Prediction of Changes That May Occur in the Neutral Cases in Conflict Theory Based on Graph Theory

Prediction of Changes That May Occur in the
Neutral Cases in Conflict Theory Based on
Graph Theory
Prof. Dr. Hussein K. Khafaji
AL-Rafidain University College/Computer Communication Eng.Dept
dr.hkm1811@yahoo.com
Huda Ahmed Abed
Iraqi Commission for Computers and Informatics/ Informatics Institute for Postgraduate
Studies
Programer8039@gmail.com
Abstract-Rough set theory is a novel mathematical tool to process uncertainty decision-making problem. It offers a
new viewpoint to study conflict analysis decision making as in Pawlak conflict analysis model.
Conflict Theory supports the political defacto such as the well-known sayings "friend of my friend is my
friend", and "enemy of my enemy is my friend", according to the feature coalition relation in Pawlak conflict theory,
it is possible to expect of indirect relationships between neutral agents based on their relationships with others. There
is no real research dedicated to implement or discuss these features.
In this paper, we attempt to develop the conflict analysis system to predict the changes that may happen in
coalitions and conflicts relations among the agents. These changes usually occur with the neutral agents, they may
change their opinions to coalition or conflict. The proposed modification of conflict model depends on suggested
operations accomplished on the graph representation of the information system, such as ORing, ANDing, XORing,
and finding the indirect coalition and conflict paths among the agents in the model.
Keywords- Conflict analysis, Rough sets theory, Conflict model.
I. Introduction
Conflict is a feature of human nature, which exists in a various situation of life. The goal of studying
conflict is to find the conflicting parties, which have an influence on the decision making. Then try to find a way
to improve the relationship between these conflicting parties [1]. So conflict has been used in various
remarkable fields like trade, economical, governmental and political contention, games, and management
negotiations, military attacks etc., especially in areas that require decision making that have uncertainty
problems. Conflict analysis which goals to find out the kind of conflict has lately attracted raised attention[2]
[3]. Rough set theory is an influential tool in treatment vague information in conflict analysis.
Generally, uncertainty in conflict situation exists in three binary relations between the objects (agents).
These relations can be classified into the coalition, neutrality, and conflict among agents[4][5].
The heart of RST is the concept of indiscernibility relation; therefore, the conflict relation is the
differing or negation of this concept. Its meaning is the discernibility relation. Thus in conflict analysis study, it
is possible to use the conflict relation which is reasonably related to indiscernibility relation [6][7].
II. PAWLAK CONFLICT THEORY
The state of conflict consists of agents, who are in struggle over particular issues. These agents may be
members of parliament, individuals within a company, or any type of agent which have influence on decision-
making. Rough sets are considered fantastic for establishing conflict model. Agents give their opinions
according to the issues raised [8].
Conflict theory can be represented by means of the matrix, where each row is considered as an agent,
while column represent issue understudy. The value of this matrix comprises opinions of agents to specific issue
restricted to one of three values: −1, 0, 1 which means disagreement, neutral, and agreement to the issue
respectively[6][7] [9] [10].
International Journal of Computer Science and Information Security (IJCSIS),
Vol. 15, No. 9, September 2017
123 https://sites.google.com/site/ijcsis/
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This matrix can be considered as an information system, IS = (Ag, Iu), which encloses two finite nonempty sets
of agents, Ag, and issues Iu, respectively. Iu is an issue set, and the set of potential values of ∈ is plain-
possible-value-of-i={against, neutral, favorable}, representing agent’s opinion, about debated issue, and
mathematically represented as V = {−1, 0, 1} or shortly V = {−, 0, +}.
v (ag, i) is a function returning the value of the opinion of agent about the issue , where ag ∈
Ag, ∈ . For ∈ , a function ( , , ): × → {−1, 0, 1}, which is as follows:
Where and ∈ Ag and ≠ , while ∈ .Three relations are distinct to Ag × Ag: alliance,
neutrality, conflict,
these relations represent to the relationships between agents:
- ( , ) ( , , ) = 1,
- ( , ) ( , , ) = 0,
- ( , ) ( , , ) = −1.
