
g
P2
P1
X
0
A2
A1
A3
BMD 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
136
INTERPOLATION OF A BASIC BIMODAL DISTRIBUTION
(TYPE 1)
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)
137
INTERPOLATION MODULE AND PROBABILITY MODULE
Prob X is Ai is Pi , i 1, , n Prob X is
A is Q
subject to
138
EXAMPLE

• Probably it will take about two hours to get from
  San Francisco to Monterey, and it will probably
  take about five hours to get from Monterey to Los
  Angeles. What is the probability of getting to
  Los Angeles in less than about seven hours?
• BMD (probably, 2) (probably, 5)

X
Y
Z XY
w
v
u
139
CONTINUED
query is ?A
qri
subject to
140
TEST PROBLEMS (PROBABILITY THEORY)

• X is a real-valued random variable. What is known
  about X is a(usually X is much larger than
  approximately a b usually X is much smaller than
  approximately b, where a and b are real numbers
  with a lt b. What is the expected value of X?
• X and Y are random variables. (X,Y) takes values
  in the unit circle. Prob(1) is approximately 0.1
  Prob(2) is approximately 0.2 Prob(3) is
  approximately 0.3 Prob(4) is approximately 0.4.
  What is the marginal distribution of X?

Y
1
4
0
X
2
3
141
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

142
OPERATIONS ON BIMODAL DISTRIBUTIONS
P(X) defines possibility distribution of g
problem a) what is the expected value of X
143
EXPECTED VALUE OF A BIMODAL DISTRIBUTION
Extension Principle
subject to
144
PERCEPTION-BASED DECISION ANALYSIS
ranking of bimodal probability distributions
PA
0
X
PB
0
X
maximization of expected utility ranking of
fuzzy numbers
145
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
146
PROBABILITY MODULE (CONTINUED)
X isp P Y f(X) Y isp f(P)
Prob X is A is P Prob f(X) is B is Q
X isp P (X,Y) is R Y isrs S
X isu A Y f(X) Y isu f(A)
147
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
148
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?

149
USUALITY SUBMODULE
150
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

151
USUALITY CONSTRAINT
X isu A X isu B X isp P (A B) ispv Q
X is A X is B X is A B
g probability density function of X ?(g)
possibility distribution function of g
subject to
subject to
152
USUALITY QUALIFIED RULES
X isu A X isun (not A)
X isu A Yf(X) Y isu f(A)
153
USUALITY QUALIFIED RULES
X isu A Y isu B Z f(X,Y) Z isu f(A, B)
154
SUMMATION

• The concept of GC-computation is the centerpiece
  of NL-computation. The point of departure in
  NL-computation is the key idea of representing
  the meaning of a proposition drawn from a natural
  language, p, as a generalized constraint. This
  mode of representation may be viewed as
  precisiation of p, with the result of
  precisiation being a precisiand, p, of p. Each
  precisiand is associated with a measure, termed
  cointension, of the degree to which the intension
  of p is a good fit to the intension of p. A
  principal desideratum of precisiation is that the
  resulting precisiand be cointensive. The concept
  of cointensive precisiation is a key to
  mechanization of natural language understanding.
• The concept of NL-computation has wide-ranging
  ramifications, especially within human-centric
  fields such as economics, law, linguistics and
  psychology

155
APPENDIX
156
DEDUCTION THE BALLS-IN-BOX PROBLEM

• Version 1. Measurement-based
• A flat box contains a layer of black and white
  balls. You can see the balls and are allowed as
  much time as you need to count them
• q1 What is the number of white balls?
• q2 What is the probability that a ball drawn at
  random is white?
• q1 and q2 remain the same in the next version

157
DEDUCTION

• Version 2. Perception-based
• You are allowed n seconds to look at the box. n
  seconds is not enough to allow you to count the
  balls
• You describe your perceptions in a natural
  language
• p1 there are about 20 balls
• p2 most are black
• p3 there are several times as many black balls
  as white balls
• PTs solution?

158
MEASUREMENT-BASED
PERCEPTION-BASED
version 2
version 1

• a box contains 20 black and white balls
• over seventy percent 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 picked at
  random is white?

• 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?

159
COMPUTATION (version 2)

• 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

• perception-based
• X number of black balls
• Y number of white balls
• X most 20
• X several Y
• X Y 20
• P Y/N

160
FUZZY INTEGER PROGRAMMING
Y
X most 20
XY 20
X several y
x
1