Vol. 6, No. 1, 2018, 1-9 ISSN: 2321 – 9238 (online) Published on 8 February 2018
www.researchmathsci.org
DOI: http://dx.doi.org/10.22457/pindac.v6n1a1
1
Progress in
Coding Theorems on New Fuzzy Information Theory of
Order
αααα
and Type
ββββ
Safeena Peerzada1, Saima Manzoor Sofi2, M.A.K Baig3 and Ashiq Hussain Bhat4 Post-Graduate Department of Statistics, University of Kashmir, Srinagar190006, India.
1
E-mail: [email protected]; 2E-mail:[email protected]
3
E-mail: [email protected]; 4E-mail: [email protected];
1
Corresponding Author
Received 28 January 2018; accepted 2 February 2018
Abstract. In this paper, we propose a new two parametric fuzzy average code-word length Lβα ( A)for a fuzzy set ‘A’ and its relationship with fuzzy information measure
) ( A
Hαβ is discussed. Also, the bounds of proposed average code-word length, in terms of fuzzy information measure, are obtained.
Keywords: Coding Theorem, Code-Word Length, Holder’s Inequality, and Kraft’s Inequality.
AMS Mathematics Subject Classification (2010): 94A17, 94A24 1. Introduction
Due to lack of sharp distinction whether a particular item belongs to a set or not, a concept of imperfect information arises i.e., fuzziness. The concept of fuzzy sets introduced by Zadeh [10] uses imprecise knowledge to define an event. A fuzzy subset ‘A’ of universe X =
{
x1,x2,...,xn}
is defined as( )
(
)
( )
[ ]
{
x x x X x i n}
A= i,
µ
A i : i ∈ ,µ
A i ∈ 0,1 & =1,2,...,where
µ
A( )
xi is a membership function which gives the degree of belongingness of the element ‘xi’ to the set ‘A’. In case,µ
A(xi)=0 orµ
A(xi)=1 for all ‘xi’ then ‘A’ is called a crisp set.We takeX =
{
x1,x2,...,xn}
a discrete random variable with respective probabilitiesn
p p
p1, 2,..., , pi≥0and 1 1
=
∑
=
n
i i
p . Shannon [9] introduced the following measure of
information and named it entropy.
∑
= − = n
i
i D
i p
p P
H
1
log )
2 n
p p
p1, 2,..., satisfying the Kraft [5] Inequality 1
1
≤
∑
= −
n
i li
D , where ‘D’ is the size of
code alphabet, then the lower bound of mean code word length i.e.,
∑
= = n
i i il
p L
1
lies between H(P) and H(P)+1 as proved by Shannon [9] in his noiseless coding theorem.
The concept of fuzziness, because of its property to consider inaccurate and ambiguous information has vast applicability that covers fields of engineering, computer science, medicine, fuzzy aircraft control and so on. Different information measures along with the basic noiseless coding theorems have been given by several authors who include Kapur [3,4], Renyi [7], Ashiq et.al. [1,2], Baig et. al. [6] etc. The lower bounds for the average code-word length of a uniquely decipherable code have been obtained in terms of Shannon’s [9] measure of entropy. Kapur [4] propounded the relation between the probabilistic entropy and coding. The probabilistic measures of entropy do not work in all situations; in such cases the idea of fuzziness can be considered.
