Computing with Words and its Applications to Information Processing, Decision and

Control

Lotfi A. Zadeh

Computer Science Division

Department of EECSUC Berkeley

February 28, 2005

University of Vienna

URL: http://www-bisc.cs.berkeley.eduURL: http://www.cs.berkeley.edu/~zadeh/

Email: Zadeh@cs.berkeley.edu

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EVOLUTION OF COMPUTATION

naturallanguage

arithmetic algebra

algebra

differentialequations

calculusdifferentialequations

numericalanalysis

symboliccomputation

computing with wordsprecisiated natural language

symboliccomputation

+ +

+ +

+ +

+

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COMPUTING WITH WORDS (CW)

In computing with words (CW), the objects of computation are words and propositions drawn from a natural language

example:X and Y are real-valued variables and f is a

function from R to R which is described in words:

f: if X is small then Y is smallif X is medium then Y is largeif X is large then Y is small

What is the maximum of f?

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COMPUTING WITH WORDS (CW)

The centerpiece of CW is the concept of a generalized constraint (Zadeh 1986)

In CW, the traditional view that information is statistical in nature is put aside. Instead, a much more general view of information is adopted: information is a generalized constraint, with statistical information constituting a special case

The point of departure in CW is representation of the meaning of a proposition drawn from a natural language as a a generalized constraint

CW is based on fuzzy logic (FL)

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FROM BIVALENT LOGIC, BL, TO FUZZY LOGIC, FL

In classical, Aristotelian, bivalent logic, BL, truth is bivalent, implying that every proposition, p, is either true or false, with no degrees of truth allowed

In multivalent logic, ML, truth is a matter of degree

In fuzzy logic, FL: everything is, or is allowed to be, a matter of

degree everything is, or is allowed to be imprecise

(approximate) everything is, or is allowed to be, granular

(linguistic)

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NUMBERS ARE RESPECTED—WORDS ARE NOT

It is a deep-seated tradition in science to equate scientific progress to progression from perceptions to measurements and from the use of words to the use of numbers.

Computing with words is a challenge to this tradition

Computing with words opens the door to a wide-ranging enlargement of the role of natural languages in scientific theories—in particular, in decision analysis, medicine and economics

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IN QUEST OF PRECISION

Reducing smog would save lives, Bay report says (San Francisco Examiner)

Expected to attract national attention, the Santa Clara Criteria Air Pollutant Benefit Analysis is the first to quantify the effects on health of air pollution in California

Removing lead from gasoline could save the lives of 26.7 Santa Clara County residents and spare them 18 strokes, 27 heart attacks, 722 nervous system problems and 1,668 cases where red blood cell production is affected

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CONTINUED

Study projects S.F. 5-year AIDS toll (S.F. Chronicle July 15, 1992)

The report projects that the number of new AIDS cases will reach a record 2,173 this year and decline thereafter to 2,007 new cases in 1997

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QUEST FOR PRECISION

Qualitative Analysis for Management

H. Bender et al

There is a 60% chance that the survey results will be positive

Prob(success) = 0.600

Throughout the text, probabilities and utilities are treated as exact numbers

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COMPUTING WITH WORDS

LEVEL 1: Computing with Wordsmost × manysmall + large

LEVEL 2: Computing with PropositionsMost Swedes are tall It is unlikely to rain in San Francisco in

midsummer

LEVEL 2: Employs generalized-constraint-based semantics of natural languages

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KEY POINTS

Computing with words is not a replacement for computing with numbers; it is an addition

Use of computing with words is a necessity when the available information is perception-based or not precise enough to justify the use of numbers

Use of computing with words is advantageous when there is a tolerance for imprecision which can be exploited to achieve tractability, simplicity, robustness and reduced cost

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BASIC POINTS

In computing with words, the objects of computation are words, propositions, and perceptions described in a natural language

A natural language is a system for describing perceptions

In CW, a perception is equated to its description in a natural language

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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

THE BALLS-IN-BOX PROBLEM

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CONTINUED

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 ballsYou 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

PT’s solution?

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CONTINUED

Version 3. Measurement-based

The balls have the same color but different sizes

You are allowed as much time as you need to count the balls

q1: How many balls are large?

q2: What is the probability that a ball drawn at random is large

PT’s solution?

