EXTENDED FUZZY LOGIC, SOFT COMPUTING AND COMPUTATIONAL INTELLIGENCE

1
EXTENDED FUZZY LOGIC, SOFT COMPUTING AND
COMPUTATIONAL INTELLIGENCE Lotfi A.
Zadeh Computer Science Division Department of
EECSUC Berkeley August 17, 2009 IASTED
CI Honolulu, Hawaii Research supported in part
by ONR N00014-02-1-0294, BT Grant CT1080028046,
Omron Grant, Tekes Grant, Chevron Texaco Grant
and the BISC Program of UC Berkeley. Email
zadeh_at_eecs.berkeley.edu
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PREVIEW
3
FROM FUZZY LOGIC TO EXTENDED FUZZY LOGIC
FLe
extended fuzzy logic
FL/FLp
FLu
precisiated fuzzy logic measurement-based
unprecisiated fuzzy logic perception-based

• precisiation graduation
• graduation specification of membership function

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KEY POINTS

• Fuzzy logic(precisiated fuzzy logic)a new
  perspective
• Unprecisiated fuzzy logica new logic
• Unprecisiated fuzzy logic is the logic of
  everyday reasoning
• Unprecisiated fuzzy logic lies in an uncharted
  territorya territory in the realm on
  quasi-mathematics
• Fuzzy logic is a precise logic of imprecise
  reasoning
• Unprecisiated fuzzy logic is an imprecise logic
  of imprecise reasoning.

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A USEFUL ANALOGY

• In bivalent logic, the writing/drawing instrument
  is a ballpoint pen. In fuzzy logic, the
  writing/drawing instrument is a spray pena
  miniature spray canwith an adjustable, precisely
  specified spray pattern and a white marker for
  the centroid of the spray patterna marker which
  serves the purpose of precisiation. Such a pen
  will be referred to as precisiated.

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A USEFUL ANALOGY

• In unprecisiated fuzzy logic, the spray pen has
  an adjustable spray pattern and a white marker
  but is not precisiated. In extended fuzzy logic,
  there are two spray pensa precisiated spray pen
  and an unprecisiated spray pen.

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WHAT IS FUZZY LOGIC?-- A NEW PERSPECTIVE
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FUZZY LOGICA NEW PERSPECTIVE

• Fuzzy logic is a precise conceptual system of
  reasoning, deduction and computation in which the
  objects  of discourse and analysis are associated
  with information which is, or is allowed to be,
  imperfect.  Imperfect information is defined as
  information which in one or more respects is
  imprecise, uncertain, vague, incomplete,
  unreliable, partially true or partially possible.

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FUZZY LOGICA NEW PERSPECTIVE

• In fuzzy logic everything is or is allowed to be
  a matter of degree. Degrees are allowed to be
  fuzzy. In multivalued logic, truth is a matter of
  degree but fuzzy degrees are not allowed.
• Fuzzy logic is not a replacement for bivalent
  logic or bivalent-logic-based probability theory.
  Fuzzy logic is an addition to bivalent logic and
  bivalent-logic-based probability theory. What
  fuzzy logic adds is a wide range of concepts and
  techniques for dealing with imperfect
  information.

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FUZZY LOGICA KEY POINT

• Fuzzy logic is designed to address problems in
  reasoning, deduction and computation with
  imperfect informationproblems which are beyond
  the reach of traditional methods based on
  bivalent logic and bivalent-logic-based probabilit
  y theory.

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DEDUCTION FROM IMPERFECT INFORMATIONSIMPLE
EXAMPLES

• Most Swedes are tall
• Most tall Swedes are blond
• What fraction of Swedes are blond?
• Most Swedes are tall
• What is the average height of Swedes?
• Most Swedes are tall
• What is the truth value of Not many Swedes are
  not tall?

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FUZZY LOGIC GAMBIT

• Although fuzzy logic is designed to deal with
  imperfect information, what may appear to be
  surprising is that in many of its applications,
  perfect information is available. In such
  applications, perfect information is deliberately
  imprecisiated, resulting in imperfect
  information. This is the key idea which underlies
  the Fuzzy Logic Gambit. At first glance, the
  Fuzzy Logic Gambit appears to be a step in the
  wrong direction. Following is the rationale.

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RATIONALE

• (a) Precision carries a cost
• (b) If there is a tolerance for imprecision, the
  tolerance can be exploited and the cost reduced
  through deliberate imprecisiation. Basically, in
  the Fuzzy Logic Gambit precisiated words are used
  in place of numbers. Precisiated words can be
  computed with through the use of the formalism of
  Computing with Words (CW).

