THE CONCEPT OF A Z-NUMBER—A NEW
DIRECTION IN UNCERTAIN COMPUTATION
Lotfi A. Zadeh
Computer Science Division
Department of EECS
UC Berkeley
IRI 2011
Aug 3, 2011
Las Vegas
Research supported in part by ONR Grant N00014-02-1-0294, Omron Grant, Tekes
Grant, Azerbaijan Ministry of Communications and Information Technology Grant,
Azerbaijan University of Azerbaijan Republic and the BISC Program of UC
Berkeley.
Email: [email protected]
URL: http://www.cs.berkeley.edu/~zadeh/
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PREAMBLE
 In
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large measure, science and
engineering dwell in the world of
measurements and numbers. In
this world, a basic question which
arises is: How reliable are the
numbers which we deal with? This
question plays a particularly
important role in decision analysis,
planning, economics, risk
assessment, design and process
analysis.
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CONTINUED
 The
concept of a Z-number is
intended to provide a basis for
computation with numbers which
are not totally reliable. More
concretely, a Z-number, Z=(A,B), is
an ordered pair of two fuzzy
numbers. The first number, A, is a
restriction on the values which a
real-valued variable, X, can take.
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CONTINUED
 The
second number, B, is a
restriction on the degree of
certainty that X is A. Typically, A
and B are described in a natural
language.
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EXAMPLES
 X=anticipated
budget deficit
 A=approximately 2 million dollars
 B=very likely
 X=travel
time by car from Berkeley
to Palo Alto
 A=approximately 1 hour
 B=usually
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CONTINUED
 (approximately
100, very sure)
 (approximately
100, not very likely)
 (low,
sure)
 (high,
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not sure)
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COMPUTATION WITH Z-NUMBERS
 (approximately
1 hr, usually) +
(approximately 45 min, usually)
 What
is the square root of
(approximately 100, very likely?)
 Computation
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with Z-numbers falls
within the province of Computing
with Words (CW or CWW).
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Z-NUMBERS AND COMPUTING WITH
WORDS
 Before
proceeding further with our
discussion of computation with Znumbers, we will discuss briefly
some of the pertinent features of
Computing with Words. The
methodology of Computing with
Words is of interest in its own
right.
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WHAT IS COMPUTING WITH WORDS (CW OR
CWW)?

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There are many misconceptions about
what Computing with Words is and
what it has to offer. A common
misconception is that CW and NLP
(Natural Language Processing) are
closely related. In fact, this is not the
case. CW and NLP have different
agendas and address different
problems. A very simple example of a
problem in CW is the following.
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EXAMPLE
CN
Dana is 25 years
old
 Tandy is 3 years
older than Dana

CW
Dana is young
 Tandy is a few
years older than
Dana
 Tandy is (young +
few) years old


Tandy is (25+3)
years old

In CW, young and few are interpreted as
labels of fuzzy numbers. Fuzzy arithmetic
is used to find the sum of young and few.
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BASIC STRUCTURE OF CW

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The point of departure in CW is a
question, q, of the form: What is the
value of a variable, Y? The answer to
this question is expected to be derived
from a collection of propositions, I,
I=(p1, …, pn), which is referred to as the
information set. In essence, I is a
collection of question-relevant
propositions.
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CONTINUED

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The terminus consists of an answer of
the form: Y is ans(q/I). Generally,
ans(q/I) is not a value of Y but a
restriction (generalized constraint) on
the values which Y is allowed to take.
(Zadeh 2006) Equivalently, ans(q/I)
identifies those values of Y which are
consistent with I. In CW, consistency is
equated to possibility, with the
understanding that possibility is a
matter of degree.
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BASIC STRUCTURE OF CW
NL q
CW
I
p
+p
engine
ans(q/I)
NL
NL
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ESSENCE OF CW

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In essence, CW is a system of
computation in which the objects of
computation are words, phrases and
propositions drawn from a natural
language. The carriers of information
are propositions. It should be noted
that CW is the only system of
computation which offers a capability
to compute with information described
in a natural language.
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KEY IDEA
A
prerequisite to computation
is precisiation of meaning. Raw
(unprecisiated) natural
language cannot be computed
with. A key idea in CW is that of
precisiating the meaning of a
proposition, p, as a restriction.
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NOTE: PRECISIATION VS.
REPRESENTATION

Precisiation and representation have
distinct meanings.

Precisiation goes beyond
representation.