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CONTINUED

Version 4. Perception-based

You are allowed n seconds to look at the box. n seconds is not enough to allow you to count the balls

Your perceptions are:

p1: there are about 20 balls

p2: most are small

p3: there are several times as many small balls as large balls

q1: how many are large?

q2: what is the probability that a ball drawn at random is large?

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CONTINUED

Version 5. Perception-based

My perceptions are:

p1: there are about 20 balls

p2: most are large

p3: if a ball is large then it is likely to be heavy

q1: how many are heavy?

q2: what is the probability that a ball drawn at random is not heavy?

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A SERIOUS LIMITATION OF PT

• Version 4 points to a serious short coming of PT • In PT there is no concept of cardinality of a fuzzy set• How many large balls are in the box?

0.5

• There is no underlying randomness

0.9

0.80.6

0.9

0.4

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MEASUREMENT-BASED

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?

PERCEPTION-BASED

version 2

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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

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FUZZY INTEGER PROGRAMMING

x

Y

1

X= several × y

X= most × 20*

X+Y= 20*

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PARTIAL EXISTENCE

X, a and b are real numbers, with a b

Find an X or X’s such that X a* and X b*

a*: approximately a; b*: approximately b

Fuzzy logic solution

Partial existence is not a probabilistic

concept

µX(u) = µ>>a*(u)^ µ<<b* (u)

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PRECISIATION OF “approximately a,” *a

x

x

x

a

a

20 250

1

0

0

1

p

fuzzy graph

probability distribution

interval

x 0

a

possibility distribution

x a0

1

s-precisiation singleton

g-precisiation

cg-precisiation

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CONTINUED

x

p

0

bimodal distribution

GCL-based (maximal generality)

g-precisiation X isr R

GC-form

*a

g-precisiation

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VERA’S AGE PROBLEM

q: How old is Vera?

p1: Vera has a son, in mid-twenties

p2: Vera has a daughter, in mid-thirties

wk: the child-bearing age ranges from about 16 to about 42

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CONTINUED

*16

*16

*16

*41 *42

*42

*42

*51

*51 *67

*67

*77

range 1

range 2

p1:

p2:

(p1, p2)

0

0

R(q/p1, p2, wk): a= ° *51 ° *67

timelines

*a: approximately a How is *a defined?

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THE PARKING PROBLEM

I have to drive to the post office to mail a package. The post office closes at 5 pm. As I approach the post office, I come across two parking spots, P1 and P2, P1 is closer to the post office but it is in a yellow zone. If I park my car in P1 and walk to the post office, I may get a ticket, but it is likely that I will get to the post office before it closes. If I park my car in P2 and walk to the post office, it is likely that I will not get there before the post office closes. Where should I park my car?

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THE PARKING PROBLEM

P0 P1 P2

P1: probability of arriving at the post office after it closes, starting in P1

Pt: probability of getting a ticket

Ct: cost of ticket

P2 : probability of arriving at the post office after it closes, starting in P2

L: loss if package is not mailed

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CONTINUED

Ct: expected cost of parking in P1

C1 = Ct + p1L

C2 : expected cost of parking in C2

C2 = p2L

• standard approach: minimize expected cost

• standard approach is not applicable when the values of variables and parameters are perception-based (linguistic)

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DEEP STRUCTURE (PROTOFORM)

Ct

L

L

0P1 P2

Gain

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THE NEED FOR NEW TOOLS

The balls-in-box problem and the parking problem are simple examples of problems which do not lend themselves to analysis by conventional techniques based on bivalent logic and probability theory. The principal source of difficulty is that, more often than not, decision-relevant information is a mixture of measurements and perceptions. Perception-based information is intrinsically imprecise, reflecting the bounded ability of human sensory organs, and ultimately the brain, to resolve detail and store information. To deal with perceptions, new tools are needed. The principal tool is the methodology of computing with words (CW). The centerpiece of new tools is the concept of a generalized constraint.