2. Generalized fuzzy average code word length Consider the measure proposed by Safeena [8] et.al. as
(
)
{
( ) 1 ( )}
;0 1,0 1& .1 log 1 ) (
1
) 1 ( )
1
(
µ
α
β
β
α
µ
α
β
β α β αβ
α < < < ≤ >
− + −
=
∑
=
− −
n
i
i A i
A
D x x
n A
H
(1) Corresponding to the above measure (1), we propose the following average code-word length
(
)
{
( ) 1 ( )}
;1 log 1
) (
1
1
) 1 ( )
1 (
− + −
=
− −
=
− −
∑
αα α
β α
β β
α
αβ
α
µ
µ
i l n
i
i A i
A
D x x D
n A
L
. & 1 0 , 1
0<
α
< <β
≤β
>α
(2) In equation (2), ‘D’ is the size of code alphabet. Now, we obtain the bounds of (2) in terms of (1) under the following condition(
)
{
( ) 1 ( )}
1.1
1
) 1 ( )
1
( + − − ≤
=
− −
∑
lin
i
i A i
A x x D
n
α β α
β
µ
µ
(3)Or we can write (3) as
(
)
{
( ), ( )}
1.1
1
≤ −
= ′
∑
lin
i
i A i
A x x D
f
n
µ
µ
(4)where
(
)
(1 )(
)
(1 ))
(
1
)
(
)
(
,
)
(
µ
µ
β αµ
β αµ
− −′ i
=
A i+
−
A iA i
A
x
x
x
x
f
. (5)Coding Theorems on New Fuzzy Information Measure of Order
α
and Typeβ
3
Theorem 2.1. For all integers(D>1), the code-word lengths l1,l2,...,ln satisfying the condition (4), then the code-word length (2) satisfies the inequality
. & 1 0 , 1 0 ; ) ( )
( βα
α
β
β
α
β
α A ≤L A < < < ≤ >
H (6)
The equality holds in (6) if
(
)
{
( ), ( )}
.1
1 log
1
− =
∑
= ′
n
i
i A i A D
i
x x
f n l
µ
µ
(7)
where
(
)
(1 )(
)
(1 )) ( 1 ) ( )
( , )
(
µ
µ
β αµ
β αµ
− −′ i = A i + − A i
A i
A x x x x
f .
Proof: By Holder’s inequality, we have
1 1 1 & ..., , 2 , 1 ; 0 , ;
1
1 1
1 1
= + =
∀
≥
∑
∑
∑
=
= = p q
n i
y x y
x y
x i i
q n
i q i p
n
i
n
i p i i
i . (8)
(
p
<
1
(
≠
0
),
q
<
0
)
or(
q
<
1
(
≠
0
),
p
<
0
)
. The equality holds if and only if there exists a positive constant ‘c’ such thatq i p i
cy
x
=
.Substituting,
(
)
lii A i A
i f x x D
n
x −
− ′
= 1 ( ), ( ) α 1
α
µ
µ
.(
)
− ′
=
µ
µ
1α1
) ( , ) ( 1
i A i A
i f x x
n
y ,
α
α
α
− = −= 1&q 1
p .
Using these values in (8) and after simplification, we get
(
)
{
}
≥
∑
=
− ′
n
i
l i A i A
i
D x x
f n
1
) ( , ) ( 1
µ
µ
(
)
{
( ), ( )}
1{
(
( ), ( ))
}
.1
1 1 1
1 1
1
−
−
∑
∑
= ′
=
− − ′
α α
α
µ
µ
µ
µ
αα n
i
i A i A n
i
l i A i
A f x x
n D
x x
f n
i
4
(
)
{
( ), ( )}
1{
(
( ), ( ))
}
1.1 1 1 1 1 1 1 ≤ − −
∑
∑
= ′ = − − ′ α α αµ
µ
µ
µ
α α n i i A i A n i l i A iA f x x
n D x x f n i
(
)
{
( ), ( )}
1{
(
( ), ( ))
}
.1 1 1 1 1 1 1 − − ≤ ⇒
∑
∑
= − − ′ = ′ α α α α αµ
µ
µ
µ
n i l i A i A n i i A i A i D x x f n x x f n (10) Applying logarithms with base ‘D’ to the both sides of (10), we get(