Each of above relation have its own features. First the alliance relation has the following features:
(1) ( , ),
(2) ( , ) ( , ),
(3) ( , ) ( , ) ( , ),
The last condition(3) can be meant as a phrase : "friend of my friend is my friend".
The features of conflict relation can be simplified as:
(4) ( , ),
(5) ( , ) ( , ),
(6) ( , ) ( , ) ( , ),
Property in (6) translate to the famous phrase "enemy of my enemy is my friend"
By the same token the neutrality relationship has the followeing features:
(8) ( , )
(9) ( , ) = ( , ) [51].
The concept of a discernibility matrix assume = ( , ), ⊆ , meant MINFS (Isu), or
M(Isu), it would mean , = | |, matrix represents as follow:
( , ) = {i ∈ Isu|i(r) ≠ i(s)} … … … … … … … . eq.2
So Is(a,b) means all attributes that distinguish agent from .
Every pair of agents and that specify by the discernibility matrix ( ) are sub-set of attributes
( , ) ⊆ , and have features [6][7] [9] [10]:
i. ( , ) = ∅,
ii. ( , ) = ( , ),
iii. ( , ) ⊆ ( , ) ( , ).
now define a conflict function based on discernibility matrix.
CON ( , ) =
| ( , )|
| |
where 0 ≤ ( , ) ≤ 1 … … … … … … eq. 3
- CON ≠ 0, indicates that and in conflict over (issues) with a degree CON ( , )
- ( , ) = 0, indicates that and in coalition about .
The distance function can be represents as: (r, s) iff CON (r, s) > 0.
If (r, s) this means that and are in conflict with degree ( , ).
The calculating of used function *, Instead of function CON.
So a distance function between agents con∗
: Ag × Ag → [0, 1] is clarified:
( , , ) =
1 ( , ) × ( , ) = 1 = ,
0 ( , ) × ( , ) = 0 ≠ ,
−1 ( , ) × ( , ) = −1.
… … … . . eq. 1
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con∗ ( , ) =
∑ ∗
(r, s, i) ∈
{ }
… … … … … … … .4
Where:
∗(r, s, i) =
1 − ( , , )
2
… … … … … … … .5
The ∗(r, s, i) based on value auxiliary function ( , , ) to obtain:
=
So the definition of the relations between agents would be clarified as a pair , ∈ that [16] [136] [47]
[87]:
- If ( , ) < 0.5, ℎ ( , ),
- If ( , ) > 0.5, ℎ ( , ),
- If ( , = 0.5, ℎ ( , ).
In many levels of processing, the information system can be represented as digraph, bipartite graph, or
weighted graph, and in this way, it can be obeyed to the graph mathematics, theories, and operations that can be
applied on the graph. In this research, many graph operations are suggested such as ORing, ANDing, and
XORing of the graphs that represent a conflict models according to selected issues. Also, an algorithm is
presented to find the indirect coalition or conflict paths among the agents that plays the main role to predict the
changes that may happened in the opinion of neutral agent. The next section explains the suggested development
that accomplished on the conflict theory.
III. MODIFIED CONFLICT MODEL
Rough set and conflict theory discover knowledge of conflict and alliances for current situation of
agents depending on the interest issues. However, Conflict Theory, supports the political in fact such as the
well-known sayings "friend of my friend is my friend", and "enemy of my enemy is my friend", According to
the feature coalition relation in Pawlak conflict theory:
− ( , ) ( , ) ( , ),
And the features of conflict relation:
− ( , ) ( , ) ( , ),
Therfore it is possible to foresee of indirect relationships between neutral agents based on their relationships
with others.
There is no real application for these features. In this section, an attempt to develop the conflict analysis
system to predict the changes may happen in coalitions and conflicts relations among the agents. These changes
usually occur with the neutral agents, they may alter their situation from coalition to conflict and vice versa.
Neutral agent may alter his opinion by the influence of his direct and/or indirect friends, (direct and/or indirect
alliances) and the behavior of his direct and/or indirect enemies, (direct and/or indirect conflicts).
