Lecture 2
Elements of Fuzzy Logic
1
BACKGROUND & DEFINITIONS
The concept of a set and set theory are powerful concepts in mathematics. However,
the principal notion underlying set theory, that an element can (exclusively) either
belong to set or not belong to a set, makes it well nigh impossible to represent much of
human discourse. How is one to represent notions like:
large profit
high pressure
tall man
wealthy woman
moderate temperature.
Ordinary set-theoretic representations will require the maintenance of a crisp
differentiation in a very artificial manner:
high, high to some extent, not quite high, very high etc.
BACKGROUND & DEFINITIONS
‘Many decision-making and problem-solving tasks are too complex to be
understood quantitatively, however, people succeed by using knowledge
that is imprecise rather than precise. Fuzzy set theory, originally
introduced by Lotfi Zadeh in the 1960's, resembles human reasoning in its
use of approximate information and uncertainty to generate decisions. It
was specifically designed to mathematically represent uncertainty and
vagueness and provide formalized tools for dealing with the imprecision
intrinsic to many problems. By contrast, traditional computing demands
precision down to each bit. Since knowledge can be expressed in a more
natural by using fuzzy sets, many engineering and decision problems can
be greatly simplified.’ http://www.emsl.pnl.gov:2080/proj/neuron/fuzzy/what.html
History


In the early 1900’s, Lukasiewicz described a
three-valued logic. The third value can be
translated as the term “possible,” and he
assigned it a numeric value between True and
False.
Later, he explored four-valued logics, five-valued
logics, and declared that in principle there was
nothing to prevent the derivation of an infinitevalued logic.
History



Knuth proposed a three-valued logic similar to
Lukasiewicz’s.
He speculated that mathematics would become
even more elegant than in traditional bi-valued
logic.
His insight was to use the integral range
[-1, 0 +1] rather than [0, 1, 2].
History



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Lotfi Zadeh, at the University of California at
Berkeley, first presented multi-valued logic-fuzzy
logic in the mid-1960's.
Zadeh developed fuzzy logic as a way of processing
data. Instead of requiring a data element to be either
a member or non-member of a set, he introduced
the idea of partial set membership.
In 1974 Mamdani and Assilian used fuzzy logic to
regulate a steam engine.
In 1985 researchers at Bell laboratories developed
the first fuzzy logic chip.
BACKGROUND & DEFINITIONS
Lotfi Zadeh introduced the theory of fuzzy sets: A fuzzy set
is a collection of objects that might belong to the set to a
degree, varying from 1 for full belongingness to 0 for full
non-belongingness, through all intermediate values
Zadeh employed the concept of a membership function
assigning to each element a number from the unit interval
to indicate the intensity of belongingness. Zadeh further
defined basic operations on fuzzy sets as essentially
extensions of their conventional ('ordinary') counterparts.
Lotdfi Zadeh, Professor in the Graduate School, Computer Science Division
Department of Elec. Eng. and Comp Sciences, University of California Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)
http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
In 1995, Dr. Zadeh was awarded the IEEE Medal of Honor "For pioneering development of fuzzy logic and
its many diverse applications." In 2001, he received the American Computer Machinery’s 2000 Allen
Newell Award for seminal contributions to AI through his development of fuzzy logic.
BACKGROUND & DEFINITIONS
Zadeh also devised the so-called fuzzy logic: This logic was
devised model 'human' reasoning processes comprising:
vague predicates:
e.g. large, beautiful, small
partial truths:
e.g. not very true, more or less false
linguistic quantifiers:
e.g. most, almost all, a few
linguistic hedges:
e.g. very, more or less.
BACKGROUND & DEFINITIONS
Charles Elkan, an assistant professor of computer science
and engineering at the University of California at San Diego,
offers the following definition:
"Fuzzy logic is a generalization of standard logic, in which a
concept can possess a degree of truth anywhere between 0.0
and 1.0. Standard logic applies only to concepts that are
completely true (having degree of truth 1.0) or completely
false (having degree of truth 0.0). Fuzzy logic is supposed to
be used for reasoning about inherently vague concepts, such
as 'tallness.' For example, we might say that 'President
Clinton is tall,' with degree of truth of 0.9.
The term fuzzy logic is used in two senses:
•Narrow sense: Fuzzy logic is a branch of fuzzy set
theory, which deals (as logical systems do) with the
representation and inference from knowledge. Fuzzy
logic, unlike other logical systems, deals with
imprecise or uncertain knowledge. In this narrow, and
perhaps correct sense, fuzzy logic is just one of the
branches of fuzzy set theory.
•Broad Sense: fuzzy logic synonymously with
fuzzy set theory
2.1 Basic concepts
Crisp (Classic、Boolean) Sets
 universe of discourse
 characteristic function
Crisp Logic
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
Crisp logic is concerned with absolutes-true or false,
there is no in-between.
Example:
Rule:
If the temperature is higher than 80F, it is hot; otherwise, it
is not hot.
Cases:
Temperature = 100F
Hot
Temperature = 80.1F
Hot
Temperature = 79.9F
Not hot
Temperature = 50F
Not hot