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SCIENTIFIC PROGRESSA PARADIGM SHIFT
traditional
precisiation
perceptions
numbers
(a)
quantification
Precisiated Natural Language
nontraditionalkey idea
PNL
NL
precisiation
unprecisiated words
precisiated words
(b)
Countertraditionalkey idea
PNL
linguistic
numbers
precisiated words
(c)
summarization
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LINGUISTIC SUMMARIZATIONA KEY IDEA

• Commonly the concept of a summary is applied to
  stories, text, scenes, etc. In fuzzy logic, the
  concept of a summary has a much broader meaning.
  In particular, linguistic summarization is
  applied to functions, relations, systems,
  perceptions and large volumes of numerical data.
  Simple example.

Y
f
linguistic
if X is small then Y is small if X is
medium then Y is large if X is large then Y
is small
summarization
0
X
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CORNERSTONES OF FUZZY LOGIC (NEW PERSPECTIVE)

• The cornerstones of fuzzy logic are graduation,
  granulation, precisiation and the concept of a
  generalized constraint.

graduation
granulation
FUZZY LOGIC
precisiation
generalized constraint
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FUZZY LOGICTHE POINT OF DEPARTURE

• The point of departure in fuzzy logicthe center
  of fuzzy logic (FL)is the concept of a fuzzy set.

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THE CONCEPT OF A FUZZY SET
class
informal (unprecisiated)
precisiation
precisiation
set
(a)
generalization
fuzzy set
fuzzy set
boundary
(b)
measure
(a) A set may be viewed as a special case of a
fuzzy set. A fuzzy set is not a set
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CONTINUED

• (b) Basic attributes of a set/fuzzy set are
  boundary and measure (cardinality, count, volume)
• Fuzzy set theory is boundary-oriented
• Probability theory is measure-oriented
• Fuzzy logic is both boundary-oriented and
  measure-oriented

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CONTINUED

• Informally, a set, A, in U is a class with a
  crisp boundary.
• A set is precisiated through association with a
  characteristic function cA U 0,1
• A fuzzy set is precisiated through graduation,
  that is, through association with a membership
  function µA U 0,1, with µA(u), ueU,
  representing the grade of membership of u in A.

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THE CONCEPT OF GRADUATION

• Graduation of a fuzzy concept or a fuzzy set, A,
  serves as a means of precisiation of A.
• Graduation involves an association of A with a
  membership function.
• Examples
• Graduation of middle-age
• Graduation of the concept of earthquake via the
  Richter Scale
• Graduation of recession?
• Graduation of mountain?

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EXAMPLEMIDDLE-AGE

• Imprecision of meaning elasticity of meaning
• Elasticity of meaning fuzziness of meaning

µ
middle-age
1
0.8
core of middle-age
40
60
45
55
0
43
definitely not middle-age
definitely not middle-age
definitely middle-age
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GRADUATION?
graduation
declaration
elicitation
verification
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DECLARATIVE GRADUATIONHONDA FUZZY LOGIC
TRANSMISSION
Not Very Low
High
Close
1
1
1
Low
High
High
Grade
Grade
Grade
Low
Not Low
0
0
0
5
30
130
180
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Throttle
Shift
Speed

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

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GRADUATIONELICITATION

• Humans have a remarkable capability to graduate
  perceptions, that is, to associate perceptions
  with degrees on a scale. It is this capability
  that is exploited for elicitation of membership
  functions.
• Example
• Robert tells me that Vera is middle-aged. What
  does Robert mean by middle-aged? More
  specifically, what is the membership function
  that Robert associates with middle-aged? I elicit
  the membership function from Robert by asking a
  series of questions.

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GRADUATIONELICITATION

• Procedure
• Typical question What is the degree to which a
  particular age, say 43, fits your perception of
  middle-aged? Please mark the degree on a scale
  from 0 to 1 using a ballpoint pen. Repetition of
  this question leads to a membership function of
  type 1. Use of a spray pen leads to a membership
  function of type 2.

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GENERATION OF FUZZY LOGIC FL-GENERALIZATION

• The concept of FL-Generalization plays a pivotal
  role in fuzzy logic, its generation and its
  applications.
• T denotes a bivalent-logic-based theory,
  formalism, algorithm or concept.
• Fuzzy T is an FL-Generalized T. Examples fuzzy
  set theory, fuzzy topology, fuzzy measure theory,
  fuzzy game theory, fuzzy control, etc.

addition (reverse)
FL
T
fuzzy T
FL-Generalization
FL-Generalization
addition (forward)
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CONTINUED

• FL-Generalization applied to T involves (a)
  adding to T concepts and techniques drawn from
  fuzzy logic (b) employing these concepts and
  techniques to generalize T, resulting in fuzzy T
  and (c) adding FL-relevant concepts and
  techniques drawn from fuzzy T to FL.
• FL is generated by applying FL-Generalization to
  various Ts in succession, starting with the
  concept of a fuzzy set and set theory.

addition (reverse)
fuzzy set
set theory
fuzzy set theory
FL-Generalization
FL-Generalization
addition (forward)
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CONTINUED

• Application of FL-Generalization to set theory is
  followed by application of FL-Generalization to
  logic, to relations and to theories related to
  knowledge representation, information,
  probability theory and possibility theory.
  FL-Generalization is a continuing process through
  which forms the basis for generation of fuzzy
  logic and its principal facets.