Result of precisiation is a
computational model of the object that
is precisiated.
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EXAMPLE
p: Most Swedes are tall
representation
p
precisiation
Prop(tall.Swedes/Swedes) is most
1/n(µtall(h1)+…+µtall(hn) is most
hi=height of ith Swede
µtall(hi)=grade of membership of ith Swede in tall
1/n(µtall(h1)+…+µtall(hn) is most
µmost(1/n(µtall(h1)+…+µtall(hn))
computational outline of p
(generalized intension of p)
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IMPORTANT POINT—THE CONCEPT OF
PROTOFORM

Protoform of p=abstracted generalized
intension of p

Protoform equivalence. Example.
Protoform (most Swedes are
tall)=protoform of (usually it takes
Robert about an hour to get home from
work)
 Protoform= mannequin

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SUMMARY
 CW= computation with information
described in a natural language.
 CW= precisiation of objects of
computation followed by
computation with precisiated
objects.
computational model
CW = precisiation
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restriction
computation
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THE CONCEPT OF A RESTRICTION—A
CLOSER LOOK
I
am asked: What is the value of a
real-valued variable X? My answer
is: I do not know the value
precisely but I have a perception
which I can express as a restriction
(generalized constraint) on the
values which X can take.
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EXAMPLES
 8≤X≤10
X
is small
X
is normally distributed with mean
9 and variance 2.
 It
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is likely that X is between 8 and
10.
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REPRESENTATION OF A RESTRICTION
A
restriction (generalized
constraint), R(X), may be
represented as:
R(X): X isr R
where X is the restricted
(constrained) variable, R is the
restricting (constraining) relation
and r is an indexical variable which
defines how R restricts X.
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EXAMPLE
 Possibilistic
restriction (r=blank):
R(X): X is A
where A is a fuzzy set in U with the
membership function µA. A plays
the role of the possibility
distribution of X
Poss(X=u)= µA(u)
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PROBABILISTIC RESTRICTION
 Probabilistic
restriction (r=p):
R(X): X isp P
where P plays the role of the
probability distribution of X.
Prob(u≤X≤u+du)=p(u)du
where p is the probability density
function of X.
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NOTE
 There
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are many different types of
restrictions. (Zadeh 2006) In the
context of natural languages,
restrictions are predominantly
possibilistic, probabilistic or
combinations of the two. For
simplicity, the indexical variable,
r, is sometimes suppressed,
relying on the context for
disambiguation.
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DIRECT AND INDIRECT RESTRICTIONS
A
restriction is direct if it is of the
form:
R(X): X isr R
 A restriction is indirect if it is of
the form:
R(X): f(X) isr R
where f is a specified function or
funtional.
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EXAMPLE OF INDIRECT RESTRICTION
R( X ) : ∫ µ( u ) p( u )du is likely
R
Is an indirect restriction on p.

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Note: The term “restriction” is
sometimes applied to R.
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PRINCIPAL LEVELS OF GENERALITY OF
RESTRICTIONS
Z-numbers
fuzzy numbers
random numbers
intervals
real numbers
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Level 3
Level 2
Level 1
ground level
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
VISUAL REPRESENTATION OF RESTRICTIONS—
THE CONCEPT OF A Z-MOUSE
A Z-mouse is an electronic implementation of a spray pen.
The cursor is a round fuzzy mark called an f-mark. The
color of the mark is a matter of choice. A dot identifies the
centroid of the mark. The cross-section of a f-mark is a
trapezoidal fuzzy set with adjustable parameters.
Age(Son)
Age
(Daughter)
Age
(Mother)
*25
Age(Vera)
1
*.8
*40
*35
Usually
*20
f-mark
0
0
0
specified
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0
0
computed
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CONTINUED

I believe that Robert is very honest
belief
honesty
1
1
precisiation
f-mark
f-mark
0
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0
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MORE ON Z-MOUSE
If I am not sure what the degree is, and
I am allowed to use a Z-mouse, I will
put a fuzzy f-mark on the scale.
 A fuzzy f-mark reflects imprecision of
my perception.
 A Z-mouse reads my f-mark and
represents it internally as a trapezoidal
fuzzy set—a fuzzy set which serves as
an object of computation for the
machinery of Computing with Words.