Remember that, Distance Function matrix DF, or its variations contain three relation, such that DF [A1]
[A2] =0 means A1 and A2 have same opinion about a specific issue(s), while DF [A1] [A2] =1 means A1 and
A2 have different opinions about a specific issue(s). DF [A1] [A2] =0.5 means that at least A1 and/or A2
have/has no opinion about a specific issue(s). The following strategy has been suggested to predict the possible
changes that may occur in agents' opinion.
After the construction of distance function for all issues, it will be utilized for analysis in terms of conflict
by changing all the values of conflict in this function to 1 (values greater than 0.5), while all the remaining
values change to 0 (by neglecting the value of the alliance and neutrality situations). In other word copy the
conflict values of DF in distance function of conflict (DFC) and replace these values by 1, i.e., create binary
matrix. In the same manner, Binary matrix distance function for alliances (DFA) is created from alliance values
0 ( , ) × ( , ) = 1 = , ( , , ) = 1
0.5 ( , ) × ( , ) = 0 ≠ , ( , , ) = 0 … … … eq. 6
1 ( ) × ( ) = −1, ( , , ) = −1
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of DF (whose values are less than 0.5, and converted them to a value 1, while all the remaining values change to
0, neglecting the value of the conflict and neutrality situations). In the same way, the creation of a binary matrix,
distance function of neutral (DFN), is accomplished from neutral values of DF (whose values are equal to 0.5,
and converted them to a value 1, while all the remaining values change to 0, neglecting the value of the conflict
and alliance situations).
Let DFC= (DFC)1
means the direct conflicts among the agents. Each element of (DFC)2
= (DFC)1 ×
(DFC)1 represents the number of conflict between two agents passing through a third agent. For example,
(DFC)2
[i][j]=3 means that there are three conflict passageways of length two between agent i and agent j. In the
same way (DFC)Agent#-1= (DFC)1 × (DFC)Agent#-2 can be obtained. DFA= (DFA)1
means the direct
alliances between any two agents. Each element of (DFA)2
=(DFA)1× (DFA)1 represents the number of
alliances between two agents passing through a third agent. For example, (DFA)2
[i][j]=3 means that there are
three alliance passageways of length two between agent i and agent j.
In the same way (DFA)Agent#-1 =(DFA)1×(DFA)Agent#-2 can be obtained. Algorithm (3.21) presents this
strategy, consider step 7 to step 14. Weights are given for each generated DF such that DF of highest power is
assigned a weight of 1. The second DF of highest power is assigned a weight of 2 and so on, this process is
illustrated in steps 16 to 21. Steps 22 to 30 select a neutral value related to a pair of agents, then sum the
corresponding values in CPMs(conflict path matrices), sum the corresponding values in APMs(alliance path
matrices), and according to their difference, the predicted value will be assigned. These steps will be repeated
for all neutral values.
An amazing by-product result of the prediction process is that the logical ORing of CPMs indicates that
there is a direct conflict or indirect conflict of length 1, 2, or N, (number of agents), consider eq.7
= ( (CPM ))
#
… . eq. 7
Same saying can be adopted for APMs, consider eq.8
= ( (
#
)) … . . eq. 8
Subsequently, for example, CPM2=CPM1 ORing CPM2 indicates the existence of indirect conflict between two
agents through another agent, and so on for higher power CPMs. Algorithm presented in figure (1.1)
accomplishes this duty by modifying steps 34 to 42 of. The mathematical operations are replaced by logical
operations and in this way the expensive way of finding powered matrices.