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Membership function of crisp logic
True
1
HOT
False
0
80F
Temperature
If temperature >= 80F, it is hot (1 or true);
If temperature < 80F, it is not hot (0 or false).
Drawbacks of crisp logic

The membership function of crisp logic fails
to distinguish between members of the same
set.
Conception of Fuzzy Logic

Many decision-making and problem-solving
tasks are too complex to be defined precisely

however, people succeed by using imprecise
knowledge

Fuzzy logic resembles human reasoning in its
use of approximate information and
uncertainty to generate decisions.
Natural Language

Consider:



Joe is tall
-- what is tall?
Joe is very tall -- what does this differ from tall?
Natural language (like most other activities in
life and indeed the universe) is not easily
translated into the absolute terms of 0 and 1.
“false”
“true”
Fuzzy Sets
• Human reasoning often uses vagueness
– Individuals cannot be classified into two groups!
(either true or false)
• Example: The set of tall men
– But… what is tall?
– Height is all relative
– As a descriptive term, tall is very subjective and relies on
the context in which it is used
• Even a 5ft7 man can be considered "tall" when he is
surrounded by people shorter than he is
Fuzzy Logic

An approach to uncertainty that combines
real values [0…1] and logic operations

Fuzzy logic is based on the ideas of fuzzy set
theory and fuzzy set membership often found
in natural (e.g., spoken) language.
Example: “Young”

Example:

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Ann is 28,
Bob is 35,
Charlie is 23,
0.8 in set “Young”
0.1 in set “Young”
1.0 in set “Young”
Unlike statistics and probabilities, the degree
is not describing probabilities that the item is
in the set, but instead describes to what
extent the item is the set.
Membership function of fuzzy logic
Fuzzy values
DOM
Degree of
Membership
Young
Middle
Old
1
0.5
0
25
40
55
Age
Fuzzy values have associated degrees of membership in the set.
Crisp set vs. Fuzzy set
A traditional crisp set
A fuzzy set
Crisp set vs. Fuzzy set
Membership Functions
fuzzy set A on the universe of discourse X can be
expressed symbolically as the set of ordered pairs
•the membership function of x on the set Α
•a mapping of the universe of discourse Χ on the closed
interval [0,1]
•simply a measure of the degree to which x belongs to the
set Α
Membership Functions
Membership functions represent distributions of possibility
rather than probability
For instance, the fuzzy set Young expresses the possibility
that a given individual be young
Membership functions often overlap with each others
– A given individual may belong to different fuzzy sets (with
different degrees)
Example
temperature of water
[0,100]
fuzzy variable - Low
Α={Low}
‘+’ represents the union operator
Fuzzy variable
expressed in terms of phrases that combine
linguistic variables, linguistic descriptors and hedges
fuzzy conditional statement
R : IF S1 THEN S2 or
symbolically as: S1 →S2
for example:
S : Χ is Α
fuzzy conditional statement
composite conditional statement
for example:
2.2 Fuzzy Algorithms
Two or more fuzzy statements can be combined with the
OR connective to form a fuzzy algorithm
2. 3 Fuzzy Operators
min (for minimum) and max (for maximum)
PLC
AND and OR functions
two elements
2. 3 Fuzzy Operators
min (for minimum) and max (for maximum)
one element
minimum (inf) or maximum (sup) of all the elements of
the set
two sets
2. 3 Fuzzy Operators
discrete sets
2. 4 Operations on Fuzzy Sets
null set
identical
membership function is zero everywhere
membership functions are identical
everywhere on the universe of discourse
2. 4 Operations on Fuzzy Sets
subset
union
intersection
product
Operations
A
AB
B
AB
A
2. 5 Algebraic Properties of Fuzzy Sets
2. 5 Algebraic Properties of Fuzzy Sets
The following properties apply only to fuzzy sets:
2. 5 Algebraic Properties of Fuzzy Sets
Example
2.6 Linguistic Variables
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
Variables used in fuzzy systems to express
qualities such as height, which can take
values such as “tall”, “short” or “very tall”.
These values define subsets of the universe
of discourse.
2.6 Linguistic Variables
terms
primary terms which are labels of fuzzy sets, such as
High, Low, Small, Medium, Zero,
negation NOT and connectives AND and OR,
hedges such as very, nearly, almost and
markers such as parentheses ( ).
2.6 Linguistic Variables
 primary terms-----continuous or discrete membership
functions
Gaussian-like membership functions
2.6 Linguistic Variables
 primary
functions
terms-----continuous or discrete membership
generic S functions
2.6 Linguistic Variables
 primary terms-----continuous or discrete membership
functions
Π function
2.6 Linguistic Variables
 primary terms-----continuous or discrete membership functions
constructed from standardized trapezoidal and
triangular functions.
2.6 Linguistic Variables
 primary terms-----continuous or discrete membership functions
Discrete fuzzy set --- sets of singletons on a finite universe of discourse
example
universe of discourse
linguistic variables
small, medium and large
2.7 Connectives
negation NOT and connectives AND and OR
Negation (NOT) and the connectives AND and OR can be defined in terms of
the complement, union and intersection operations respectively.
AND
different universes of discourse
2.7 Connectives
negation NOT and connectives AND and OR
OR
NOT operator
same universes of discourse
negation
Hedges



