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National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012)
Proceedings published in International Journal of Computer Applications® (IJCA)

Uncertainty Classification of Expert Systems - A Rough
Set Approach
B. S. Panda

Rahuk Abhishek

S. S. Gantayat

Asst. Professor
MITS Eng. College
Rayagada, Odisha.

Research Student
MITS Eng. College
Rayagada, Odisha.

Associate Professor
GMR Institute of Technology,
Rajam, Andhra Pradesh.

ABSTRACT
In this paper, we discussed about the un certainity classifications
of the Expert Systems using a Rough Set Approach. It is a
Softcomputing technique using this we classified the types of
Expert Systems. An expert system has a unique structure,
different from traditional programs. It is divided into two parts,
one fixed, independent of the expert system: the inference
engine, and one variable: the knowledge base. To run an expert
system, the engine reasons about the knowledge base like a
human. In the 80's a third part appeared: a dialog interface to
communicate with users. This ability to conduct a conversation
with users was later called "conversational". Rough set theory is
a technique deals with uncertainty.

Key Words
Expert System, Rough Sets, Lower and Upper approximations,
Uncertainity

1. INTRODUCTION
Expert systems are designed to solve complex problems by
reasoning about knowledge, like an expert, and not by following
the procedure of a developer as is the case in conventional
programming. The first expert systems were created in the 1970s
and then proliferated in the 1980s. [3, 4, 8, 19]. Expert systems
were among the first truly successful forms of AI software.

2. EXPERT SYSTEMS
Inference engine + Knowledge = Expert system
Expert systems are computer programs that are derived from a
branch of computer science research called Artificial
Intelligence (AI) [7, 17, 22]. AI's scientific goal is to understand
intelligence by building computer programs that exhibit
intelligent behavior. It is concerned with the concepts and
methods of symbolic inference, or reasoning, by a computer, and
how the knowledge used to make those inferences will be
represented inside the machine. AI programs that achieve
expert-level competence in solving problems in task areas by
bringing to bear a body of knowledge about specific tasks are
called knowledge-based or expert systems. . The area of human
intellectual endeavor to be captured in an expert system is called
the task domain. Task refers to some goal-oriented, problemsolving activity. Domain refers to the area within which the task
is being performed. Typical tasks are diagnosis, planning,
scheduling, configuration and design. Expert system have a
number of major system components and interface with
individuals who interact with the system in various roles. These
are illustrated below in fig. 1.

fig: (1) Component of expert systems
Expert systems are designed to facilitate tasks in the fields of
accounting, medicine, process control, financial service,
production, human resources, among others. Typically, the
problem area is complex enough that a more simple traditional
algorithm cannot provide a proper solution. The foundation of a
successful expert system depends on a series of technical
procedures and development that may be designed by
technicians and related experts. As such, expert systems do not
typically provide a definitive answer, but provide probabilistic
recommendations [12].
Expert systems provide an advantage when dealing with
uncertainty as compared to decision trees. With decision trees,
all the facts must be known to arrive at an outcome. Probability
theory is devoted to dealing with theories of uncertainty.
There are many theories of probability – each with advantages
and disadvantages.

12
National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012)
Proceedings published in International Journal of Computer Applications® (IJCA)


For

any X  U and

an

equivalence

relation R  IND( K ) , there associate two subsets:


Lower

approximation

of

X

in

A

is

the

set

RX  {Y  U / R : Y  X }

U which can
of X with the

The elements of RX are those elements of
be certainly classified
knowledge of R .


as elements

Upper

of

approximation

X

in

A

is

the

set

RX  {Y  U / R : Y  X  }



fig: (2) Development of an Expert system

The elements of RX are those elements of U, which can
certainly be classified as elements of X, and the elements of
RX are those elements of U, which can possibly be
classified as elements of X, employing knowledge of R.

3. ROUGH SETS
3.1 Definitions and Notations
Rough set theory is a technique deals with uncertainty. In this
section we reintroduce some basic notations of Rough set theory
[1, 2, 13, 18].