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PRINCIPAL FACETS OF FUZZY LOGIC

• Starting with the concept of a fuzzy set,
  successive application of FL-Generalization to
  related theories leads to the principal facets of
  fuzzy logic.

Fuzzy Logic (wide sense) (FL)

FLl
logical (narrow sense)
FLs
set-theoretic
FLr
relational
fuzzy set
epistemic
FLe
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BASIC STRUCTURE OF FL
fuzzy logic
applied fuzzy logic
theoretical fuzzy logic
epistemic facet
relational facet
logical facet
set-theoretic facet
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CONTINUED

• A facet of FL consists of FL-generalization of a
  theory or FL-generalization of a collection of
  related theories.
• The principal facets of FL are logical, FLl set
  theoretic, FLs epistemic, FLe and relational,
  FLr.

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NOTESPECIALIZATION VS. GENERALIZATION

• Consider a concatenation of two words, MX, with
  the prefix, M, playing the role of a modifier of
  the suffix, X, e.g., small box.
• Usually M specializes X, as in convex set
• Unusually, M generalizes X. The prefix fuzzy
  falls into this category. Thus, fuzzy set
  generalizes the concept of a set. The same
  applies to fuzzy topology, fuzzy measure theory,
  fuzzy control, etc. Many misconceptions about
  fuzzy logic are rooted in misinterpretation of
  fuzzy as a specializer rather than a generalizer.

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CORNERSTONES OF FUZZY LOGIC

• The cornerstones of fuzzy logic are graduation,
  granulation, precisiation and the concept of a
  generalized constraint.

graduation
granulation
FUZZY LOGIC
precisiation
generalized constraint
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THE CONCEPT OF GRANULATION

• The concept of granulation is unique to fuzzy
  logic and plays a pivotal role in its
  applications. The concept of granulation is
  inspired by the way in which humans deal with
  imprecision, uncertainty and complexity.
• Granulation serves as a means of imprecisiation
  (coarsening of information).

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GRADUATION / GRANULATION
A
graduation/precisiation
granulation/imprecisiation
A
A

• graduation precisiation
• granulation imprecisiation

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BASIC CONCEPTSGRANULE

• Informally a granule in a universe of discourse,
  U, is a clump of elements of U drawn together by
  indistinguishability, equivalence, similarity,
  proximity or functionality.
• A granule is precisiated through association with
  a generalized constraint.

U
A
granule
universe of discourse
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BASIC CONCEPTSSINGULAR AND GRANULAR VALUES
U
A
granular value of X
singular value of X
A
universe of discourse
singular
granular
7.3 high
.8 high
160/80 high
unemployment
probability
blood pressure
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BASIC CONCEPTSSINGULAR AND GRANULAR VARIABLES
A singular variable, X, is a variable which takes
values in U, that is, the values of X are
singletons in U. A granular variable, X, is a
variable whose values are granules in U. A
linguistic variable, X, is a granular variable
with linguistic labels for granular values. A
quantized variable is a special case of a
granular variable.
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EXAMPLE

• Age as a singular variable takes values in the
  interval 0,120.
• Age as a granular (linguistic) variable takes as
  values fuzzy subsets of 0,120 labeled young,
  middle-aged, old, not very young, etc.

middle-aged
µ
µ
old
young
1
1
0
Age
0
quantized
Age
granulated
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BASIC CONCEPTSGRANULATION

• Granulation is closely related to coarsening of
  information, and to summarization.
• Granulation is a transformation which may be
  applied to any object, A
• A A
• (a) Granulation applied to a singular value

granulation
U
A
granular value of X
A
singular value of X
universe of discourse
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BASIC CONCEPTSGRANULATION

• (b) Granulation applied to a singular variable,
  X, transforms X into a granular variable, X.
• X X
• (c) Granulation of a function

granulation
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(c) GRANULATION OF A SET/FUZZY SET, A
A
A
A
granule
granule
granule
partition
covering
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(d) GRANULATION OF A FUNCTION GRANULATIONSUMMARIZ
ATION
Y
f
0
X
Y
medium large
perception
f (fuzzy graph)
f f
summarization
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
0
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(e) GRANULAR VS. GRANULE-VALUED DISTRIBUTIONS
distribution
p1
pn

granules
probability distribution of possibility
distributions
possibility distribution of probability
distributions
46
GRANULATION
granulation
forced
deliberate

• Forced singular values of variables are not
  known.
• Deliberate singular values of variables are
  known. There is a tolerance for imprecision.
  Precision carries a cost. Granular values are
  employed to reduce cost.