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A fuzzy set qualifier, such as “very”, “quite”,
“extremely”, or “somewhat”
When applied to fuzzy set, the new set will be produce
 Eg: apply “very” to “tall people” – subset of “tall
people” will produce that called “very tall people”
The meanings of these hedges are fairly subjective
usually use a systematic mathematic definition for the
hedges so that it can be applied logically.
Often a hedges is applied by raising the set’s
membership function to an appropriate power
2.7 Connectives
hedges
Hedge is an intensifier
 Example:
LV = tall, LV1 = very tall,
LV2 = somewhat tall
 ‘very’ operation:
μvery tall(x) = μ2tall(x)
 ‘somewhat’ operation:
μsomewhat tall(x) = √(μtall(x))

somewhat tall
tall
1
μtall(h)
0
very tall
h
An Example: Consider a set of numbers: X = {1, 2, ….. 10}. Johnny’s
understanding of numbers is limited to 10, when asked he suggested
the following. Sitting next to Johnny was a fuzzy logician noting :
Comment
‘Degree of membership’
10
‘Surely’
1
9
‘Surely’
1
8
‘Quite poss.’
0.8
7
‘Maybe’
0.5
6
‘In some cases, not usually’
0.2
5, 4, 3, 2, 1
‘Definitely Not’
0
‘Large Number’
We can denote Johnny’s notion of ‘large number’ by the
fuzzy set
A =0/1+0/2+0/3+0/4+0/5+ 0.2/6 + 0.5/7 + 0.8/8 + 1/9 + 1/10
Fuzzy (sub-)sets: Membership Functions
For the sake of convenience, usually a fuzzy set is
denoted as:
A = A(xi)/xi + …………. + A(xn)/xn
that belongs to a finite universe of discourse:
A

~
{x1, x2 ,.......,xn }
where A(xi)/xi (a singleton) is a pair “grade of
membership element”.
Johnny’s large number set membership function
can be denoted as:
‘Large Number’
 (.)
10
A (10) = 1
9
A (9) = 1
8
A (8) = 0.8
7
A (7) = 0.5
6
A (6) = 0.2
5, 4, 3, 2, 1
A (5) = A (4) = A (3) = A (2) = A (10) = 0
Johnny’s large number set membership function can
be used to define ‘small number’ set B, where
B (.)= NOT (A (.) ) = 1 - A (.):
‘Small Number’
 (.)
10
B(10) = 0
9
B (9) = 0
8
B (8) = 0.2
7
B (7) = 0.5
6
B (6) = 0.8
5, 4, 3, 2, 1
B (5) = B (4) = B (3) = B (2) = B (10) = 1
Concept of Hedge
Johnny’s large number set membership function can be used to define ‘very
large number’ set C, where
C (.)= A (.)* A (.)
Number
Very Large (C (.))
10
1
9
1
8
0.64
7
0.25
6
0.04
5, 4, 3, 2, 1
0
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