U (  ) is the universe and be a finite set of objects.
R is the indiscernibility relation, or equivalence relation
over U .



Indiscernibility is the inability to distinguish between two
or more values.
A= (U, R) an ordered pair is called an approximation space.



[ x] R

denotes the equivalence class or R containing an

element x  U .


For any subset (  )   , the intersection of all
equivalence relations in






is denoted by IND() and is

called the indiscernibility relation over  .
Elementary sets in A – the equivalence classes of R.
Definable set in A–Any finite union of elementary sets in
A.

RX is the set of elements of X which can be possibly
classified as elements of X employing knowledge of R .
The boundary of X is, RX  RX .



The borderline region is the undecidable area of the
universe.

We say that X is rough with respect to R if and only
if RX  RX , equivalently BN R ( X )   . X is said to be Rdefinable if and only if RX  RX , or BN R ( X )   .

3.2 Application of Rough Set
We briefly highlight few applications of RST [5, 9, 14]. We
again feel discussing on various applications will increase the
length of paper.
a. Representation of uncertain or imprecise knowledge.
b. Empirical learning and knowledge acquisition from
experience.
c. Knowledge analysis.
d. Analysis of conflicts.
e. Evaluation of the quality of the available information with
respect to its consistency and the presence or absence of
repetitive data patterns.
f. Identification and evaluation of data dependencies.
g. Approximate pattern classification.
h. Reasoning with uncertainty.
i. Information-preserving data reduction.

3.3 Equivalence relations of Rough Sets and
Classification of Expert Systems
As discussed through the definitions and notations, the rough set
philosophy is based upon the notion of equivalence relations.
This describes the partitions made of classes of indiscernible
objects with common lower and upper approximations.
Incomplete information can be analyzed by means of classical
approximation spaces. Such spaces consist of a universe and an
indiscernibility relation, which is always equivalence relation.
Knowledge granules determined by these spaces are equivalence
classes.
The equivalence relation divides the universe U into pair-wise
disjoint subsets, called equivalence classes of objects. That is,
for an object x∈.U, the equivalence class of objects containing x
is given by: [x] = {x’|xE(V)x’}
In expert system technology, the knowledge base is expressed
with natural language rules IF ... THEN ... For examples :
 "IF it is living THEN it is mortal"

fig: (3) Rough Set Approach
13
National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012)
Proceedings published in International Journal of Computer Applications® (IJCA)



certainity with which the information is known. There are
different ways in which these numbers can be defined, and how
they are combined during the inference process [15, 21].
An important features of expdrt systems is their ability to
explain themselves. Given that the system knows which rules
were used during the inference process, the system can provide
those rules to the user as means for explaaining the results.
By looking explanations, the knowledge engineer can seen how
the system is behaving, and how the rules and data are
inteacting. This is very valueable diagnositc tool during
development.

4. UNCERTAINITY WITH EXPERT
SYSTEMS AND CLASSIFICATIONS
4.1 What is Uncertainty?

5. CONCLUSIONS

"IF his age = known THEN his year of birth = date of today
- his age in years"
 "IF the identity of the germ is not known with certainty
AND the germ is gram-positive AND the morphology of
the organism is "rod" AND the germ is aerobic THEN there
is a strong probability (0.8) that the germ is of type
enterobacteriacae".
This formulation has the advantage of speaking in everyday
language which is very rare in computer science (a classic
program is coded). Rules express the knowledge to be exploited
by the expert system [20].






Uncertainty is essentially lack of information to formulate a
decision.
Uncertainty may result in making poor or bad decisions.
As living creatures, we are accustomed to dealing with
uncertainty – that’s how we survive.
Dealing with uncertainty requires reasoning under
uncertainty along with possessing a lot of common sense
[6, 10, 11].