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FUZZY LOGIC GAMBIT

• Fuzzy Logic Gambit deliberate granulation
  followed by graduation
• The Fuzzy Logic Gambit is employed in most of the
  applications of fuzzy logic in the realm of
  consumer products

Y
f
granulation
if X is small then Y is small if X is
medium then Y is large if X is large then Y
is small
summarization
0
X
48
The Concepts of Precisiation and Cointensive Preci
siation
49
PREAMBLE

• In one form or another, precisiation of meaning
  has always played an important role in science.
  Mathematics is a quintessential example of what
  may be called a meaning precisiation language
  system.

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SEMANTIC IMPRECISION (EXPLICIT)
EXAMPLES
WORDS/CONCEPTS

• Recession
• Civil war
• Very slow
• Honesty
• It is likely to be warm tomorrow.
• It is very unlikely that there will be a
  significant decrease in the price of oil in the
  near future.
• What is the probability that Obama will succeed
  in solving the financial crisis?

• Arthritis
• High blood pressure
• Cluster
• Hot

PROPOSITIONS AND QUESTIONS
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SEMANTIC IMPRECISION (IMPLICIT)
EXAMPLES

• Speed limit is 65 mph
• Checkout time is 1 pm

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PRECISIATION OF IMPRECISION

• Can you explain to me the meaning of Speed limit
  is 65 mph?
• No imprecise numbers and no probabilities are
  allowed
• Imprecise numbers are allowed. No probabilities
  are allowed.
• Imprecise numbers are allowed. Precise
  probabilities are allowed.
• Imprecise numbers are allowed. Imprecise
  probabilities are allowed.

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NECESSITY OF IMPRECISION

• Can you precisiate the meaning of arthritis?
• Can you precisiate the meaning of recession?
• Can you precisiate the meaning of beyond
  reasonable doubt?
• Can you precisiate the meaning of causality?
• Can you precisiate the meaning of near?

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PRECISION IN VALUE AND PRECISION IN MEANING

• The concept of precision has a position of
  centrality in scientific theories. And yet, there
  are some important aspects of this concept which
  have not been adequately treated in the
  literature. One such aspect relates to the
  distinction between precision of value
  (v-precision) and precision of meaning
  (m-precision).
• The same distinction applies to imprecision,
  precisiation and imprecisiation.

55
CONTINUED
PRECISE
v-precise
m-precise

• precise value
• p X is in the interval a, b. a and b are
  precisely defined real numbers
• p is v-imprecise and m-precise
• 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

precise meaning
56
PRECISIATION AND IMPRECISIATION

• A proposition, predicate, query or command may be
  precisiated or imprecisiated
• Data compression and summarization are instances
  of imprecisiation

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MODALITIES OF m-PRECISIATION
m-precisiation
mh-precisiation
mm-precisiation
machine-oriented (mathematically well-defined)
human-oriented
Example bear market mh-precisiation declining
stock market with expectation of further
decline mm-precisiation 30 percent decline
after 50 days, or a 13 percent decline after 145
days. (Robert Shuster)
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BASIC CONCEPTS
precisiation language system
p object of precisiation
p result of precisiation
precisiend
precisiation
precisiand
cointension

• precisiand model of meaning
• precisiation modelization
• intension attribute-based meaning
• cointension measure of proximity of meanings
• measure of proximity of the model and the
  object of modelization
• precisiation translation into a precisiation
  language system

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COINTENSIVE PRECISIATION
description/definition of perception
perception
precisiend
precisiation
precisiand
cointension

• Cointension qualitative measure of the proximity
  of precisiand to precisiend (closeness of fit).
• Cointensive precisiation cointension of
  precisiand is high.

middle-age
precisiation
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COINTENSION PRINCIPLE

• Cointensive precisiation (fuzzy precisiand
  fuzzy precisiend)
• Achievement of cointension precisiation
  necessitates that if the precisiend is fuzzy so
  must be the precisiand.
• Crisp definitions of fuzzy concepts is the norm
  in science. What is widely unrecognized is that
  crisp definitions of fuzzy concepts are generally
  not cointensive.
• In fuzzy logic one writes/draws with a spray pen
  which has an adjustable precisiated spray pattern.

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MM-PRECISIATION OF approximately a, a(MODELS
OF MEANING OF a)
Bivalent Logic
?
1
number
0
x
a
?
1
interval
0
a
x
p
probability
0
a
x
It is a common practice to ignore imprecision,
treating what is imprecise as if it were precise.
62
CONTINUED
Fuzzy Logic Bivalent Logic
1
fuzzy interval
0
a
x
1
fuzzy interval type 2
0
a
x
1
fuzzy probability
0
x
a
Fuzzy logic has a much higher expressive power
than bivalent logic.
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GOODNESS OF MODEL OF MEANING
goodness of model (cointension, computational
complexity) a approximately a
x
cointension
best compromise
computational complexity
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v-IMPRECISIATION
v-imprecisiation
Imperative (forced)
Intentional (deliberate)

• imperative value is not known precisely
• intentional value need not be known precisely
• data compression and summarization are instances
  of v-imprecisiation

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THE CONCEPT OF COINTENSIVE PRECISIATION

• m-precisiation of a concept or proposition, p, is
  cointensive if 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

66
mm-PRECISIATION

• Basic question
• Given a proposition, p, how can p be cointesively
  mm-precisiated?
• Key idea
• In generalized-constraint-based semantics,
  mm-precisiation is carried out through the use of
  the concept of a generalized constraint.
• The concept of a generalized constraint opens the
  door to computation with information described in
  natural languageComputing with Words (CW).