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BL

PT PNL

CW+ +bivalent logic

probability theory

precisiated natural language

computing with words

GTU

CTP: computational theory of perceptionsPFT: protoform theoryPTp: perception-based probability theoryTHD: theory of hierarchical definabilityGTU: Generalized Theory of uncertainty

CTP THD PFT

PTp

EXISTING TOOLS NEW TOOLS

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PERCEPTIONS

• Perceptions play a key role in human cognition. Humans—but not machines—have a remarkable capability to perform a wide variety of physical and mental tasks without any measurements and any computations. Everyday examples of such tasks are driving a car in city traffic, playing tennis and summarizing a book.

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BASIC FACET OF HUMAN COGNITION

perceptions are intrinsically imprecise principal reasons:

a) Bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information

b) Incompleteness of information

perception

*a (granule)

approximately a

perception of XX: attribute

singleton a

domain of X

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•it is 35 C°

•Over 70% of Swedes are taller than 175 cm

•probability is 0.8

•

•

•It is very warm

•Most Swedes are tall

•probability is high

•it is cloudy

•traffic is heavy

•it is hard to find parking near the campus

INFORMATION

measurement-based numerical

perception-based linguistic

• measurement-based information may be viewed as a special case of perception-based information

• perception-based information is intrinsically imprecise

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BASIC PERCEPTIONS / F-GRANULARITY

temperature: warm+cold+very warm+much warmer+…

time: soon + about one hour + not much later +… distance: near + far + much farther +… speed: fast + slow +much faster +… length: long + short + very long +…

1

0 size

small medium large

a granule is a clump of attribute-values which are drawn together by indistinguishability, equivalence, similarity, proximity or functionality

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CONTINUED

similarity: low + medium + high +… possibility: low + medium + high + almost

impossible +… likelihood: likely + unlikely + very likely +… truth (compatibility): true + quite true + very

untrue +… count: many + few + most + about 5 (5*) +…

subjective probability = perception of likelihood

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COMPUTING WITH PERCEPTIONS

One of the major aims of CW is to serve as a basis for equipping machines with a capability to operate on perception-based information. A key idea in CW is that of dealing with perceptions through their descriptions in a natural language. In this way, computing and reasoning with perceptions is reduced to operating on propositions drawn from a natural language.

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BASIC PERCEPTIONS

attributes of physical objects

•distance•time•speed•direction

•length•width•area•volume

•weight•height•size•temperature

sensations and emotions•color•smell•pain

•hunger•thirst •cold

•joy•anger•fear

concepts•count•similarity•cluster

•causality•relevance•risk

•truth•likelihood•possibility

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DEEP STRUCTURE OF PERCEPTIONS

perception of likelihood perception of truth (compatibility) perception of possibility (ease of attainment or

realization) perception of similarity perception of count (absolute or relative) perception of causality perception of risk

subjective probability = quantified perception of likelihood

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PERCEPTION OF RISK

perception of

function

perception of likelihood

perception of lossperception of risk

• Conventional definition of risk as the expected value of the loss function is an oversimplification

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PERCEPTION OF MATHEMATICAL CONCEPTS: PERCEPTION OF FUNCTION

if X is small then Y is small if X is medium then Y is large if X is large then Y is small0 X

0

Y

f f* :perception

Y

f* (fuzzy graph)

medium x large

f

0

S M L

L

M

S

granule

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BIMODAL DISTRIBUTION (PERCEPTION-BASED PROBABILITY

DISTRIBUTION)

A1A2 A3

P1

P2

P3

probability

P(X) = Pi(1)\A1 + Pi(2)\A2 + Pi(3)\A3

Prob {X is Ai } is Pj(i)

0X

P(X)= low\small+high\medium+low\large

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TEST PROBLEM

A function, Y=f(X), is defined by its fuzzy graph expressed as

f1 if X is small then Y is small

if X is medium then Y is large

if X is large then Y is small

(a) what is the value of Y if X is not large?