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EXAMPLE OF APPLICATION OF Z-MOUSE

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I am scheduled to fly from San
Francisco to Los Angeles. My flight is
scheduled to leave at 5pm. I have to be
at the airport about an hour before
departure. Usually it takes about forty
five minutes to get to the airport from
my home. I would like to be pretty sure
that I arrive at the airport in time. At
what time should I leave my home?
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CONTINUED
Time of
departure
Time of
arrival
Usually
1
Time of
departure
from home
0
0
*.8
5pm
*3:50pm
*4pm
f-mark
0
specified
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0
specified/
computed
specified
trial
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PRECISIATION OF MEANING IN CW—A KEY
IDEA
Point of departure
Information=restriction
A proposition, p, is a carrier of
information.
 In CW, a proposition, p, is precisiated
by representing p as a restriction
p: X is R
where X, R and r are variables which
typically are implicit in p. X is R is
referred to as a canonical form of p.

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EXAMPLES
Age(Robert)=young
X
R
 p: Most Swedes are tall
Proportion(tall Swedes/Swedes) is most
X
R
 p: Usually it takes Robert about an hour
to get home from work
Travel time
from office to home (Robert) is a Znumber, (approximately 1 hr, usually)

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p: Robert is young
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CONTINUED
X
need not be a scalar variable.
Example:
 p: Robert gave a ring to Anne, X
may be represented as the 3-tuple
(Giver, Recipient, Object), with the
corresponding values of R being
(Robert, Anne, Ring).
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NOTE
A
semantic network may be viewed
as a canonical form of p, with X as
an n-ary variable. The same applies
to FrameNet and related
representations.
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PHASES OF CW
CW= [PRECISIATION COMPUTATION]
Granular computing
CW
Phase 1
q
q*
precisiation
I
precisiation
module
Phase 2
computation
I*
Ans(q/I)
computation
module
fuzzy logic

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Precisiation and computation employ the
machinery of fuzzy logic.
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FROM PRECISIATION TO COMPUTATION
Phase 1
I
p1
p1* : X1 isr1 R1
pn-1
Pn
q
I*
pn-1*: X isr R
n-1
n-1 n-1
pn*: X isr R
n
n n
q*
.
.
.
.
precisiation
Phase 2
X1 isr1 R1
I*
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.
.
Xn-1 isrn-1 Rn-1
Xn isrn Rn
q*
computation
with
ans(q/I)
restrictions
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COMPUTATION WITH RESTRICTIONS
(GENERALIZED CONSTRAINTS)
 In
computation with restrictions,
the principal rule is the Extension
Principle (Zadeh 1965, 1975 a, b
and c). A basic version of the
Extension Principle may be stated
as:
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CONTINUED
f(X) is A
g(X) is ?B
The answer to the question is the solution of
a mathematical program expressed as:
µ B ( w ) = supu µ A ( f ( u ))
subject to
w = g( u )
where µA and µB are the membership
functions of A and B, respectively.
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Z-NUMBERS—PRINCIPAL CONCEPTS AND
IDEAS
RECOLLECTION

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Decisions are based on information. To
be useful, information must be reliable.
Basically, the concept of a Z-number
relates to the issue of reliability of
information. A Z-number, Z, has two
components, Z=(A,B). The first
component, A, is a restriction
(constraint) on the values which a realvalued uncertain variable, X, is allowed
to take.
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CONTINUED

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The second component, B, is a
measure of reliability (certainty) of the
first component. Closely related to
certainty are the concepts of sureness,
confidence, reliability, strength of
belief, probability, possibility, etc.
Typically, A and B are described in a
natural language. Example: (about 45
minutes, very sure). An important issue
relates to computation with Z-numbers.
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EXAMPLES
 What
is the sum of (about 45
minutes, very sure) and (about 30
minutes, sure)?
 What
is the square root of
(approximately 100, likely)?
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CONTINUED
 The
concept of a Z-number has a
potential for many applications,
especially in the realms of
computation with probabilities and
events described in a natural
language.
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CONTINUED
 Of
particular importance are
applications in economics,
decision analysis, risk
assessment, prediction,
anticipation, planning, biomedicine
and rule-based manipulation of
imprecise functions and relations.
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CONTINUED
 The
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ordered triple (X,A,B) is
referred to as a Z-valuation. A Zvaluation is equivalent to an
assignment statement, X is (A,B). X
is an uncertain variable if A is not a
singleton. In a related way,
uncertain computation is a system
of computation in which the
objects of computation are not
values of variables but restrictions
on values of variables.
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CONTINUED
 In
the following, unless stated to
the contrary, X is assumed to be a
random variable. Simple examples
of Z-valuations are:
 (anticipated budget deficit, close to
2 million dollars, very likely)
 (population of Spain, about 45
million, quite sure)
 (degree of Robert's honesty, very
high, absolutely)
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CONTINUED
 (degree
of Robert's honesty, high,
not sure)
 (travel time by car from Berkeley to
San Francisco, about 30 minutes,
usually)
 (price of oil in the near future,
significantly over 100
dollars/barrel, very likely)
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CONTINUED
 (Height(John),
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tall, probable)
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NOTE
 It
is important to note that, in large
measure, computation with
probabilities and events described
in a natural language, the
information set, I, consists of a
collection of Z-valuations.
 Z-information
= collection of Z-
valuations.
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CONTINUED