67
FUZZY LOGIC IN A NEW PERSPECTIVEA KEY IDEA

• The concept of a generalized constraint serves as
  a bridge between linguistics and mathematics by
  providing a means of precisiation of propositions
  and concepts drawn from a natural language.

Linguistics
Mathematics
p
X isr R
precisiation
generalized constraint

• The concept of a generalized constraint is the
  
• centerpiece of FL-bases semantics of natural
• languages.

68
THE CONCEPT OF A GENERALIZED CONSTRAINT A BRIEF
INTRODUCTION
69
PREAMBLE

• The concept of a generalized constraint is the
  centerpiece of generalized-constraint-based
  semantics.
• In scientific theories, representation of
  constraints is generally oversimplified.
  Oversimplification of constraints is a necessity
  because bivalent-logic-based constraint
  definition languages have a very limited
  expressive power.

70
CONTINUED

• The concept of a generalized constraint is
  intended to provide a basis for construction of a
  maximally expressive meaning precisiation
  language for natural languages.
• Generalized constraints have elasticity.
• Elasticity of generalized constraints is a
  reflection of elasticity of meaning of words in a
  natural language.

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GENERALIZED CONSTRAINT (Zadeh 1986)

• Bivalent constraint (hard, inelastic,
  categorical)

X ? C
constraining bivalent relation

• Generalized constraint on X GC(X) (elastic)

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)

72
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
73
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

74
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

75
THE CONCEPT OF GENERALIZED CONSTRAINT AS A BASIS
FOR PRECISIATION OF MEANING

• Meaning postulate
• Equivalently, mm-precisiation of p may be
  realized through translation of p into GCL.

mm-precisiation
p X isr R
76
EXAMPLES POSSIBILISTIC

• Lily is young Age (Lily) is young
• most Swedes are tall
• Count (tall.Swedes/Swedes) is most

annotation
X
R
R
X
77
TOWARD EXTENDED FUZZY LOGIC A FIRST STEP
78
EVOLUTION
fuzzy logic (FL)
FLs
G/G
bivalent logic
multivalued logic
FLl
FLr
FLe
beyond FL
FL
FLe

• FLe is an extension of fuzzy logic

79
A CLOSER LOOK

• Over the years, fuzzy logic has been enriched
  through introduction of a long list of concepts,
  ideas and techniques.
• The concepts of extended fuzzy logic, FLe, and
  f-validity which are sketched in the following
  represent a more radical development. In essence,
  extended fuzzy logic may be viewed as an attempt
  at legitimizing the concept of fuzzy theorem
  (Zadeh 1975) and fuzzy validity.

80
STRUCTURE OF EXTENDED FUZZY LOGIC
FLe
extended fuzzy logic
FL/FLp
FLu
precisiated fuzzy logic measurement-based
unprecisiated fuzzy logic perception-based

• precisiation graduation
• graduation specification of membership function

81
BACKDROP

• Science deals not with reality but with models of
  reality. In large measure, scientific progress is
  driven by a quest for better models of
  reality. In constructing better models of
  reality, a problem that has to be faced is that
  as the complexity of a system, S, increases, it
  becomes increasingly difficult to construct a
  model, M(S), which is both cointensive, that is,
  close-fitting, and precise.

82
IMPOSSIBILITY PRINCIPLE

• This applies, in particular, to systems in which
  human judgment, perceptions and emotions play a
  prominent role. Economic systems, legal systems
  and political systems are cases in point.
• As the complexity of a system increases further,
  a point is reached at which construction of a
  model which is both cointensive and precise is
  not merely difficultit is impossible.

83
CONTINUED

• At this point, a key idea comes into play. The
  idea is that of constructing a fuzzy logic, FLu,
  which, in contrast to FL, is unprecisiated. What
  this means is that in FLu membership functions
  and generalized constraints are not specified,
  and are a matter of perception rather than
  measurement. To stress the contrast between FL
  and FLu, FL may be written as FLp, with p
  standing for precisiated.

84
CONTINUED

• A question which arises is What is the point of
  constructing FLua logic in which provable
  validity (p-validity) is off the table? But what
  is not off the table is what may be called fuzzy
  validity, or f-validity for short. A model of FLu
  is f-geometrya geometry in which figures are
  drawn by hand with a spray pen, without the use
  of a ruler or compass.

85
CONTINUED

• Everyday human reasoning is preponderantly
  f-valid reasoning. Humans have a remarkable
  capability to perform a wide variety of physical
  and mental tasks without any measurements and any
  computations. In this context, f-valid reasoning
  is perception-based. In FLu, there are no formal
  definitions, no theorems and no p-valid proofs.

