Fuzzy Logic
Introduction:
In Artificial Intelligence (AI) the ultimate goal is to create machines that think like
humans. Human beings make decisions based on rules. Although, we may not be
aware of it, all the decisions we make are all based on computer like if-then
statements. If the weather is fine, then we may decide to go out. If the forecast
stays the weather will be bad today, but fine tomorrow, then we make a decision
not to go today, and postpone it till tomorrow.
By abandoning the rigid idea of true or false, Lofti Zadeh, redefined how we think
about logic. Constant researches lead to the invention of Fuzzy Logic.
For instance, consider the given three statements
Dinosaurs’ have ruled on this planet for a long time (for a million of years)
It hasn’t rained since a long time (for a couple of months)
He waited for his turn for a long time (for a couple of hours)
In these statements we cannot infer the true value of ‘for a long time’. There is no
way to represent this concept in standard binary set theory. In a standard set theory
an object is either a member of a set or it is not. Either the value for that object is
true or false. There is no in-between.
This amounts to the use of a characteristic function f for a set A, where f(A)=1 if
the element belongs to A, otherwise it is 0;
Definition:
Fuzzy logic is a form of many valued logic, superset of Boolean logic that has
been extended to handle the concept of partial truth- truth values between
"completely true" and "completely false" to deal with reasoning.
Consider a Universal set U of which a subset called fuzzy subset A(bar) is defined
by function f. If uA(x)=1 or, f(x)=1 signify that x is completely contained in
A(bar). If f(x)=0 signify that x is not a member of A(bar). Values of 0<f(x)<1
signify that x is a partial member of A(bar).
To illustrate this let's talk about people and "youthness". In this case the set U (the
universe of discourse) is the set of people. A fuzzy subset YOUNG is also defined,
which answers the question "to what degree is person x young?" To each person in
the universe of discourse, we have to assign a degree of membership in the fuzzy
subset YOUNG. The easiest way to do this is with a membership function based on
the person's age.
young(x) = { 1, if age(x) <= 20,
(30-age(x))/10, if 20 < age(x) <= 30,
0, if age(x) > 30 }
A graph of this looks like:
Given this definition, here are some example values:
Person
Karim
Rafiq
Safiq
Rafi
Rabbi
Ratul
Age
10
21
25
26
28
83
Degree of youth
1.00
0.90
0.50
0.40
0.20
00
Importance:
Fuzzy sets and fuzzy logic is an important and a practical mathematical tool for the
processing of uncertain and vague information. In our earlier work, a fuzzy logic
with similarity was proposed and the soundness and completeness were proved
(Jiabing Wang et al. 2002). In this paper, some properties of fuzzy inference based
on the resolution principle and paramodulation are discussed: first, the significance
of fuzzy logical inference by the resolution principle is discussed in the context of
fuzzy first-order logic; second, it is shown that the fuzzy logical inference by the
paramodulation is always significant; finally, the relation between a resolvent or
paramodulant and ambiguity is discussed.
The following rules which are common in classical set theory also apply to Fuzzy
set theory.
𝐴̅ U (𝐵̅ ∩ 𝐶̅ ) = (𝐴̅ U 𝐵̅) ∩ (𝐴̅ U 𝐶̅ )
distributivity
𝐴̅ ∩ (𝐵̅ U 𝐶̅ ) = (𝐴̅ ∩ 𝐵̅) U (𝐴̅ ∩ 𝐶̅ )
(𝐴̅ U 𝐵̅ ) U 𝐶̅ = 𝐴̅ U ( 𝐵̅ U 𝐶̅ )
associativity
(𝐴̅ ∩ 𝐵̅) ∩ 𝐶̅ = 𝐴̅ ∩ ( 𝐵̅ ∩ 𝐶̅ )
𝐴̅ ∩ 𝐵̅ = 𝐵̅ ∩ 𝐴̅ , 𝐴̅ U 𝐵̅ = 𝐵̅ U 𝐴̅
commutativity
𝐴̅ ∩ 𝐴̅ = 𝐴̅ , 𝐴̅ U 𝐵̅ = 𝐴̅
idempotency
There is also a form of DeMorgan’s laws:
𝑢(𝐴∩𝐵)′ (x) = 𝑢𝐴′∩𝐵′ (x)
𝑢(𝐴𝑈𝐵)′ (x) = 𝑢𝐴′𝑈𝐵′ (x)
A number of operation that are unique to fuzzy sets have been defined. A few of
the more common operation include.
1. Dilation
2. Concentration
3. Normalization
Figure: (a) Normalization; (b) Concentration ; (c) Dilation
Dilation: The dilation 𝐴̅ defined as
DIL(𝐴̅)= [uA(x)]1/2
for all x in U
Dilation tends to increase the degree of membership of all particular member x by
spreading out the characteristic function curve. In the Figure c we illustrated
Dilation.
Concentration: The concentration 𝐴̅ defined as
CON(𝐴̅)= [uA(x)]2
for all x in U
The concentration is the opposite of dilation. It tends to decrease the degree of
membership of all partial member and concentrates the characteristic function
curve. In the Figure b we illustrated Concentration.
Normalization: The normalization 𝐴̅ defined as
NORM(𝐴̅)= uA(x)/ maxx{ uA(x)}
for all x in U
we illustrated normalization in the figure a. Normalization provides a means of
normalizing all characteristic functions to the same base much the same as vectors
can be normalized.