If X is a random variable, then X is A
represents a fuzzy event in R, the real
line. The probability measure of this
event, p, may be expressed as (Zadeh
1968):
p = ∫ µ A ( u ) p X ( u )du ,
R
where pX is the underlying (hidden)
probability density of X. More
compactly, p may be represented as
the scalar product of µA and pX,
p=µA·pX
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KEY RELATION
 In
effect, a Z-valuation (X,A,B) may
be viewed as a restriction
(generalized constraint) on X
defined by:
Prob(X is A) is B
where Prob(X is A) is the
probability measure of the event X
is A.
(X,A,B) = Prob(X is A) is B
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MEMBERSHIP FUNCTION OF A AND
PROBABILITY DENSITY FUNCTION OF X
μp
p
X
1
0

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µA
R
compatibility
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Z-RULES
A Z-rule is an if-then rule in which the
antecedent and/or the consequent are
Z-valuations. Examples:
 if (anticipated budget deficit, about two
million dollars, very likely) then
(reduction in staff, about ten percent,
very likely)
 if (degree of Robert’s honesty, high, not
sure) then (offer a position, not, sure)
 if (X, small) then (Y, large, usually.)

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RELATED CONCEPTS—Z+-NUMBERS AND Z-NUMBERS
A Z+-number carries more information
than a Z-number.
 A Z+-number is an ordered pair,
Z+=(A,p)
 As in the case of a Z-number, A is a
fuzzy number which is a restriction on
the value of a variable, X. p is the
underlying (hidden) probability
distribution of X.

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CONTINUED
Example.
(approximately 9, normal distribution
with mean 9 and variance 2)
 A Z+-number is a bimodal distribution
which may be expressed as (µ,p),
where µ is the membership function of
A. µ and p define, respectively, the
possibility and probability distribution
of X. Equivalently, a Z+-number may be
viewed as an ordered pair of a fuzzy
LAZ 5/31/2011
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
THE CONCEPT OF A Z--NUMBER
A Z--number, Z-=(A,C) carries less
information than a Z-number.
 In a Z--number, as in a Z-number, A is a
fuzzy number which represents a
restriction on the values of a variable,
X. C is a fuzzy number which is a
restriction on the probability of A,
rather than on the probability measure
of A. In the case of a Z--number, X is a
fuzzy-set-value random variable.

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CONTINUED
X
C
A

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Z--numbers and Z-numbers are
identical in appearance. A common
error is that of misinterpreting a
proposition as a Z--number rather than
a Z-number.
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EXAMPLE OF A Z--NUMBER

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A box contains 20 black and white
balls. A ball is picked at random. If the
ball is white, I get approximately 10
dollars. If it is black, I lose
approximately 20 dollars. I am not
given enough time to count the number
of white balls, but my perception is that
most are white. In this case, most/20 is
the probability of winning
approximately 10 dollars. The Z-number is (approximately 10, most/20).
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SUMMARY
Z-valuation, (X, A, B)= Prob(X is A) is B,
where A is a fuzzy restriction on X,
Prob is a probability measure, and B is
a fuzzy restriction on Prob(X is A).
 Z+-valuation, (X, A, p), where A is a
fuzzy restriction on X, and p is the
probability distribution of X.
 Z--valuation, (X, A, C), where A is a
fuzzy restriction on X, and C is a fuzzy
restriction on Prob(X is A), with Prob
standing for probability (not probability
measure).
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
CORELATION

Z-valuations are corelated if their
associated variables are identical and
their restrctions are non-conflicting.
A SIMPLE EXAMPLE OF COMPUTATION
WITH CORELATED Z-VALUATIONS
(X, A, B)
(X, C, is ?D)
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Prob (X is A) is B
Prob(X is C) is ?D
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CORELATION
∫ µ A ( u ) p X ( u )du is B
f(X) is B
∫ µC ( u ) p X ( u )du is ?D
g(X) is ?D
R
R
Extension Principle
µ D ( w ) = supv µ B ( f ( v ))
subject to w = g ( v )
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CONTINUED
Extension Principle
µ D ( w ) = sup p X µ B ( ∫ µ A ( u ) p X ( u )du )
R
subject to w = ∫ µC ( u ) p X ( u )du
R
∫ p X ( u )du = 1
R
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A PROBLEM IN PROBABILITY THEORY

Probably John is tall. What is the
probability that John is short?

