1. Restriction- Centred Theory (Nontraditional view of fuzzy logic)
Morteza Saberi Young Researcher Club, .
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1th Fuzzy & Soft Computing Workshop March 1, 2013 Tafresh , Iran
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2. Fuzzy Logic: Basic Concept Fuzzy Logic Misunderstanding
Contents
Introduction
Fuzzy Logic: Basic Concept
Fuzzy Logic Misunderstanding
Restriction- Centred Theory
A Note on Z-numbers
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3. Lotfi A. Zadeh: Father of Fuzzy Logic
Timeline
Lotfi A. Zadeh: Father of Fuzzy Logic
- 1921
Born in Baku, Azerbaijan from an Iranian father and a Russian mother
Moved to Iran
With his family, after Stalin took over
- 1942
Placed 2nd in Iran’s national university exams and graduated from the University of Tehran in electrical engineering
Moved to the US during WW II
Master's degree from MIT in 1946
Ph.D. from Columbia in 1949
- Since 1959
taught at Berkeley, became Chair of EE (later EECS) in 1963
-1965 – invented Fuzzy Logic
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4. Fuzzy Logic: Basic Concept
Fuzzy logic does is bridge the gap between classical logic and the real world.
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5. Fuzzy Logic: Basic Concept
Classical logic, imposes that an element should strictly belong or not belong to a clearly demarcated set, like for instance the set of even numbers. But reality is a lot more complex.
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6. Fuzzy Logic: Basic Concept
We have groups, classes and sets whose boundaries are blurred, like that of “good basketball players.” To belong to this set, a basketball player must “be tall” and “shoot well”, but these concepts are imprecise.
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7. Fuzzy Logic Misunderstanding
When it comes to practical application of fuzzy logic, there is a major source of misunderstanding. Fundamentally, fuzzy logic is aimed at precisiation of what is imprecise. But in many of its applications fuzzy logic is used, paradoxically to imprecisiate what is precise.
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8. Fuzzy Logic Misunderstanding
In such applications, there is a tolerance for imprecision, which is exploited through the use of fuzzy logic
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9. Fuzzy Logic Gambit
Precisiation carries a cost. Imprecisiation reduces cost and enhances tractability. This is what I call the Fuzzy Logic Gambit. What is important to note is that precision has two different meanings: precision in value and precision in meaning. In the Fuzzy Logic Gambit what is sacrificed is precision in value, but not precision in meaning.
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10. Fuzzy Logic Gambit
What is important to note is that precision has two different meanings: precision in value and precision in meaning. In the Fuzzy Logic Gambit what is sacrificed is precision in value, but not precision in meaning.
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11. Fuzzy Logic Gambit
More concretely, in the Fuzzy Logic Gambit imprecisiation in value is followed by precisiation in meaning. An example is Yamakawa's inverted pendulum. In this case, differential equations are replaced by fuzzy if-then rules in which words are used in place of numbers. What is precisiated is the meaning of words.
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12. Fuzzy Versus Vagueness
Basically, vagueness connotes insufficient specificity, Whereas fuzziness connotes unsharpness of class boundaries.
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13. Fuzziness Versus Vagueness
I will be back in a few minutes, is fuzzy but not vague. I will be back sometime, is fuzzy and vague. A shadow is fuzzy but not vague. Inappropriate use of the term vague is still a common practice in the literature of philosophy.
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14. Fuzziness Versus Vagueness
Sufficient Information : Numbers that are near to zero. Just Fuzzy. Insufficient Information : How are you? I do not know: sometimes good, sometimes bad!! . Fuzzy, vagueness
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15. Selected papers Fuzzy sets, Inf. Control 8, 338-353, 1965.
Probability measures of fuzzy events, 1968.
Decision-making in a fuzzy environment, (with R. E. Bellman), 1970.
Fuzzy languages and their relation to human and machine intelligence, 1972
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16. Selected papers
The concept of a linguistic variable and its application to approximate reasoning, 1975.
Local and fuzzy logics, (with R.E. Bellman), 1976.
A fuzzy-algorithmic approach to the definition of complex or imprecise concepts 1976.
Fuzzy sets as a basis for a theory of possibility,1978.
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17. Selected papers A theory of approximate reasoning, 1979.
A computational approach to fuzzy quantifiers in natural languages, 1983.
The role of fuzzy logic in the management of uncertainty in expert systems, 1983
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18. Selected papers A theory of commonsense knowledge, 1984.
Fuzzy probabilities, 1984.
Outline of a computational approach to meaning and knowledge representation based on a concept of a generalized assignment statement, 1986.