4.2 Theories to Deal with Uncertainty







Bayesian Probability
Hartley Theory
Shannon Theory
Dempster-Shafer Theory
Markov Models
Zadeh’s Fuzzy Theory

4.3 Dealing with Uncertainty





Deductive reasoning – deals with exact facts and exact
conclusions
Inductive reasoning – not as strong as deductive premises
support the conclusion but do not guarantee it.
There are a number of methods to pick the best solution in
light of uncertainty.
When dealing with uncertainty, we may have to settle for
just a good solution.

4.4 Uncertainty and Rules




There are several sources of uncertainty in rules:
Uncertainty related to individual rules
Uncertainty due to conflict resolution
Uncertainty due to incompatibility of rules

In this paper, we have reviewed Expert Systems and rough set
theory. We discussed the classifications and uncertainity in
Expert Systems data, and took an initiative to combat new
approach is roughsets. It is observed that some theories
previously dealing with uncertainity classifications. We focused
a new technique is rough set theory to clasify the Expert
Systems systematically and developed a practically feasible
approach. For future work, we believe that the discussed
approach will become suitability for classify the expert systems.

6. REFERENCES
[1] Cheeseman, P. (1986) :“Probabilistic vs. Fuzzy
Reasoning”, in Uncertainity in AI, L.N. Kanal and J.F.
Lemmer, eds., Elsevier Scince Publichers, New York,
N.Y., pp. 85-102.
[2] Chelsea, MI.(1989), Dynamic Modelling and Expert
Systems in Wastewater Engineering. Lewis Publishers,
Inc., pp. 167-192.
[3] Finn, G.A. (1989). Applications of Expert Systems in the
Process Industry. In: G.G. Patry and D. Chapman (eds.).
[4] Grzymala-Busse, J.(1988): LERS - A system for learning
from examples based on rough sets, J. Intelligent and
Robotics Systems, 1, pp. 3-16.
[5] Grzymala-Busse, J. (1988): Knowledge Acquisition Under
Uncertainty – A Rough Set Approach, J. Intelligent and
Robotics Systems, 1, pp. 3-16.
[6] Hayes-Roth, F., D.A. Watchman, and D.B. Lenat, eds.
(1983), Building Expert Systems, Addison-Wesley,
Reading, Mass.
[7] Hyde, Andrew Dean (Sept 28, 2010), “The future of
Artificial Intelligence”.
[8] Ishizuka, M., K.S.Fu, and J.T.P. Ya, (1982) “A Rule-Based
Inference with Fuzzy Set for Structural Damage
Assessment”, in Approximate Reasonin in Decision
Analysis, M.M. Gupta and E. Sanchez, eds., Elsevier
North-Holland, New York, N.Y, pp. 261-268.
[9] Jackson, Peter (1998), Introduction to Expert Systems (3rd
Ed.), Addison Wesley, New-Delhi.
[10] Nikolopoulos, Chris (1997), “Expert Systems: Introduction
To First And Second Generation And Hybrid Knowledge
Based Systems”, Mercell Dekker INC.

fig: (4)Uncertainity and Rules
Often the Knowledge is imperfect which causes uncertainty. To
work in the real world, Expert systems must be able to deal with
uncertainity[16].
One simple way is to associate anumeric value with each piece
of information in the system. The numeric value represents the

[11] Panda, G. K., Mitra, A. (2008): Rough Set Application in
Social Network using Rule Induction. In: Proceedings of
NCETIT, India, pp. 59-64.
[12] Panda, G. K., Panda, B. S.(2009), “Preserving privacy in
social networks – A Rough Set Approach”, In: Proceedings
of ICONCT-09, MEPCO Shlenck Eng. College, Sivakasi,
India, pp. 315-320,

14
National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012)
Proceedings published in International Journal of Computer Applications® (IJCA)
[13] Patterson, Dan W, (2007), “Introduction to Artificial
Intelligence & Expert Systems”, Prentice-Hall India, New
Delhi.
[14] Pawlak, Z. (1982): Rough Sets. J. Inf. & Comp. Sc. II, 341356.
[15] Pawlak, Z. (1991): Rough Sets-Theoretical Aspects of
Reasoning about Data. Kluwer Acad Publ.
[16] Pawlak, Z., Skowron, A.: Rough sets- Some Extensions. J.
Information Sciences. 177(1), 28-40, 2007.