(b) what is the maximum value of Y

0

S M L

LM

SX

Y

M × L

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KEY MOTIVATION

Basic objectiveMechanization of reasoning

PrerequisitePrecisiation of meaning

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PRECISIATION OF MEANING

Use with adequate ventilation Speed limit is 100km/hr Most Swedes are tall Take a few steps Monika is young Beyond reasonable doubt Overeating causes obesity Relevance Causality Mountain Most Usually

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KEY IDEA

The point of departure in PNL is the key idea: A proposition, p, drawn from a natural language,

NL, is precisiated by expressing its meaning as a generalized constraint

In general, X, R, r are implicit in p precisiation of p explicitation of X, R, r

p X isr R

constraining relation

Identifier of modality (type of constraint)

constrained (focal) variable

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SIMPLE EXAMPLE

Monika is young Age(Monika) is young

Annotated representation

X/Age(Monika) is R/young

X

Rr (blank)

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BASIC POINTS

A proposition is an answer to a question

example:p: Monika is young

is an answer to the questionq: How old is Monika?

The concept of a generalized constraint serves as a basis for generalized-constraint-based semantics of natural languages

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GENERALIZED CONSTRAINT (Zadeh 1986)

• Bivalent constraint (hard, inelastic, categorical:)

X Cconstraining bivalent relation

X isr R

constraining non-bivalent (fuzzy) relation

index of modality (defines semantics)

constrained variable

Generalized constraint:

r: | = | | | | … | blank | p | v | u | rs | fg | ps |…

bivalent non-bivalent (fuzzy)

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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: X=f(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), and each member of the group is associated with an attribute-value, A1. X may be expressed symbolically as

X: (Name1/A1+…+Namen/An)

Basically, X is a relation

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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 Vera’s age

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GENERALIZED CONSTRAINT—MODALITY r

X isr R

r: = equality constraint: X=R is abbreviation of X is=Rr: ≤ inequality constraint: X ≤ Rr: subsethood constraint: X Rr: blank possibilistic constraint; X is R; R is the possibility

distribution of Xr: v veristic constraint; X isv R; R is the verity

distribution of Xr: p probabilistic constraint; X isp R; R is the

probability distribution of X

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CONTINUED

r: rs random set constraint; X isrs R; R is the set-valued probability distribution of X

r: fg fuzzy graph constraint; X isfg R; X is a function and R is its fuzzy graph

r: u usuality constraint; X isu R means usually (X is R)

r: g group constraint; X isg R means that R constrains the attributed-values of the group

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GENERALIZED CONSTRAINT—SEMANTICS

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(X=u) = µR(u)

ts(u) = µR(u)

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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 (X is small) is likely

(X is small) isv very true ( X is small) is very true

(X isu R) Prob(X is R) is usually

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GENERALIZED CONSTRAINT LANGUAGE (GCL)

GCL is an abstract language GCL is generated by combination, qualification and

propagation of generalized constraints examples of elements of GCL

(X isp R) and (X,Y) is S) (X isr R) is unlikely) and (X iss S) is likely If X is A then Y is B

the language of fuzzy if-then rules is a sublanguage of GCL

deduction= generalized constraint propagation

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PRECISIATION = TRANSLATION INTO GCL

annotationp X/A isr R/B GC-form of p

examplep: Carol lives in a small city near San FranciscoX/Location(Residence(Carol)) is R/NEAR[City] SMALL[City]

p p*

NL GCL

precisiation

translationGC-formGC(p)

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GENERALIZED-CONSTRAINT-FORM(GC(p))

annotation

p X/A isr R/B annotated GC(p)

suppression

X/A isr R/B

X isr R is a deep structure (protoform) of p

instantiation

abstractionX isr R

A isr B

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PRECISIATION VIA TRANSLATION INTO GCL

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

X

R

R

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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\u1+p2\u2+p3\u3)

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EXAMPLES: VERISTIC

Robert is half German, quarter French and quarter Italian

Ethnicity (Robert) isv (0.5|German + 0.25|French + 0.25|Italian)

Robert resided in London from 1985 to 1990

Reside (Robert, London) isv [1985, 1990]

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INFORMATION AND GENERALIZED CONSTRAINTS—KEY POINTS

In CW, the carriers of information are propositions

p: proposition

GC(p): X isr R

p is a carrier of information about X

GC(p) is the information about X carried by p

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BASIC STRUCTURE OF PNL

p• • •p* p**

NL PFLGCL

abstraction

world knowledge

module

GCL: Generalized Constraint LanguagePFL: Protoform LanguageDictionary 1: NL GCLDictionary 2: GCL PFLDictionary 3: NL to PFL

deduction module

summarization

D2D1

D3

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WORLD KNOWLEDGE

Examples

icy roads are slippery big cars are safer than small cars usually it is hard to find parking near the campus

on weekdays between 9 and 5 most Swedes are tall overeating causes obesity Ph.D. is the highest academic degree Princeton usually means Princeton University A person cannot have two fathers Netherlands has no mountains