)
{
}
log 1{
(
( ), ( ))
}
.1 ) ( , ) ( 1 log 1 1 1 1 1 − ≤ −
∑
∑
= − − ′ = ′ n i l i A i A D n i i A i A D i D x x f n x x f n α αµ
µ
α
α
µ
µ
α
(11)As
0
<
β
≤
1
, multiplying both sides of (11) byβ
>0 , we get(
)
{
}
log 1{
(
( ), ( ))
}
.1 ) ( , ) ( 1 log 1 1 1 1 − ≤ −
∑
∑
= − − ′ = ′ n i l i A i A D n i i A i A D i D x x f n x x f n α αµ
µ
α
αβ
µ
µ
α
β
(12) Using (5) in (12), we get(
)
{
}
≤ − + −∑
= − − n i i A i AD x x
n 1 ) 1 ( ) 1 ( ) ( 1 ) ( 1 log 1 α β α β
µ
µ
α
β
(
)
{
}
− + −∑
= − − − − n i l i A i A D i D x x n 1 1 ) 1 ( ) 1 ( ) ( 1 ) ( 1 log 1 α α α β α βµ
µ
α
αβ
.We can write the above as
) ( )
(A L A
Hαβ ≤ βα . Hence, the result is established. We have from equation (7),
Coding Theorems on New Fuzzy Information Measure of Order
α
and Typeβ
5 Or
(
)
{
( ), ( )}
.1
1
1
=
∑
= ′
−
n
i
i A i A l
x x
f n D i
µ
µ
(13)
Raising both sides of (13) to the power
−
α
α
1, we get
Or
(
)
{
( ), ( )}
.1
1
1
1 1
−
= ′
− −
=
∑
α α
α α
µ
µ
n
i
i A i A l
x x
f n D i
Or 1
{
(
( ), ( ))
}
.1
1
1
−
= ′
− −
=
∑
αα α
α
µ
µ
n
i
i A i A l
x x
f n
D i (14)
Multiplying both sides of equation (14) by1
{
f(
A(xi), A(xi))
}
n
µ
µ
′ , and then summingoveri=1,2,...,n, we get
(
)
{
( ), ( )}
1{
(
( ), ( ))
}
.1
1
1 1
1
α α
α
µ
µ
µ
µ
∑
∑
= ′
=
− −
′
=
n
i
i A i A n
i
l i A i
A f x x
n D
x x
f n
i
(15)
Applying logarithms to the base ‘D’ on both sides of (15), and then multiplying both sides by
α
αβ
−
1 , we get
(
)
{
}
log 1{
(
( ), ( ))
}
.1 )
( , ) ( 1
log
1 1 1
1
− =
−
∑
=∑
= ′ − − ′
n
i
i A i A D
n
i
l i A i A
D f x x
n D
x x
f n
i
µ
µ
α
β
µ
µ
α
αβ
αα
We can write the above equation as
) ( )
(A H A
Lβα = αβ .
Hence, the result is established.
Theorem 2.2. For every code with lengths l1,l2,...,lnsatisfying the condition (4),
) ( A
6
α α
Proof: From thetheorem 2.1, we have
) ( )
(A H A
Lβα = αβ . (17)
(17) holds if and only if
(
)
{
( ), ( )}
;0 1,0 1& .1
1
1
α
β
β
α
µ
µ
< < < ≤ >
=
∑
= ′
−
n
i
i A i A l
x x
f n D i
(
)
{
( ), ( )}
.1 log
1
=
⇒
∑
= ′
n
i
i A i A D
i f x x
n
l
µ
µ
We chose the code-word lengths li(i=1,2,...,n)in such a way that they satisfy the
inequality
(
)
{
( ), ( )}
log 1{
(
( ), ( ))
}
1.1 log
1 1
+
< ≤
∑
∑
= ′
= ′
n
i
i A i A D
i n
i
i A i A
D f x x
n l
x x
f
n
µ
µ
µ
µ
(18) Considering the interval of length unity as
(
)
{
( ), ( )}
, log 1{
(
( ), ( ))
}
1 . 1log
1 1
+
=
∑
∑
= ′
= ′
n
i
i A i A D
n
i
i A i A D