This problem may be represented as
computation with Z-valuations.
(Height(John), tall, probable)
(Height(John), short, ?D)
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COMPUTATION WITH Z-NUMBERS

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Computation with Z-numbers involves,
in the main, computation with functions
whose arguments are Z-numbers.
Example: f is a function whose
arguments are real numbers, Z=f(X,Y).
Assume that what we know are not the
values of X and Y but restrictions, R(X)
and R(Y), respectively. Restrictions on
X and Y induce a restriction on Z, R(Z).
The problem is that of computing R(Z)
given f, R(X) and R(Y).
LAZ 5/31/2011
CONTINUED
For example: if f is the sum of X and Y
then the problem is that of computing
R(X+Y), given R(X) and R(Y).
 In computation with Z-numbers the
principal tool is the Extension
Principle—a principle which was
discussed in an earlier section.
 The following is a brief description of
application of the Extension Principle
to computation of the sum of ZLAZ 5/31/2011
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
NOTE

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Computation with Z-numbers in a more
general setting is discussed in “A note
on Z-numbers,” Information Sciences,
2011.
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COMPUTATION OF THE SUM OF Z-NUMBERS

Let X=(AX,BX) and Y=(AY,BY). The sum
of X and Y is a Z-number, Z =(AZ,BZ).
The sum of (AX,BX) and (AY,BY) is
defined as:
(AX,BX) + (AY,BY)= (AX+AY,BZ)
where AX+AY is the sum of fuzzy
numbers AX and AY computed through
the use of fuzzy arithmetic. The main
problem is computation of BZ.
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CONTINUED

Let pX and pY be the underlying
probability density functions in the Zvaluations (X,AX,BX) and (Y,AY,BY),
respectively. If pX and pY where known,
the underlying probability density
function in Z would be the convolution
of pX and pY, pZ= pX °pY, expressed as:
p X +Y ( v ) = ∫ p X ( u ) pY ( v − u )du
R
where R is the real line
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CONTINUED

What we know are not pX and pY but
restrictions on pX and pY which are
expressed as:
∫ µ AX ( u ) p X ( u )du is BX
R
∫ µ AY ( u ) pY ( u )du is BY
R
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CONTINUED

Using the Extension Principle we can
compute the restriction on pZ. It reads:
µ pZ ( pZ ) = sup p X , pY ( µ BX ( ∫ µ AX ( u ) p X ( u )du ) ∧
R
µ BY ( ∫ µ AY ( u ) pY ( u )du ))
R
subject to
pZ = p X  pY
∫ p X ( u )du = 1 , ∫ pY ( u )du = 1
R
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R
LAZ 5/31/2011
CONTINUED

If pZ were known, BZ would be given
by:
BZ = ∫ µ AZ ( u ) pZ ( u )du ,
R
where
µ AZ ( u ) = supv ( µ AX ( v ) ∧ µ AY ( u − v ))
Since what we know at this point is a
restriction on pZ, it is necessary to
employ the Extension Principle to
compute the restriction on BZ. The
result may be expressed as:
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LAZ 5/31/2011
CONTINUED
µ BZ ( w ) = sup pZ µ pZ ( pZ )
subject to
w = ∫ µ AZ ( u ) pZ ( u )du
R
where µpZ(pZ) was derived earlier. In
principle, this completes computation
of the sum of Z-numbers, ZX and ZY.
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CONCLUDING REMARK

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Computation with Z-numbers is a move
into uncharted territory. A variety of
issues remain to be explored. One
such issue is that of informativeness of
results of computations. To enhance
informativeness and reduce complexity
of computations it may be expedient to
make simplifying assumptions about
the underlying probability
distributions. A discussion of this
issue may be found in Zadeh 2011, A
LAZ 5/31/2011
Note on Z-numbers.