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19. Selected papers
Test-score semantics as a basis for a computational approach to the representation of meaning, 1986.
A computational theory of dispositions, 1987.
Knowledge representation in fuzzy logic, 1989.
Fuzzy Logic and the Calculus of Fuzzy If-Then Rules, 1991.
Uncertainty in Knowledge Bases, 1991.
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20. Selected papers
An Introduction to Fuzzy Logic Applications in Intelligent Systems, 1991.
Fuzzy Logic and the Calculus of Fuzzy If-Then Rules, 1991.
The Calculus of Fuzzy If-Then Rules, 1992.
Fuzzy Logic for the Management of Uncertainty, 1992.
Fuzzy Logic, Neural Networks and Soft Computing, Communications of the ACM, 1994.
The Role of Fuzzy Logic in Modeling, Identification and Control, 1994.
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21. Selected papers
Why the Success of Fuzzy Logic is not Paradoxical,
Probability Theory and Fuzzy Logic are Complementary rather than Competitive,
Fuzzy Logic = Computing with Words,
Fuzzy Logic and the Calculi of Fuzzy Rules and Fuzzy Graphs, 1996
From Computing with Numbers to Computing with Words - From Manipulation of Measurements to Manipulation of Perceptions, 1999.
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22. Selected papers
Outline of Computational Theory of Perceptions Based on Computing with Words, 2000.
Fuzzy Logic and the Calculi of Fuzzy Rules and Fuzzy Graphs, 1996
From Computing with Numbers to Computing with Words -- From Manipulation of Measurements to Manipulation of Perceptions, 1999.
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23. Selected papers Is there a need for fuzzy logic?, 2008
Toward a perception-based theory of probabilistic reasoning with imprecise probabilities, 2002
Toward extended fuzzy logic—A first step, 2009
Toward a generalized theory of uncertainty (GTU)––an outline, 2005
A Note on Z-numbers,2011
Toward a perception-based theory of probabilistic reasoning with imprecise probabilities,2003
A note on web intelligence, world knowledge and fuzzy logic,2004
Generalized theory of uncertainty (GTU)—principal concepts and ideas,2006
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24. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
Now, Fuzzy logic is not fuzzy. Basically, fuzzy logic is a precise logic of imprecision and approximate reasoning.
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25. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
Fuzzy logic : an attempt at formalization/mechanization of two remarkable human capabilities. 1- The capability to converse, reason and make rational decisions in an environment of imprecision, uncertainty, incompleteness of information, conflicting information, partiality of truth and partiality of possibility – in short, in an environment of imperfect information. 2- The capability to perform a wide variety of physical and mental tasks without any measurements and any computations
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26. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
Duality of meaning of fuzzy logic : one of source of confusion In a narrow sense, fuzzy logic is a logical system :a generalization of multivalued logic In a wide sense, (which is in dominant use today), fuzzy logic, FL, is much more than a logical system
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27. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
The principal facets of FL are:
The logical facet, FLl;
The fuzzy-set-theoretic facet, FLs;
The epistemic facet, Fle;
the relational facet, FLr
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28. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
FLl, is fuzzy logic in its narrow sense. FLl may be viewed as a generalization of multivalued logic.
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29. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
The fuzzy-set-theoretic facet, FLs, is focused on fuzzy sets, that is, on classes whose boundaries are un-sharp. The theory of fuzzy sets is central to fuzzy logic. Historically, the theory of fuzzy sets preceded fuzzy logic in its wide sense.
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30. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
The epistemic facet of FL, FLe, is concerned with Knowledge representation, Semantics of natural languages and Information analysis. In FLe, a natural language is viewed as a system for describing perceptions. An important branch of FLe is possibility theory . Another important branch of FLe is the computational theory of perceptions.
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31. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
The relational facet, FLr, is focused on fuzzy relations and, more generally, on fuzzy dependencies. The concept of a linguistic variable play pivotal roles in almost all applications of fuzzy logic .
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32. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
The concepts of precisiation:
v-precision : precision of value
m-precision: precision of meaning
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33. In this event, the proposition a< X≤ b
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
Specifically, consider a variable X, whose value is not known precisely.
In this event, the proposition a< X≤ b
where a and b are precisely specified numbers,
is v-imprecise and m-precise.
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34. “L. A Zadeh, Is there a need for fuzzy logic
“L.A Zadeh, Is there a need for fuzzy logic?, Information Science, 2008 “: Review
X is small, is both v-imprecise and m-imprecise :
if small is not defined precisely.
If small is defined precisely as a fuzzy set,
then the proposition in question is
v-imprecise and m-precise.
