1. Fuzzy Expert System Fuzzy Logic
دكترمحسن كاهاني

2. Introduction Experts rely on common sense when they solve problems.
How can we represent expert knowledge that uses vague and ambiguous terms in a computer?
Fuzzy logic is not logic that is fuzzy, but logic that is used to describe fuzziness.
Fuzzy logic is the theory of fuzzy sets, sets that calibrate vagueness.
Fuzzy logic is based on the idea that all things admit of degrees.
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3. Fuzzy Logic Boolean logic uses sharp distinctions.
Fuzzy logic reflects how people think. It attempts to model our sense of words, our decision making and our common sense. As a result, it is leading to new, more human, intelligent systems.
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4. Fuzzy Logic Histroty
Fuzzy, or multi-valued logic was introduced in the 1930s by Jan Lukasiewicz, a Polish philosopher. This work led to an inexact reasoning technique often called possibility theory.
Later, in 1937, Max Black published a paper called “Vagueness: an exercise in logical analysis”. In this paper, he argued that a continuum implies degrees.
In 1965 Lotfi Zadeh, published his famous paper “Fuzzy sets”.
Zadeh extended possibility theory into a formal system of mathematical logic.
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5. Why fuzzy?
As Zadeh said, the term is concrete, immediate and descriptive.
Why logic?
Fuzziness rests on fuzzy set theory, and fuzzy logic is just a small part of that theory.
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6. Definition
Fuzzy logic is a set of mathematical principles for knowledge representation based on degrees of membership.
Unlike two-valued Boolean logic, fuzzy logic is multi-valued.
It deals with degrees of membership and degrees of truth.
Fuzzy logic uses the continuum of logical values between 0 (completely false) and 1 (completely true).
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7. Range of logical values in Boolean and fuzzy logic
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8. Fuzzy sets The concept of a set is fundamental to mathematics.
However, our own language is also the supreme expression of sets. For example, car indicates the set of cars. When we say a car, we mean one out of the set of cars.
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9. Tall men example
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10. Crisp and fuzzy sets of “tall men”
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11. A fuzzy set is a set with fuzzy boundaries
The x-axis represents the universe of discourse
The y-axis represents the membership value of the fuzzy set.
In classical set theory, crisp set A of X is defined as
fA(x): X → {0, 1}, where
In fuzzy theory, fuzzy set A of universe X is defined
μA(x): X → [0, 1], where μA(x) = 1 if x is totally in A;
μA(x) = 0 if x is not in A;
0 < μA(x) < 1 if x is partly in A.
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12. fuzzy set representation
First, we determine the membership functions. In our “tall men” example, we can obtain fuzzy sets of tall, short and average men.
The universe of discourse − the men’s heights − consists of three sets: short, average and tall men.
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13. Crisp and fuzzy sets
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14. Representation of crisp and fuzzy subsets
Typical functions : sigmoid, gaussian and pi.
However, these functions increase the time of computation. Therefore, in practice, most applications use linear fit functions.
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15. Membership Functions (MFs)
Characteristics of MFs:
Subjective measures
Not probability functions
“tall” in Asia
MFs
“tall” in NBA
Here I like to emphasize some important properties of membership functions. First of all, it subjective measure; my membership function of all?is likely to be different from yours. Also it context sensitive. For example, I 5?1? and I considered pretty tall in Taiwan. But in the States, I only considered medium build, so may be only tall to the degree of .5. But if I an NBA player, Il be considered pretty short, cannot even do a slam dunk! So as you can see here, we have three different MFs for all?in different contexts. Although they are different, they do share some common characteristics --- for one thing, they are all monotonically increasing from 0 to 1. Because the membership function represents a subjective measure, it not probability function at all. .8
“tall” in the US .5 .1
5’10’’
Heights
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16. A fuzzy set is totally characterized by a membership function (MF).
Fuzzy Sets
Formal definition:
A fuzzy set A in X is expressed as a set of ordered pairs:
Membership
function
(MF)
Universe or
universe of discourse
Fuzzy set
A fuzzy set is totally characterized by a
membership function (MF).
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17. Fuzzy Sets with Discrete Universes
Fuzzy set C = “desirable city to live in”
X = {SF, Boston, LA} (discrete and nonordered)
C = {(SF, 0.9), (Boston, 0.8), (LA, 0.6)}
Fuzzy set A = “sensible number of children”
X = {0, 1, 2, 3, 4, 5, 6} (discrete universe)
A = {(0, .1), (1, .3), (2, .7), (3, 1), (4, .6), (5, .2), (6, .1)}
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18. Fuzzy Sets with Cont. Universes
Fuzzy set B = “about 50 years old”
X = Set of positive real numbers (continuous)
B = {(x, mB(x)) | x in X}
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19. Alternative Notation
A fuzzy set A can be alternatively denoted as follows:
X is discrete
X is continuous
Note that S and integral signs stand for the union of membership grades; “/” stands for a marker and does not imply division.