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WORLD KNOWLEDGE

world knowledge—and especially knowledge about the underlying probabilities—plays an essential role in disambiguation, planning, search and decision processes

what is not recognized to the extent that it should, is that world knowledge is for the most part perception-based

KEY POINTS

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WORLD KNOWLEDGE: EXAMPLES

specific: if Robert works in Berkeley then it is likely that

Robert lives in or near Berkeley if Robert lives in Berkeley then it is likely that

Robert works in or near Berkeley

generalized:

if A/Person works in B/City then it is likely that A lives in or near B

precisiated:

Distance (Location (Residence (A/Person), Location (Work (A/Person) isu near

protoform: F (A (B (C)), A (D (C))) isu R

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THE CONCEPT OF A PROTOFORM AND ITS BASIC ROLE IN KNOWLEDGE REPRESENTATION,

DEDUCTION AND SEARCH

Informally, a protoform—abbreviation of prototypical form—is an abstracted summary. More specifically, a protoform is a symbolic expression which defines the deep semantic structure of a construct such as a concept, proposition, command, question, scenario, case or a system of such constructs

Example:Monika is young A(B) is C

instantiation

abstraction

young C

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CONTINUED

object

p

object space

summarization abstractionprotoform

protoform spacesummary of p

S(p) A(S(p))

PF(p)

PF(p): abstracted summary of pdeep structure of p

• protoform equivalence• protoform similarity

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EXAMPLES

Monika is young Age(Monika) is young A(B) is C

Monika is much younger than Robert(Age(Monika), Age(Robert) is much.youngerD(A(B), A(C)) is E

Usually Robert returns from work at about 6:15pmProb{Time(Return(Robert)} is 6:15*} is usuallyProb{A(B) is C} is D

usually6:15*

Return(Robert)Time

abstraction

instantiation

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PROTOFORMS

at a given level of abstraction and summarization, objects p and q are PF-equivalent if PF(p)=PF(q)

examplep: Most Swedes are tall Count (A/B) is Qq: Few professors are rich Count (A/B) is Q

PF-equivalenceclass

object space protoform space

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EXAMPLES

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?

Y

X0

object

Y

X0

i-protoform

option 1

option 2

01 2

gain

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BASIC POINTS annotation: specification of class or type

Monika is young A(B) is CA/attribute of B, B/name, C/value of A

abstraction has levels, just as summarization doesmost Swedes are tall most A’s are tallmost A’s are B QA’s are B’s

P and q are PF-equivalent (at level ) iff they have identical protoforms (at level )most Swedes are tall = few professors are rich

The concepts of cluster and mountain are PF-equivalent

The concepts of risk and obesity are PF-equivalent

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PF-EQUIVALENCE

Scenario A:

Alan has severe back pain. He goes to see a doctor. The doctor tells him that there are two options: (1) do nothing; and (2) do surgery. In the case of surgery, there are two possibilities: (a) surgery is successful, in which case Alan will be pain free; and (b) surgery is not successful, in which case Alan will be paralyzed from the neck down. Question: Should Alan elect surgery?

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PF-EQUIVALENCE

Scenario B:Alan needs to fly from San Francisco to St. Louis and has to get there as soon as possible. One option is fly to St. Louis via Chicago and the other through Denver. The flight via Denver is scheduled to arrive in St. Louis at time a. The flight via Chicago is scheduled to arrive in St. Louis at time b, with a<b. However, the connection time in Denver is short. If the flight is missed, then the time of arrival in St. Louis will be c, with c>b. Question: Which option is best?

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THE TRIP-PLANNING PROBLEM

I have to fly from A to D, and would like to get there as soon as possible

I have two choices: (a) fly to D with a connection in B; or (b) fly to D with a

connection in C

if I choose (a), I will arrive in D at time t1

if I choose (b), I will arrive in D at time t2

t1 is earlier than t2 therefore, I should choose (a) ?