i f x x
n x
x f
n
µ
µ
µ
µ
δ
In every
δ
i, there lies one positive integer li such that(
)
{
( ), ( )}
log 1{
(
( ), ( ))
}
1.1 log 0
1 1
+
< ≤
<
∑
∑
= ′
= ′
n
i
i A i A D
i n
i
i A i A
D f x x
n l
x x
f
n
µ
µ
µ
µ
(19) We will first show that the sequence l1,l2,...,ln satisfies the generalized fuzzy Kraft [5] inequality (4). Now, taking L.H.S. of (19), we have
(
)
{
}
in
i
i A i A
D f x x l
n ≤
∑
=1 ′
) ( , ) ( 1
Coding Theorems on New Fuzzy Information Measure of Order
α
and Typeβ
7
(
)
{
( ), ( )}
.1
1
1
≤
⇒
∑
= ′
−
n
i
i A i A l
x x
f n D i
µ
µ
(20)
Multiplying both sides of (20) by 1
{
f(
A(xi), A(xi))
}
n
µ
µ
′ and then summing over ni=1,2,..., on both sides, we get
(
)
{
( ), ( )}
11
1
≤ −
= ′
∑
lin
i
i A i
A x x D
f
n
µ
µ
.which is the generalized fuzzy Kraft [5] inequality. Now, taking R.H.S. of (19), we have
(
)
{
( ), ( )}
11 log
1
+
<
∑
= ′
n
i
i A i A D
i f x x
n
l
µ
µ
.(
)
{
( ), ( )}
.1
1
D x x
f n D
n
i
i A i A li
<
⇒
∑
= ′
µ
µ
(21)Since, 0<
α
<1thus (1−α
)>0and 1 >0
−
α
α
. Now, raising both sides of (21) to the
power
−
α
α
1
, we have
(
)
{
( ), ( )}
.1
1
1
1
−
= ′
−
<
∑
αα α
α
µ
µ
x x D fn D
n
i
i A i A li
(
)
{
( ), ( )}
.1 1
1
1 1
− −
= ′
− −
<
⇒
∑
αα α
α α
α
µ
µ
x x D fn D
n
i
i A i A li
(22)
Multiplying both sides of equation (22) by 1
{
f(
A(xi), A(xi))
}
n
µ
µ
′ , and then summingover i=1,2,...,non both sides, we get
(
)
{
( ), ( )}
1{
(
( ), ( ))
}
.1
1
1 1
1
1
∑
∑
=
− ′
=
− −
′
<
n
i
i A i A n
i
l i A i
A f x x D
n D
x x
f n
i α
α α α
α
µ
µ
µ
µ
8
(
)
{
( ), ( )}
1log 1{
(
( ), ( ))
}
1 .1 log
1 1
−
+
<
∑
=∑
= ′ ′
α
α
µ
µ
α
µ
µ
αi
i A i A D
i
i A i A
D f x x
n D
x x
f n
i
(24) Since 0<
α
<1&0<β
≤1thus (1−α
)>0& 01 >
−
α
αβ
.Multiplying both sides of
equation (24) by
−
α
αβ
1 , we get
orLβα(A)<Hαβ(A)+
β
;0<α
<1,0<β
≤1&β
>α
. Hence, the result is established.3. Conclusion
In this article, we propose a new generalized fuzzy code word length Lβα( A) and develop the coding theorems corresponding to this code word length and also show that
α
β
β
α
β
β α β
α β
α (A)≤L (A)<H (A)+ ; 0< <1,0< ≤1& >
H .
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(
)
{
}
log 1{
(
( ), ( ))
}
.1 )
( , ) ( 1
log
1 1 1
1
β
µ
µ
α
β
µ
µ
α
αβ
αα +
− <
−
∑
=∑
= ′ − − ′
n
i
i A i A D
n
i
l i A i A
D f x x
n D
x x
f n
Coding Theorems on New Fuzzy Information Measure of Order
α
and Typeβ
9
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