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35. The concept of a generalized constraint
In most realistic settings, constraints have some elasticity and are not precisely defined.
Check-out time is l pm. A constraint on check-out time.
Speed limit is l00 km/h. A constraint on speed.
Vera has a son in mid-twenties and a daughter in mid-thirties. A constraint on Vera’s age.
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36. Restriction-Centered Theory
Restriction-Centered Theory (RCC, 5th international workshop on soft computing application , , Zadeh L.A, 2012 ) First Version: Generalized theory of uncertainty (Computational Statistics & Data Analysis 51, 15-46, Zadeh L.A, 2006 ) “The point of departure in RCC is the concept of a restriction (generalized constraint) “. e.g: Peiman is tall
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37. Restriction-Centered Theory (Continue)
A restriction is an answer to this question:
What is the value of a variable, X?
Simple example :
Question: How long will it take me to drive from Tehran City center to airport? (Zadeh adapted example)
Answers:
1hr and 10 min (arithmetic)
Usually about 1hr and 10min (RRC)
During pick traffic usually it takes about an hour and a half (RRC).
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38. Restriction-Centered Theory (Continue)
A restriction is a : carrier of information about X
Precisiated restriction : when the limitation is mathematically well defined;
Otherwise: UnPrecisiated
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39. Restriction Examples
2≤Y≤6 (possibilistic) Y is heavy (possibilistic) Usually Y is heavy (possibilistic/probabilistic) It is very likely that there will be a significant change increase in the gold world price by next month (possibilistic/probabilistic)
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40. Restrictions In everyday discourse
Restrictions are described and used, in everyday discourse. We can define a natural language as a system of restrictions. “Imprecision of natural languages is in conflict with precision of bivalent logic” (Zadeh, 2012) So: Bivalent logic do not work in this case!
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41. Fuzzy Logic-Generalization
“The canonical form of a restriction on X, R(X), may be represented as: R(X): X isr R, where X is the restricted variable, R is the restricting relation r is an indexical variable which defines how R restricts X.”(Zadeh, 2012)
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42. Direct & Indirect Restriction
R(X): X isr R,
Indirect:
R(X): f(X) isr R,
e.g: “ is likely
“ ( zadeh Example)
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43. Conditional Restrictions
A restriction is conditioned on another restriction. If X isr R then Y iss T Example: If Price is high then demand is low.
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44. Possibilistic Restriction
R(Y): Y is B, (r=blank)
where
B is a fuzzy set
r is blank as fuzzy set shows the relation
Example:
Moscow is a cold city
Moscow : Restricted Variable
Cold: Restricting Relation (As a fuzzy set)
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45. Probabilistic Restriction
R(X): X isp P, where P : X probability density function
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46. Z-Number
The ordered pair, (A,B), is referred to as a Z-number (Zadeh, 2011)
A, B: Fuzzy sets
“B is a possibility restriction on the certainty (probability) that X is A” (Zadeh, 2012)
Usually Tehran is hot in summer.
Temperature (Tehran) iz (hot, usually)
Usually temperature is low (Zadeh Example)
Temperature iz (low, usually)
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47. Z-Restriction R(X): X iz Z,
Z is a combination of possibilistic and probabilistic restrictions :
Z: Prob(X is A) is B,
A & B : Fuzzy sets
Usually Tehran is hot in summer.
Temperature (Tehran) iz (hot, usually)
Usually temperature is low (Zadeh Example)
Temperature iz (low, usually)
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48. “Computational problems which are stated in a natural language”
q: What is the average height of Swedes?
p: Most Swedes are tall.
µ=?
John has a son who is in mid-twenties, and a daughter, who is in mid-thirties.
What is John’s age?
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49. Computational steps 1- Precisiation of imprecise terms: Most, Tall
2- Precisiation of the question (q)
3- Precisiation of the given information ? (p)
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50. Steps 1 & 2 Suppose h is the height density function
Average height of Swedes
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51. Steps 3 Proportion of tall Swedes : Precisiation of p : p* = is most
µtall is the membership function of tall.
Precisiation of p : p* =
is most
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52. Steps 3 (continue)
Precisiation of p : p* =
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53. q* : p* : Extension principle is most
Applying the basic, indirect, possibilistic version of the extension principle
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54. Extension principle (continued)
Subject to
and
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55. Computational steps
It is important to note that the solution is a fuzzy set which is a restriction on the values which have can take. The fuzzy set may be viewed as the set of all values of have which are consistent with the given information, p, with the understanding that consistency is a matter of degree.
Note that the solution involves reasoning and computation of Type 2.
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