A

C

B

D

(a)

(b)

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PROTOFORM EQUIVALENCE

options

gain

c

1 2

a

b

0

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PROTOFORM EQUIVALENCE

Backpain= trip planning= divorce= job change

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PROTOFORMAL SEARCH RULES

example

query: What is the distance between the largest city in Spain and the largest city in Portugal?

protoform of query: ?Attr (Desc(A), Desc(B))

procedure

query: ?Name (A)|Desc (A)

query: Name (B)|Desc (B)

query: ?Attr (Name (A), Name (B))

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PNL AS A DEFINITION / DESCRIPTION / SPECIFICATION LANGUAGE

X: concept, description, specification

• Describe X in a natural language• Precisiate description of X

KEY IDEA

Test: What is the definition of a mountain?

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FUZZY CONCEPTS

Relevance Causality Summary Cluster Mountain Valley

In the existing literature, there are no operational definitions of these concepts

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DIGRESSION: COINTENSION

C

human perception of Cp(C)

definition of Cd(C)

intension of p(C) intension of d(C)

CONCEPT

cointension: coincidence of intensions of p(C) and d(C)

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PROTOFORMAL DEDUCTION

p

qp*

q*

NL GCL

precisiation p**

q**

PFL

summarization

precisiation abstraction

answera

r** World KnowledgeModule

WKM DM

deduction module

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protoformal rule

symbolic part computational part

FORMAT OF PROTOFORMAL DEDUCTION RULES

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Rules of deduction in the Deduction Database (DDB) are protoformal

examples: (a) compositional rule of inference

PROTOFORMAL DEDUCTION

X is A

(X, Y) is B

Y is A°B

symbolic

))v,u()u(sup()v( BAB

computational

(b) extension principle

X is A

Y = f(X)

Y = f(A)

symbolic

Subject to:

))u((sup)v( Auy

)u(fv

computational

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PROTOFORM DEDUCTION RULE: GENERALIZED MODUS PONENS

X is AIf X is B then Y is CY is D

D = A°(B×C)

D = A°(BC)

computational 1

computational 2

symbolic

(fuzzy graph; Mamdani)

(implication; conditional relation)

classical

AA B B

fuzzy logic

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X is A(X, Y) is BY is AB

))v,u()u((max)v( BAuBA ∧=

symbolic part

computational part

))du)u(g)u(((max)u( AU

BqD ∫=

du)u(g)u(vU

C∫ =subject to:

1du)u(gU

=∫

Prob (X is A) is BProb (X is C) is D

PROTOFORMAL RULES OF DEDUCTION

examples

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COUNT-AND MEASURE-RELATED RULES

Q A’s are B’s

ant (Q) A’s are not B’sr0

1

1

ant (Q)

Q

Q A’s are B’s

Q1/2 A’s are 2B’sr0

1

1

Q

Q1/2

most Swedes are tall ave (height) Swedes is ?h

Q A’s are B’s ave (B|A) is ?C

))a(N(sup)v( iBiQaave ∑1

)(1

ii aNv

),...,( 1 Naaa ,

crisp

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CONTINUED

not(QA’s are B’s) (not Q) A’s are B’s

Q1 A’s are B’sQ2 (A&B)’s are C’sQ1 Q2 A’s are (B&C)’s

Q1 A’s are B’sQ2 A’s are C’s(Q1 +Q2 -1) A’s are (B&C)’s

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PROTOFORMAL CONSTRAINT PROPAGATION

Dana is youngAge (Dana) is young X is A

p GC(p) PF(p)

Tandy is a few years older than Dana

Age (Tandy) is (Age (Dana)) Y is (X+B)

X is AY is (X+B)Y is A+B

Age (Tandy) is (young+few)

)uv(+)u((sup=)v( BAuB+A -

+few

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THE TALL SWEDES PROBLEM

p: Most Swedes are tallq: What is the average height of SwedesPNL-based solution

1. PF(p): Count(A/B) is QPF(q): H(A) is Cno match in Deduction Databaseexcessive summarization

2. PF(p): Count(P[H is A] / P) is Q

Have is C

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CONTINUED

Name H

Name 1 h1

. . . .