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Figure .1 Predicted conflict and alliances algorithm
00 Algorithm opinion change prediction algorithm
01 Input: Distance function matrix//
N Agents# // number of agents
02 Output: // Conflict path matrices-CPM1.. CPMn-1
03 CPM1 , CPM2, … , CPMn-1
04 //Alliance path matrices- APM1..APMn-1
05 APM1,APM2,..APMn-1
06 Prediction_matrix =IS;
07 {
08 construct a matrix of conflict values only and zero other values; CPM1
09 construct a matrix of alliance values only and zero other values; APM1
10 construct a matrix of neutral values only and zero other values; DFN
11 for (i=2; i<N;I++)
12 { indirect_alliance_or_conflict(CPM1, CPMi-1, CPMi);
13 indirect_alliance_or_conflict(APM1, APMi-1, APMi);
14 }
15 // assign weight for indirect conflict matrices and alliances matrices
16 int w=N;
17 for (i=1; i<N;I++)
18 { multiply(w,CPMi, CPMi);
19 multiply(w,APMi, APMi);
20 w--;
21 }
22 //find prediction matrix
23 for each pair of neutral agents in IS, A1 and A2
24 { predicted_conflict_value= CPM1 [A1][A2] +…+ CPMn-1 [A1][A2];
25 predicted_alliance_value= APM1 [A1][A2] +…+APMn-1 [A1][A2];
26 if (predicted_conflict_value> predicted_alliance_value)
27 predicted_matrix[A1][A2]=-1;
28 else if (predicted_conflict_value< predicted_alliance_value)
29 predicted_matrix[A1][A2]=1;
30 // else do nothing; It is already 0 i.e., neutral
31 }
32 } // of the algorithm
33 // to find indirect conflict or coalition
34 indirect_alliance_or_conflict(one, two, three);
35 { for(int i=0; i<n; i++)
36 for(int j=0; j<n; j++)
37 { buffer=0;
38 for(int k=0; k<n; k++)
39 buffer=buffer+one[i][k]*two[k][j];
40 three[i][j]=buffer;
41 }// of for j
42 } // of the indirect_alliance_or_conflict
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Figure.2 Existence of indirect alliances and conflicts.
For more elucidation, consider the following example, which is designed to explain most aspects presented
in this section. Consider the following information system presented in Table .1. The values of this table
represent opinions of agents about specific issues restricted to values of (1; agreement, 0; neutrality, -1;
disagreement). This information system contains five agents (1, 2, 3, 4, and 5) with four issues (a, b, c, and d).
The distance function DF between agents has been computed and the results are shown in Table .2. conflicts
and alliances.
After computing the distance function, its graph is obtained as presented in Figure.3 for more clarity.
Each node represents agent. The dotted line, which connects between any two nodes, represents the alliance
situation existing between two agents. The solid line represents the conflict situation.
Figure.3 a graphical representation of
distance between agents for all issues
As it is evident from the graph, there is no direct relation between agent# 4 and agent# 3, agent# 5 and
agent# 2, or between agent# 1 and agent# 2. According to the feature coalition relation in Pawlak conflict
theory :
( , ) ( , ) ( , ),
This condition can be translated as a phrase: "friend of my friend is my friend". Therfore it is possible
to expect of indirect relationships between neutral agents based on their relationships with other agents.
After applying algorithms in Figure.1 and Figure.2, from step 8 and step 9, the APM1 that denotes
DFA (Distance Function of Alliance), the CPM1 denotes DFC (Distance Function of Conflict) and finally
A/U a b c d
1 -1 0 1 0
2 0 1 0 -1
3 -1 0 1 -1
4 -1 1 0 1
5 1 1 -1 1
Table.1 Information System
agents 1 2 3 4 5
1 0 0 0 0 0
2 0.5 0 0 0 0
3 0.250 0.375 0 0 0
4 0.375 0.500 0.5 0 0
5 0.750 0.5 0.875 0.375 0
Table.2 Distance Function
……
01 // to find the matrix of indirect conflict or coalition existence
02 existence_of_indirect_alliance_or_conflict(one, two, three);
03 { for(int i=0; i<n; i++)
04 for(int j=0; j<n; j++)
05 { buffer=0;
06 for(int k=0; k<n; k++)
07 buffer=buffer ORING one[i][k] ANDING two[k][j];
08 three[i][j]=buffer;
09 }
10 }
…..
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DFN (Distance Function of Neutral) for all issues are presented in Table.3 , Table.4, and Table.5
respectively.
After that DFA2, DFA3, …., DFAn-1 have been calculated to extract indirect alliance passageway
between agents. Table.6, Table.7, and Table.8 with Figure.4, Figure.5, and Figure.6 represent indirect alliance
passageway with various numbers and lengths respectively.
