This paper presents a new five-valued knowledge representation of bipolar information. This representation is related to a five-valued logic that uses two logical values of truth (true, false) and three logical values of uncertainty (incomplete, inconsistent and fuzzy). The new approach is based on the concept of saturation function and ignorance function. In the framework of five-valued representation new formulae for union and intersection are constructed. Also, the paper presents a short application related to fuzzy preference modeling and decision making.
Let X be a set of objects. We consider a property A , an
object x X and the following sentence PA (x) : x has
the property A . We want to know if the sentence PA (x)
is true or false. After an evaluation, the information about
logical value of sentence PA (x) is described by a scalar
TA (x) [0,1] . For the considered sentence, TA (x)
represents its truth degree. In the same time, the function
TA : X [0,1] defines a Zadeh fuzzy set associated to the
property A
VA (x) TA (x), FA (x)
TA (x) FA (x) 1 and supplementary these two components verify the inequality: The information represented by vector VA (x) is not normalized but, Atanassov has introduced the intuitionistic index:
U A (x) 1 TA (x) FA (x) Using the vector VA (x) , we have obtained an intuitionistic representation of information about sentence PA (x) . (4) (5) (6) WA (x) (TA (x),U A (x), FA (x)) (7) This information is normalized because the components of vector WA (x) verify the condition of partition of unity: TA (x) U A (x) FA (x) 1 (8) The representation (8) is related to a three-valued logic based on true, neutral and false.
In this paper we will consider the bipolar representation
Saturation, Ignorance and Bi-fuzziness
Functions In this section, firstly, we introduce the concepts of saturation function and ignorance function. These two functions are complementary.
Both
functions
are
essentially characterized by symmetry, boundary and
monotonicity properties. Secondly, we introduce the
concept of bi-fuzziness related to the index
of
indeterminacy
Definition 1: A saturation function is a mapping S : [0,1]2 [0,1] such that: i) ii)
S(x, y) S( y, x)
S(x, y) 0 if and only if (x, y) (0,0) iii) S(x, y) 1 if and only if (x, y) (1,1) iv) S (x, y) increases with respect to x and y The property a) describes the commutativity and the property d) describes the monotonicity. From property b) it results that the saturation value is low if and only if both arguments have low value and from property c) it results that the saturation value is high if and only if both arguments have high value.
U : [0,1]2 [0,1] such that:
U (x, y) U ( y, x) i) ii)
U (x, y) 0 if and only if (x, y) (1,1) iii) U (x, y) 1 if and only if (x, y) (0,0) iv) U (x, y) decreases with respect to x and y is a saturation function.
Proof: It is evident because in the new saturation function construction it was used the scalar multiplication based on the uninorm function.
Proposition 5: Let S be a saturation function. Then (9) (10) (12)
S(x, y) R(x, y) (13)
S(x, y) S(1 x,1 y) is a saturation function.
Proof: It is results immediately that the new saturation function verifies the properties i), ii), iii) and iv). Definition 3: A bi-fuzziness function is a mapping I :[0,1]2 [0,1] such that: i) I (x, y) I ( y, x) ii) I (x, y) I (1 x, y) iii) I (x, y) I (x,1 y) iv) I (x, y) 0 if and only if x, y {0,1} v)
I (x, y) 1 if and only if (x, y) (0.5,0.5)
vi) I (x, y) increases with x if x 0,5 and I (x, y)
decreases with x if x 0,5
vii) I (x, y) increases with y if y 0,5 and I (x, y)
decreases with y if y 0,5
The bi-fuzziness function represents a
measure of
similarity between the point (x, y) [0,1]2 and the center
of unit square, the point (0.5,0.5) . The index of
bifuzziness verifies, for each argument x and y , the
properties considered by De Luca and Termini for fuzzy
entropy definition
I (x, y) (1 | S(x, y) S(x, y) |) (1 | S(x, y) S(x, y) |) is a bi-fuzziness function.