Name N hn

P:

Name H µa

Namei hi µA(hi)P[H is A]:

))((sup)( iAihave hN1

v ),...,( N1 hhh ,

subject to: ii h

N1

v

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Rules of deduction are basically rules governing generalized constraint propagation

The principal rule of deduction is the extension principle

RULES OF DEDUCTION

X is A

f(X,) is B Subject to:

))u((sup)v( AuB

computationalsymbolic

)u(fv

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GENERALIZATIONS OF THE EXTENSION PRINCIPLE

f(X) is A

g(X) is B

Subject to:

))u(f((sup)v( AuB

)u(gv

information = constraint on a variable

given information about X

inferred information about X

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CONTINUED

f(X1, …, Xn) is A

g(X1, …, Xn) is B Subject to:

))u(f((sup)v( AuB

)u(gv

(X1, …, Xn) is A

gj(X1, …, Xn) is Yj , j=1, …, n

(Y1, …, Yn) is B

Subject to:

))u(f((sup)v( AuB

)u(gv n,...,1j =

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MODULAR DEDUCTION DATABASE

POSSIBILITY

MODULEPROBABILITY MODULE

SEARCH MODULE

FUZZY LOGIC MODULE

agent

FUZZY ARITHMETIC MODULE

EXTENSION PRINCIPLE MODULE

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SUMMATION

humans have a remarkable capability—a capability which machines do not have—to perform a wide variety of physical and mental tasks using only perceptions, with no measurements and no computations

perceptions are intrinsically imprecise, reflecting the bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information

KEY POINTS

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CONTINUED

imprecision of perceptions stands in the way of constructing a computational theory of perceptions within the conceptual structure of bivalent logic and bivalent-logic-based probability theory

this is why existing scientific theories—based as they are on bivalent logic and bivalent-logic-based probability theory—provide no tools for dealing with perception-based information

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CONTINUED

in computing with words (CW), the objects of computation are propositions drawn from a natural language and, in particular, propositions which are descriptors of perceptions

computing with words is a methodology which may be viewed as (a) a new direction for dealing with imprecision, uncertainty and partial truth; and (b) as a basis for the analysis and design of systems which are capable of operating on perception-based information

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January 26, 2005

Factual Information About the Impact of Fuzzy Logic

 

 PATENTS

Number of fuzzy-logic-related patents applied for in Japan: 17,740

Number of fuzzy-logic-related patents issued in Japan:  4,801

Number of fuzzy-logic-related patents issued in the US: around 1,700

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PUBLICATIONS

 Count of papers containing the word “fuzzy” in title, as cited in INSPEC and MATH.SCI.NET databases.

Compiled by Camille Wanat, Head, Engineering Library, UC Berkeley, December 22, 2004 Number of papers in INSPEC and MathSciNet which have "fuzzy" in their

titles: INSPEC - "fuzzy" in the title1970-1979: 5691980-1989: 2,4041990-1999: 23,2072000-present: 14,172Total: 40,352 MathSciNet - "fuzzy" in the title1970-1979: 4431980-1989: 2,4651990-1999: 5,4832000-present: 3,960Total: 12,351

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JOURNALS (“fuzzy” or “soft computing” in title) 1. Fuzzy Sets and Systems 2. IEEE Transactions on Fuzzy Systems 3. Fuzzy Optimization and Decision Making 4. Journal of Intelligent & Fuzzy Systems 5. Fuzzy Economic Review 6. International Journal of Uncertainty, Fuzziness and

Knowledge-Based Systems7. Journal of Japan Society for Fuzzy Theory and Systems 8. International Journal of Fuzzy Systems 9. Soft Computing 10. International Journal of Approximate Reasoning--Soft

Computing in Recognition and Search 11. Intelligent Automation and Soft Computing 12. Journal of Multiple-Valued Logic and Soft Computing 13. Mathware and Soft Computing 14. Biomedical Soft Computing and Human Sciences 15. Applied Soft Computing

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APPLICATIONS

The range of application-areas of fuzzy logic is too wide for exhaustive listing. Following is a partial list of existing application-areas in which there is a record of substantial activity.