TABLE.6 DFA2
:NUMBER OF INDIRECT ALLIANCE
PASSAGEWAYS OF LENGTH TWO BETWEEN AGENTS
Passageways between agent# 1&2
Passageways between agent# 1&5 New Predicted pasageways
Figure.4 DFA2
:number of indirect alliance passageways of length two between agents
Table.3 distance function of
1 2 3 4 5
1 0 0 1 1 0
2 0 0 1 0 0
3 1 1 0 0 0
4 1 0 0 0 1
5 0 0 0 1 0
alliance DFA
conflict DFC
1 2 3 4 5
1 0 0 0 0 0
2 0 0 0 0 0
3 0 0 0 0 0
4 0 0 0 0 0
5 1 0 1 0 0
1 2 3 4 5
1 0 0 0 0 0
2 1 0 0 0 0
3 0 0 0 0 0
4 0 1 1 0 0
5 0 1 0 0 0
neutral DFN
agents 1 2 3 4 5
1 0 1 0 0 1
2 1 0 0 0 0
3 0 0 0 1 0
4 0 0 1 0 0
5 1 0 0 0 0
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agents 1 2 3 4 5
1 0 0 1 1 0
2 0 0 0 1 0
3 1 1 0 0 1
4 1 1 0 0 1
5 0 0 1 0 0
TABLE.7 DFA3
: NUMBER OF INDIRECT ALLIANCE
PASSAGEWAYS OF LENGTH THREE BETWEEN AGENTS
Passageways between agent# 1&3 Passageways between agent# 1&4
New Predicted pasagew
Figure.5 DFA3
: number of indirect alliance passageways of length three between agents
agents 1 2 3 4 5
1 0 2 0 0 2
2 1 0 0 0 1
3 0 0 0 2 0
4 0 0 2 0 0
5 1 1 0 0 0
Table.8 DFA4
: number of indirect alliance
passageways of length four between agents
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For example the value of DFA2 in position (1, 2), i.e., DFA2 [1] [2] =1, means that there is only one indirect
alliance passageway between agent#1 and agent#2. Actually, the claim is true, recall Figure.3 because there is
passageway through agent#3 as illustrated in Figure.7. In other word the neutrality relation between agent# 1
and agent# 2 can be changed to alliance relation because both agent# 1 and agent# 2 have alliance relation with
agent# 3.
In same way DFA3[2][4]=1, which mean that there is one passageway between agent#2 and agent#4 of length
three as shown in Figure.8.
New Predicted pasagew
Figure.6 DFA4: number of indirect alliance passageways of length four between agents
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After finding all the indirect passageways between the neutral agents, these matrices are combined now
using eq.7, and converted to a single matrix representing all direct and indirect alliance passageways, regardless
of their length, then the resulted matrix, graph, will be converted to a binary matrix as shown in Table.9.
TABLE.9 ALL DIRECT AND INDIRECT ALLIANCE
PASSAGEWAYS
agents 1 2 3 4 5
1 0 0 0 0 0
2 1 0 0 0 0
3 1 1 0 0 0
4 1 1 1 0 0
5 1 1 1 1 0
Similarly, all previous operations are repeated to the conflict situation. All indirect alliance
passageways through conflict that may lead to alliance are calculated (Table.10, Table.11, Table.12) and then a
binary matrix of conflict is found as in Table.13.