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20. Fuzzy Partition
Fuzzy partitions formed by the linguistic values “young”, “middle aged”, and “old”:
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21. MF Terminology MF 1 .5 a Core Crossover points a - cut Support X
Core X
Crossover points
a - cut
Support
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22. MF Formulation Triangular MF: Trapezoidal MF: Gaussian MF:
Generalized bell MF:
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23. MF Formulation
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24. Linguistic variables and hedges
At the root of fuzzy set theory lies the idea of linguistic variables.
A linguistic variable is a fuzzy variable. For example, the statement “John is tall” implies that the linguistic variable John takes the linguistic value tall.
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25. Example
In fuzzy expert systems, linguistic variables are used in fuzzy rules. For example:
IF wind is strong
THEN sailing is good
IF project_duration is long
THEN completion_risk is high
IF speed is slow
THEN stopping_distance is short
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26. Hedge
A linguistic variable carries with it the concept of fuzzy set qualifiers, called hedges.
Hedges are terms that modify the shape of fuzzy sets. They include adverbs such as very, somewhat, quite, more or less and slightly.
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27. Fuzzy sets with the hedge very
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28. Representation of hedges
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29. Representation of hedges (cont.)
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30. Operations of fuzzy sets
The classical set theory developed in the late 19th century by Georg Cantor describes how crisp sets can interact. These interactions are called operations.
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31. Cantor’s sets
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32. μØA(x) = 1 − μA(x) Complement
Crisp Sets: Who does not belong to the set?
Fuzzy Sets: How much do elements not belong to the set?
The complement of a set is an opposite of this set.
μØA(x) = 1 − μA(x)
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33. Containment Crisp Sets: Which sets belong to which other sets?
Fuzzy Sets: Which sets belong to other sets?
A set can contain other sets. The smaller set is called subset.
In crisp sets, all elements of a subset entirely belong to a larger set.
In fuzzy sets, each element can belong less to the subset than to the larger set. Elements of the fuzzy subset have smaller memberships in it than in the larger set.
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34. Intersection μA∩B(x) = min [μA(x), μB(x)] = μA(x) ∩ μB(x)
Crisp Sets: Which element belongs to both sets?
Fuzzy Sets: How much of the element is in both sets?
In classical set theory, an intersection between two sets contains the elements shared by these sets
In fuzzy sets, an element may partly belong to both sets with different memberships. A fuzzy intersection is the lower membership in both sets of each element.
μA∩B(x) = min [μA(x), μB(x)] = μA(x) ∩ μB(x)
where xÎX
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35. Union μAÈB(x) = max [μA(x), μB(x)] = μA(x) È μB(x)
Crisp Sets: Which element belongs to either set?
Fuzzy Sets: How much of the element is in either set?
The union of two crisp sets consists of every element that falls into either set.
In fuzzy sets, the union is the reverse of the intersection. That is, the union is the largest membership value of the element in either set.
μAÈB(x) = max [μA(x), μB(x)] = μA(x) È μB(x)
where xÎX
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36. Operations of fuzzy sets
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37. Fuzzy rules
In 1973, Lotfi Zadeh published his second most influential paper. This paper outlined a new approach to analysis of complex systems, in which Zadeh suggested capturing human knowledge in fuzzy rules.
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38. What is a fuzzy rule?
A fuzzy rule can be defined as a conditional statement in the form:
IF x is A
THEN y is B
where x and y are linguistic variables; and A and B are linguistic values determined by fuzzy sets on the universe of discourses X and Y, respectively.
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39. classical vs. fuzzy rules?
A classical IF-THEN rule uses binary logic
Rule: 1
IF speed is > 100
THEN stopping_distance is long
Rule: 2
IF speed is < 40
THEN stopping_distance is short
Representing the stopping distance rules in a fuzzy form:
Rule: 1
IF speed is fast
THEN stopping_distance is long
Rule: 2
IF speed is slow
THEN stopping_distance is short
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40. Fuzzy Rules Fuzzy rules relate fuzzy sets.
In a fuzzy system, all rules fire to some extent, or in other words they fire partially.
If the antecedent is true to some degree of membership, then the consequent is also true to that same degree
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41. Fuzzy sets of tall and heavy men
These fuzzy sets provide the basis for a weight estimation model. The model is based on a relationship between a man’s height and his weight:
IF height is tall
THEN weight is heavy
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42. monotonic selection
The value of the output or a truth membership grade of the rule consequent can be estimated directly from a corresponding truth membership grade in the antecedent. This form of fuzzy inference uses a method called monotonic selection.
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43. Fuzzy Rule A fuzzy rule can have multiple antecedents, for example:
IF project_duration is long
AND project_staffing is large
AND project_funding is inadequate
THEN risk is high
IF service is excellent
OR food is delicious
THEN tip is generous
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44. Fuzzy Rule
The consequent of a fuzzy rule can also include multiple parts, for instance:
IF temperature is hot
THEN hot_water is reduced;
cold_water is increased
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