86
UNPRECISIATED VS. PRECISIATED PERCEPTIONS
perception
NL-description (unprecisiated)
NL-description (unprecisiated)
p-perception (precisiated)
u-perception (unprecisiated)
Computational theory of perceptions (CTP)
Human-centered reasoning, decision-making and
discourse
87
f-VALID VS. p-VALID REASONING

• f-valid reasoning is not admissible in FL.
  f-valid reasoning is admissible in FLe when
  p-valid reasoning is infeasible, carries an
  excessively high cost or is unneeded. In many
  realistic settings, this is the norm rather than
  exception. The following very simple example is a
  case in point.

88
TAXI CAB EXAMPLE

• I hail a taxi and ask the driver to take me to
  address A. There are two versions (a) I ask the
  driver to take me to A the shortest way and (b)
  I ask the driver to take me to A the fastest way.
  Based on his/her experience, the driver chooses
  route (a) for (a) and route (b) for (b). Routes
  (a) and (b) is an f-valid solution which in some
  sense is good enough.

89
TAXI CAB EXAMPLE

• In version (a) if there is a map of the area it
  is possible to construct the shortest way to A.
  This would be a p-valid solution. Thus, for
  version (a) there exists a p-valid solution but
  the drivers choice of route (a) may be viewed as
  an f-valid solution which in some sense is good
  enough.

90
TAXI CAB EXAMPLE

• In version (b), it is not possible to construct a
  cointensive model of the system and hence it is
  not possible to construct a p-valid solution. The
  problem is rooted in uncertainties related to
  traffic conditions, timing of lights, etc. In
  fact, if the driver had asked me to define what I
  mean by the fastest way, I could not come up
  with an answer to his/her question. Thus, in
  version (b) there exists an f-valid solution, but
  a p-valid solution does not exist.

91
THE CAB DRIVER PROBLEMTHE POINT OF DEPARTURE
A
shortest route (a)




fastest route (b)
starting point
92
f-VALID REASONING AND f-GEOMETRY

• A model of f-valid reasoning is f-geometry.
• The underlying logic in f-geometry is
  unprecisiated fuzzy logic, FLu.
• f-geometry is unrelated to Postons fuzzy
  geometry (Poston, 1971), coarse geometry (Roe,
  1996), fuzzy geometry of Rosenfeld (Rosenfeld,
  1998), fuzzy geometry of Buckley and Eslami
  (Buckley and Eslami, 1997), fuzzy geometry of
  Mayburov (Mayburov, 2008), and fuzzy geometry of
  Tzafestas (Tzafestas et al, 2006). The underlying
  logic in these fuzzy geometries is FL(FLp).

93
f-TRANSFORMATION AND f-GEOMETRY
World of Euclidean Geometry
World of Fuzzy Geometry
Weg
Wfg
f-C or C
C
f-transformation
prototype of f-C
Note that fuzzy figures, as shown, are not hand
drawn. They should be visualized as hand drawn
figures.
94
f-TRANSFORMATION
Informally, in the context of f-geometry, an
f-transform of C is the result of execution of
the instruction Draw C by hand with a spray pen.
95
f-CONCEPTS IN f-GEOMETRY

• f-point
• f-line
• f-triangle
• f-parallel
• f-similar
• f-circle
• f-median
• f-perpendicular

• f-bisector
• f-altitude
• f-concurrence
• f-tangent
• f-definition
• f-theorem
• f-proof

96
f-TRANSFORMATION

• The cointension of f-C is a qualitative measure
  of the proximity of f-C to its prototype, C. A
  fuzzy transform, f-C, is cointensive if its
  cointension is high. Unless stated to the
  contrary, f-transforms are assumed to be
  cointensive.
• A key idea in f-geometry is the following if C
  is p-valid then its f-transform, f-C, is f-valid
  with a high validity index. As a simple example,
  consider the definition, D, of parallelism in
  Euclidean geometry.

97
f-TRANSFORMATION OF DEFINITIONS

• D Two lines are parallel if for any
  transversal that cuts the lines the corresponding
  angles are congruent.
• f-transform of this definition reads
• f-D Two f-lines are f-parallel if for any
  f-transversal that cuts the lines the
  corresponding f-angles are f-congruent.

98
f-TRANSFORMATION OF DEFINITIONS

• In Euclidean geometry, two triangles are similar
  if the corresponding angles are congruent.
  Correspondingly, in f-geometry two f-triangles
  are f-similar if the corresponding angles are
  f-congruent.

A
A
B
B
C
C
99
f-TRANSFORMATION OF PROPERTIES

• Simple example
• P if the triangles A, B, C and A, B, C are
  similar, then the corresponding sides are in
  proportion.

A
A
B
B
C
C
AB AB
BC BC
CA CA


100
f-TRANSFORMATION OF PROPERTIES

• P if the f-triangles A, B, C and A, B, C
  are f-similar, then the corresponding sides are
  in f-proportion.