TABLE.10 DFC2
:NUMBER OF INDIRECT ALLIANCE
PASSAGEWAYS THROUGH CONFLICT RELATION OF LENGTH
TWO BETWEEN AGENTS
agents 1 2 3 4 5
1 0 0 1 0 0
2 0 0 0 0 0
3 1 0 0 0 0
4 0 0 0 0 0
5 0 0 0 0 0
TABLE.11 DFC3: NUMBER OF INDIRECT ALLIANCE
TWO BETWEEN AGENTS
agents 1 2 3 4 5
1 0 0 0 0 0
2 0 0 0 0 0
3 0 0 0 0 0
4 0 0 0 0 0
5 1 0 1 0 0
TABLE.12 DFC4: NUMBER OF INDIRECT ALLIANCE
TWO BETWEEN AGENTS
agents 1 2 3 4 5
1 0 0 1 0 0
2 0 0 0 0 0
3 1 0 0 0 0
4 0 0 0 0 0
5 0 0 0 0 0
TABLE.13 ALL DIRECT AND INDIRECT ALLIANCE
PASSAGEWAYS
agents 1 2 3 4 5
1 0 0 1 0 1
2 0 0 0 0 0
3 1 0 0 0 1
4 0 0 0 0 0
5 1 0 1 0 0
Figure.7 indirect alliance passageway of length two between
agent#1& agent#2
Figure.8 indirect alliance passageway of length three between
agent#2& agent#4
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Finally, this module will compared the binary conflict matrix and the binary alliance matrix. It regards the
higher priority for the direct connection, i.e., when there is a direct conflict agent #n and agent #m in binary
conflict matrix and there is indirect connection between same agents in the binary alliance matrix, then the
priority for the direct conflict. Then farther operations used in this module are:
1. The XORING process was performed between the binary alliance matrix, Table.9, and the original
conflict matrix, Table.4, to obtain direct and new predicted alliance passageways. The result is shown
in Figure.9.
2. The ANDING process between the neutrality matrix in Table.5 was carried out with the binary alliance
matrix in Table.9, to show only a new predicted passageways that previously were neutral as shown in
Figure.10.
Figure 9 direct and new predicted alliance passageways Figure.10 New predicted passageways
Consequently, new alliances have been predicted in future relations between the agents as shown in
Figure.11, through applying of the famous saying "friend of my friend is my friend".
Figure.11 Existing and predicted relations
IV. Conclusions
This research presented unprecedented algorithm to develop an aspect of conflict theory in which there
is no considerable progress was attained. The following are some conclusions obtained from this research:
- The escalation in neutral opinions of agents in the information system leads to ambiguity in relations
among agents and lack of clarity of vision.
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- It is possible to benefit from having relationships of neutral agents with other agents, which have
alliance or conflict relations to discover indirect passageways, predict of future relationships between
neutral agents who have a limiting relationships, and to expect the tendencies of their opinions through
their limited relationship with other agents.
- The research produces a practical method to implement the extension of "Enmity" and "friendship"
concepts. However, there is no way to prove the credibility of the proposed modification except the
actual occurrence of the predicted changes of the model in the real life conflict problem, and this fact
matches the properties of the original conflict theory.
- An important by-product achievement of the proposed algorithm is that it can be used to find indirect
paths of different lengths in any graph or digraph.
- The proposed operations; graph ORing, ANDing, and XORing, can be used for different purposes
such as social networks.
REFERENCES
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theory.” Information Sciences, 2017.
[2] L. G. D. O. Silva and A. T. De Almeida-Filho, “A multicriteria approach for analysis of conflicts in evidence theory,” Information
Sciences, vol. 346–347. pp. 275–285, 2016.
[3] B. Sun, W. Ma, and H. Zhao, “Rough set-based conflict analysis model and method over two universes,” Information Sciences, vol.
372. pp. 111–125, 2016.
[4] X. Han, T. Dleu, L. Nguyen, and H. Xu, “Conflict Analysis Based on Rough Set in E‐ commerce,” International Journal of Advances
in Management Science, vol. 2, no. 1. 2013.
[5] Z. Pawlak, “On conflicts,” Int. J. Man. Mach. Stud., vol. 21, no. 2, pp. 127–134, 1984.
[6] Z. Pawlak and A. Skowron, “Rough sets and conflict analysis,” E-Service Intell., no. April , 2007.
[7] Z. Pawlak, “Some Issues on Rough Sets.” Springer, 2004.
[8] Z. Pawlak, “An inquiry into anatomy of conflicts,” Inf. Sci. Elsevier scinece, vol. 109, no. 1–4, pp. 65–78, 1998.
[9] Y. Yao, “The two sides of the theory of rough sets.” Knowledge-Based Systems journal, 2015.
[10] A. Skowron, S. Ramanna, and J. F. Peters, “Conflict analysis and information systems: a rough set approach,” Proceedings of the First
international conference on Rough Sets and Knowledge Technology. pp. 233–240, 2006.
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