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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: [email protected]
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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 [email protected] and Masoud Nikravesh [email protected] .
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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 [email protected] and Masoud Nikravesh [email protected] , with cc to me.