A
A
B
B
C
C
AB AB
BC BC
CA CA


101
f-TRANSFORMATION OF THEOREMS

• An f-theorem in f-geometry is an f-transform of a
  theorem in Euclidean geometry.
• Simple example
• an elementary theorem, T, in Euclidean geometry
  is
• T the medians of a triangle are concurrent.
• A corresponding theorem, f-T, in f-geometry is
• f-T the f-medians of an f-triangle are
  f-concurrent.

102
THE CONCEPT OF f-PROOF
A logical f-proof is an f-transform of a proof in
Euclidean Geometry.
103
LOGICAL f-PROOFA SIMPLE EXAMPLE
A
D
E
H
F
G
B
C
I
D, E are f-midpoints DE is f-parallel to BC FH is
f-parallel to BC AGI is an f-line passing through
f-point G f-triangles EGH and EBC are f-similar
f-triangles DFG and DBC are f-similar f-proportion
ality of corresponding sides of f-triangles
implies that G is f-midpoint of FH G is
f-midpoint of FH implies that I is f-midpoint of
BC I is f-midpoint of BC implies that the
f-medians are f-concurrent
104
A KEY OBSERVATION

• The f-theorem and its f-proof are f-transforms of
  their counterparts in Euclidean geometry. But
  what is important to note is that the f-theorem
  and its f-proof could be arrived at without any
  reference to their counterparts in Euclidean
  geometry.

105
A KEY OBSERVATION

• This suggests an intriguing possibility of
  constructing, in various fields, independently
  arrived at systems of f-concepts, f-definitions,
  f-theorems, f-proofs and, more generally,
  f-reasoning and f-computation. In the conceptual
  world of such systems, p-validity has no place.

106
f-GEOMETRY AND BEYOND

• In summary, f-geometry may be viewed as the
  result of application of f-transformation to
  Euclidean geometry.
• Beyond f-geometry lies an expanse of various
  fields to which f-transformation may be applied.
  Following are a few examples.

107
SIMPLE EXAMPLE DRAWN FROM SET THEORY

• Convex Sets
• D A is a convex set in U if for any points x
  and y in A every point in the segment xy is in
  A. The f-transform of this definition is the
  definition of an f-convex set, f-A.
  Specifically,
• f-D f-A is an f-convex set in U if for any
  f-points x and y in f-A every f-point in the
  f-segment xy is in f-A.

108
CONTINUED

• An elementary property of convex sets is
• T if A and B are convex sets, so is their
  intersection AnB.
• An f-transform of T reads
• f-T if A and B are f-convex sets, so is their
  intersection f-A n f-B.

109
CONTINUED

• More generally,
• T if A and B are convex fuzzy sets, so is
  their intersection.
• Applying f-transformation to T, we obtain the
  f-theorem
• f-T if A and B are f-convex fuzzy sets, so is
  their intersection.

110
AN f-THEOREM IN f-SET THEORY

• A, B and A are f-sets.
• A is f-contained in B
• A is f-equal to A, A A
• A is f-contained in B
• Note this theorem may be viewed as a model of a
  generalized modus ponens.

111
COMPUTATION WITH f-TRANSFORMSFUNCTIONS OF
f-TRANSFORMS

• A basic problem which arises in computation of
  f-transforms is the following. Let g be a
  function, a functional or an operator. Using
  the star notation, let an f-transform, C, be an
  argument of g. The problem is that of computing
  g(C). Generally, computing g(C) is not a
  trivial problem.

112
CONTINUED

• An f-valid approximation to g(C) may be derived
  through application of an f-principle which is
  referred to as precisiation/imprecisiation
  principle or P/I principle, for short (Zadeh
  2005). More specifically, the principle may be
  expressed as 
• g(C)g(C)
• where should be read as approximately equal.
  In words, g(C) is approximately equal to the
  f-transform of g(C).

113
EXAMPLE

• If g is the operation of differentiation and C
  is an f-function, f, then the f-derivative of
  this function is an f-function.

f
Y
df dX
df dX
df dX
differentiation
f precisiation (centroids) of f
0
0
X
X
114
AN IMPORTANT CONCLUSION

• In many real-world settings, an f-valid solution
  based on a realistic model may be better than a
  p-valid solution based on an unrealistic model.
• Warren Buffett said
• It is better to be approximately right than
  precisely wrong

115
CONCLUDING REMARK

• The concept of f-transformation opens the door to
  the use of extended fuzzy logic in a wide variety
  of fields.
• In particular, f-transformation may be applied to
  soft computing and computational intelligence.
• Soft computing and computational intelligence are
  closely related.

116
CONCLUDING REMARK

• Soft computing fuzzy logic neurocomputing
  evolutionary computing probabilistic computing
  (1991).
• Computational intelligence fuzzy logic
  neurocomputing evolutionary computing (1986).
• Example

117
F-Newton-Raphson Algorithm(Suggested by
Professor J. Rokne, University of Calgary)
f(x)
x
x0
x1

• x1x0 - f(xo)/f(xo)
• Note the curb and the tangents should be
  visualized as fuzzy and hand-drawn.