Proposition 7: Let S be a saturation function. Then
I (x, y) 1 S| 2x 1|, | 2 y 1| is a bi-fuzziness function.
I (x, y) (1 | x y |)(1 | x y 1|) . (14) (15) (16) (17) (18) (19) (20) (21) Five-Valued Representation of Bipolar
Information
Let S be a saturation function. For any pair (T , F ) , we
define the net truth and the definedness by:
(T , F ) S (T , F ) S (T , F )
(T , F ) S (T , F ) S (T , F )
The uncertainty or the entropy
g | | The two functions define a partition with two fuzzy sets X G and X H : one related to the certainty and the other to the uncertainty.
The index of bi-fuzziness will be computed by difference between uncertainty and non-fuzziness: The non-fuzziness and bi-fuzziness define two subsets of X H , namely: X Z and X I . We compute the incompleteness (undefinedness) and inconsistency (contradiction) using the non-fuzziness: where x max( 0,x) and x max( 0, x) .
The incompleteness and inconsistency define two subsets of X Z , namely: XU and X C . Notice that because c u 0 it results:
XU X C Next we compute the index of truth and falsity using the net truth function : u c t f The index of truth and index of falsity define two subsets of X G , namely: X T and X F . Notice that because t f 0 it results: XT X F (22) The index of truth (20), the index of falsity (21), the index of bi-fuzziness (16), the index of incompleteness (17) and the index of inconsistency (18) define a partition of unity: t f i u c 1 In the construction method presented above, it was used the schema shown in figure 1.
T, F
t
f
Non-fuzzy
Fuzzy i Incomplete u
Inconsistent
c
This five-valued logic is a new one, but is related to our
previous work presented in
The main differences between the proposed logic and the Belnap logic are related to the logical values u and c . We have defined c u i and c u i while in the Belnap logic there were defined c u f and c u t . defines the union (disjunction) d a b by the formula: td ta tb cd (ca f a ) (cb fb ) f a fb ud (ua f a ) (ub fb ) f a fb f d f a fb id 1 (td cd ud fd ) where one (28) The Intersection: For two vectors a, b [0,1]5 one defines the intersection (conjunction) c a b by the formula: We remark that after union or intersection the certainty increases and uncertainty decreases.
The Complement: For x (t, c, i, u, f ) ∈ [0,1]5 one defines the complement xc by formula: The Negation: For x (t, c, i, u, f ) ∈ [0,1]5 one defines the negation xn by formula: The Dual: For x (t, c, i, u, f ) ∈ [0,1]5 one defines the dual xd by formula: In the set {0,1}5 there are five vectors having the form x (t, c, i, u, f ) , which verify the condition t f c i u 1: T (1,0,0,0,0) (True), F (0,0,0,0,1) (False), C (0,1,0,0,0) (Inconsistent), U (0,0,0,1,0) (Incomplete) and I (0,0,1,0,0) (Fuzzy).
Using the operators defined by (28), (29), (30), (31) and
(32), the same truth table results as seen in Tables 1, 2, 3, 4,
5, 6 and 7.
ic 1 (tc cc uc fc )
In formulae (28) and (29), the symbols “ ” and “ ”
represent the maximum and the minimum, namely:
x, y [0,1],
x y max( x, y)
x y min( x, y)
The union “ ” and intersection “ ” operators preserve
de properties t c u f 1 , t f 0 and u c 0 ,
namely:
tab cab uab fab 1
tab f ab 0
cab uab 0
tab cab uab fab 1
tab f ab 0
cab uab 0
xc ( f , c, i, u, t)
x n ( f , u, i, c, t)
x d (t, u, i, c, f )
(29)
(30)
(31)
(32)
Fuzzy Preference Relation in The Framework
of Five-Valued Representation
A fuzzy preference relation A on a set of alternatives
X {x1, x2 ,..., x } is a fuzzy set on the product set
X X , that is characterized by a membership function
P : X X [0,1]
pij P(xi , x j ) indicating that xi preferred to x j but x j is not preferred to xi .