1. Industrial control2. Quality control3. Elevator control and scheduling4. Train control5. Traffic control6. Loading crane control7. Reactor control8. Automobile transmissions9. Automobile climate control10. Automobile body painting control11. Automobile engine control12. Paper manufacturing13. Steel manufacturing14. Power distribution control15. Software engineerinf16. Expert systems17. Operation research18. Decision analysis

19. Financial engineering20. Assessment of credit-worthiness21. Fraud detection22. Mine detection23. Pattern classification24. Oil exploration25. Geology26. Civil Engineering27. Chemistry28. Mathematics29. Medicine30. Biomedical instrumentation31. Health-care products32. Economics33. Social Sciences34. Internet35. Library and Information Science

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Product Information Addendum 1 

This addendum relates to information about products which employ fuzzy logic singly or in combination. The information which is presented came from SIEMENS and OMRON. It is fragmentary and far from complete. Such addenda will be sent to the Group from time to time.

SIEMENS:

    * washing machines, 2 million units sold    * fuzzy guidance for navigation systems (Opel, Porsche)    * OCS: Occupant Classification System (to determine, if a place in a car is occupied by

a person or something else; to control the airbag as well as the intensity of the airbag). Here FL is used in the product as well as in the design process (optimization of parameters). * fuzzy automobile transmission (Porsche, Peugeot, Hyundai) OMRON:

    * fuzzy logic blood pressure meter, 7.4 million units sold, approximate retail value $740 million dollars

Note: If you have any information about products and or manufacturing which may be of relevance please communicate it to Dr. Vesa Niskanen vesa.a.niskanen@helsinki.fi and Masoud Nikravesh Nikravesh@cs.berkeley.edu .

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Product Information Addendum 2

This addendum relates to information about products which employ fuzzy logic singly or in combination. The information which is presented came from Professor Hideyuki Takagi, Kyushu University, Fukuoka, Japan. Professor Takagi is the co-inventor of neurofuzzy systems. Such addenda will be sent to the Group from time to time.

  Facts on FL-based systems in Japan (as of 2/06/2004)

1. Sony's FL camcorders

Total amount of camcorder production of all companies in 1995-1998 times Sony's market share is the following. Fuzzy logic is used in all Sony's camcorders at least in these four years, i.e. total production of Sony's FL-based camcorders is 2.4 millions products in these four years.

     1,228K units X 49% in 1995     1,315K units X 52% in 1996     1,381K units X 50% in 1997     1,416K units X 51% in 1998

2. FL control at Idemitsu oil factories

Fuzzy logic control is running at more than 10 places at 4 oil factories of Idemitsu Kosan Co. Ltd including not only pure FL control but also the combination of FL and conventional control.

They estimate that the effect of their FL control is more than 200 million YEN per year and it saves more than 4,000 hours per year.

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3. Canon

Canon used (uses) FL in their cameras, camcorders, copy machine, and stepper alignment equipment for semiconductor production. But, they have a rule not to announce their production and sales data to public.

Canon holds 31 and 31 established FL patents in Japan and US, respectively.

4. Minolta cameras

Minolta has a rule not to announce their production and sales data to public, too.

whose name in US market was Maxxum 7xi. It used six FL systems in acamera and was put on the market in 1991 with 98,000 YEN (body pricewithout lenses). It was produced 30,000 per month in 1991. Its sistercameras, alpha-9xi, alpha-5xi, and their successors used FL systems, too.But, total number of production is confidential.

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5. FL plant controllers of Yamatake Corporation

Yamatake-Honeywell (Yamatake's former name) put FUZZICS, fuzzy software package for plant operation, on the market in 1992. It has been used at the plants of oil, oil chemical, chemical, pulp, and other industries where it is hard for conventional PID controllers to describe the plan process for these more than 10 years.

They planed to sell the FUZZICS 20 - 30 per year and total 200 million YEN.

As this software runs on Yamatake's own control systems, the software package itself is not expensive comparative to the hardware control systems.

6. Others

Names of 225 FL systems and products picked up from news articles in 1987 - 1996 are listed at http://www.adwin.com/elec/fuzzy/note_10.html in Japanese.)

Note: If you have any information about products and or manufacturing which may be of relevance please communicate it to Dr. Vesa Niskanen vesa.a.niskanen@helsinki.fi and Masoud Nikravesh Nikravesh@cs.berkeley.edu , with cc to me.