118
SUMMATION

• In large measure, existing scientific theories
  are based on bivalent logica logic in which
  everything is black or white, with no shades of
  gray allowed
• What is not recognized, to the extent that it
  should, is that bivalent logic is in fundamental
  conflict with reality
• Fuzzy logic is not in conflict with bivalent
  logicit is a generalization of bivalent logic in
  which everything is, or is allowed to be, a
  matter of degree

119
CONTINUED

• Fuzzy logic, does not replace formalisms based on
  bivalent logic with formalisms based on fuzzy
  logic.
• Fuzzy logic adds to and generalizes formalisms
  based on bivalent logic.
• One of the principal contributions of fuzzy logic
  is its high power of cointensive precisiation.
  The concept of precisiation has a position of
  centrality in precisiated fuzzy logic.

120
CONTINUED

• Fuzzy logic, FL, is precisiated. In unprecisiated
  fuzzy logic, FLu, membership functions are not
  precisiatedthey are perception-based.
• Extended fuzzy logic results from addition to
  fuzzy logic of unprecisiated fuzzy logic.
• Extended fuzzy logic adds to fuzzy logic an
  important capabilitya capability to reason and
  compute with unprecisiated objects.

121
CONTINUED

• A model of unprecisiated fuzzy logic is
  f-geometry.
• In f-geometry, figures are drawn by hand with a
  spray pen.
• f-geometry may be viewed as f-transform of
  Euclidean geometry.
• f-geometry may be developed on its own, without
  reference to Euclidean geometry.

122
RELATED PAPERS BY L.A.Z IN REVERSE CHRONOLOGICAL
ORDER

• Toward Extended Fuzzy LogicA First Step, Fuzzy
  Sets and Systems, Elsevier, 2009.
• Fuzzy logic, Encyclopedia of Complexity and
  Systems Science, Springer, 2009.
• Toward human level machine intelligenceis it
  achievable? The need for a paradigm shift, IEEE
  Computational Intelligence Magazine, Special
  Issue, August 2008.
• Is there a need for fuzzy logic? Information
  Sciences, Vol. 178, No. 13, 2751-2779, 2008.
• Generalized theory of uncertainty (GTU)principal
  concepts and ideas, Computational Statistics and
  Data Analysis 51, 15-46, 2006.

123
RELATED PAPERS BY L.A.Z IN REVERSE CHRONOLOGICAL
ORDER

• Toward a generalized theory of uncertainty
  (GTU)an outline, Information Sciences, Elsevier,
  Vol. 172, 1-40, 2005.
• Precisiated natural language (PNL), AI Magazine,
  Vol. 25, No. 3, 74-91, 2004.
• Probability theory and fuzzy logica radical
  view, Journal of the American Statistical
  Association, Vol. 99, No. 467, 880-881, 2004.
• Toward a perception-based theory of probabilistic
  reasoning with imprecise probabilities, Journal
  of Statistical Planning and Inference, Elsevier
  Science, Vol. 105, 233-264, 2002.

124
CONTINUED

• A new direction in AItoward a computational
  theory of perceptions, AI Magazine, Vol. 22, No.
  1, 73-84, 2001.
• From computing with numbers to computing with
  words --from manipulation of measurements to
  manipulation of perceptions, IEEE Transactions on
  Circuits and Systems 45, 105-119, 1999.
• Some reflections on soft computing, granular
  computing and their roles in the conception,
  design and utilization of information/intelligent
  systems, Soft Computing 2, 23-25, 1998.

125
CONTINUED

• Toward a theory of fuzzy information granulation
  and its centrality in human reasoning and fuzzy
  logic, Fuzzy Sets and Systems 90, 111-127, 1997.
• Outline of a computational approach to meaning
  and knowledge representation based on the concept
  of a generalized assignment statement,
  Proceedings of the International Seminar on
  Artificial Intelligence and Man-Machine Systems,
  M. Thoma and A. Wyner (eds.), 198-211.
  Heidelberg Springer-Verlag, 1986.
• Precisiation of meaning via translation into
  PRUF, Cognitive Constraints on Communication, L.
  Vaina and J. Hintikka, (eds.), 373-402.
  Dordrecht Reidel, 1984.

126
CONTINUED

• Fuzzy probabilities and their role in decision
  analysis, Proc. MIT/ONR Workshop on C\u3\d, MIT,
  Cambridge, MA., 1981.
• Fuzzy sets vs. probability, (correspondence
  item), Proc. IEEE 68, 421, 1980.
• Fuzzy sets and information granularity, Advances
  in Fuzzy Set Theory and Applications, M. Gupta,
  R. Ragade and R. Yager (eds.), 3-18. Amsterdam
  North-Holland Publishing Co., 1979.