Indifference: iij I (xi , x j ) indicating that xi and x j are considered equal in the sense that xi is as good as x j .
jij J (xi , x j ) which occurs if neither
aij nor a ji .
Taking into account the five-valued representation of bipolar information, we define five relations that characterize the following five fundamental attitudes: Strict preference tij T (xi , x j ) is a measure of strict preference of xi over x j , indicating that xi preferred to x j but x j is not preferred to xi .
Indifference: cij C(xi , x j ) is a measure of the simultaneous fulfillment of aij and a ji .
uij U (xi , x j ) is a measure of the incomparability of xi and x j , which occurs if neither aij nor a ji .
Strict aversion: fij F(xi , x j ) that is a measure of strict preference of x j over xi , indicating that xi is not preferred to x j .
Undecidability: iij I (xi , x j ) is a measure of undecidability between xi and x j which occurs when aij 0.5 and a ji 0.5 .
Next, we consider a decision making problem where, an expert supply the preferences over a set of n alternatives: (38) (39) (40) (41) (42) where aij [0,1] .
The algorithm that we propose to obtain the best alternative is the next: Step 0: Initialize the matrix A and define the saturation function S .
Step 1: Compute the function tij , cij , uij , fij and iij using formulae (20), (21), (16), (17) and (18).
Step 2: Compute the relative score function by: rij
tij cij 0.5 iij tij 2 cij 1.5 iij uij 3 fij Step 3: Compute the total score function by:
n Ri rij j 1 j i Step 4: Choose xoptim arg max Rk
k{1,2,...,n} In the presented algorithm, the next five items hold: If aij 1 and a ji 0 , then rij 1 .
If aij 1 and a ji 1 , then rij 0.5 .
If aij 0.5 and a ji 0.5 , then rij 0.33 .
If aij 0 , and a ji 0 , then rij 0 .
If aij 0 , and a ji 1 , then rij 0 .
X {x1, x2 ,...xn} . The preferences are represented by the following fuzzy relation: Numerical example: Let X {x1, x2 , x3, x4 , x5} be a set of alternatives. Consider the fuzzy preference relation: , it results: tij aij a ji cij aij a ji 1 uij 1 aij a ji fij a ji aij iij 1 | aij a ji | | aij a ji 1| Using the presented algorithm one obtains: R1 1.36 , R2 1.26 , R3 1.41 , R4 1.24 , R5 1.46
xoptim x5 .
In this paper, we propose a different functional approach to model the bipolar information. The new approach is based on two new information concepts: saturation function and ignorance function. Saturation function can be seen as way of generalizing t-conorms dropping out associativity. We must underline that the associativity is not crucial for the construction of five-valued representation. More than that, in our framework, the saturation function has only two arguments: the degree of truth and degree of falsity. Finally, we are dealing with a class of functions different from that of the t-conorms.
The saturation function measures the excess of information, while, the ignorance function measures the lack of information that an estimator suffers when trying to determine if a given sentence is true or false.
The third concept, bi-fuzziness function can be understood as an extension from fuzzy sets to bipolar fuzzy sets of the concept of fuzziness defined by Zadeh. In addition, the index of bi-fuzziness can be understood as a measure of partial uncertainty of bipolar information. Both saturation function and ignorance function are related. Each of them can be recovered in a functional way from the other. If suitable saturation or ignorance functions are known that fit well for a given problem, they can be used to build a five-valued knowledge representation. In this way, we are able to provide a theoretical framework which is different from the usual one to represent truth, falsity, incompleteness, inconsistency and bi-fuzziness. In this framework, a new five-valued logic was presented based on five logical values: true, false, incomplete, inconsistent and fuzzy. It was identified two components for certainty and three components for uncertainty. Based on this logic, (43) (44) (45) (46) (47) new union and intersection operators were defined for the existing five-valued structure of information.
We also propose an application in preferences under a novel score function. The using of the proposed five fundamental attitudes provides a new perspective in decision making and it offers a simple way to produce a comprehensive judgment.