[19] Wise, B. P. and M. Henrion, (1986), “A Framework for
Comparing Uncertain Inference Systems to Probability”, in
Uncertainty in AI, L.N. Kanal and J.F. Lemmer, eds.,
Elsevier Science Publishers, New York, N.Y., pp. 69-83.
[20] Zadeh, L.A, , (1986) “Is probability Theory Sufficient for
Dealing with Uncertainty in AI: A Negative View,” in
Uncertainty in AI, Elsevier Science Publishers, New York,
NY, pp. 103-116.

[17] Rich, Elaine, and Kevin Knight, (2006), “Artificial
Intelligence”, McGraw Hills Inc.

[21] Zadeh, L.A., “Is Probability Theory Sufficient for Dealing
with Uncertainty in AI: A Negative View” in Uncertainty
in AI, L.N. Kanal and J.F. Lemmer, eds., Elsevier Science
Publishers, New York, N.Y., 1986, pp. 103-116.

[18] Sweeney, L. (2002): K-anonymity: A model for protecting
privacy. Int. J. Uncertainty, Fuzziness and Knowledgebased System, vol. 10, no. 5, pp. 557-570.

[22] Zhu, W (2007): Topological Approaches to Covering
Rough Sets., J. Information Sciences.(USA), 177, 14991508.

15

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Uncertainty classification of expert systems a rough set approach

  • 1. National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012) Proceedings published in International Journal of Computer Applications® (IJCA) Uncertainty Classification of Expert Systems - A Rough Set Approach B. S. Panda Rahuk Abhishek S. S. Gantayat Asst. Professor MITS Eng. College Rayagada, Odisha. Research Student MITS Eng. College Rayagada, Odisha. Associate Professor GMR Institute of Technology, Rajam, Andhra Pradesh. ABSTRACT In this paper, we discussed about the un certainity classifications of the Expert Systems using a Rough Set Approach. It is a Softcomputing technique using this we classified the types of Expert Systems. An expert system has a unique structure, different from traditional programs. It is divided into two parts, one fixed, independent of the expert system: the inference engine, and one variable: the knowledge base. To run an expert system, the engine reasons about the knowledge base like a human. In the 80's a third part appeared: a dialog interface to communicate with users. This ability to conduct a conversation with users was later called "conversational". Rough set theory is a technique deals with uncertainty. Key Words Expert System, Rough Sets, Lower and Upper approximations, Uncertainity 1. INTRODUCTION Expert systems are designed to solve complex problems by reasoning about knowledge, like an expert, and not by following the procedure of a developer as is the case in conventional programming. The first expert systems were created in the 1970s and then proliferated in the 1980s. [3, 4, 8, 19]. Expert systems were among the first truly successful forms of AI software. 2. EXPERT SYSTEMS Inference engine + Knowledge = Expert system Expert systems are computer programs that are derived from a branch of computer science research called Artificial Intelligence (AI) [7, 17, 22]. AI's scientific goal is to understand intelligence by building computer programs that exhibit intelligent behavior. It is concerned with the concepts and methods of symbolic inference, or reasoning, by a computer, and how the knowledge used to make those inferences will be represented inside the machine. AI programs that achieve expert-level competence in solving problems in task areas by bringing to bear a body of knowledge about specific tasks are called knowledge-based or expert systems. . The area of human intellectual endeavor to be captured in an expert system is called the task domain. Task refers to some goal-oriented, problemsolving activity. Domain refers to the area within which the task is being performed. Typical tasks are diagnosis, planning, scheduling, configuration and design. Expert system have a number of major system components and interface with individuals who interact with the system in various roles. These are illustrated below in fig. 1. fig: (1) Component of expert systems Expert systems are designed to facilitate tasks in the fields of accounting, medicine, process control, financial service, production, human resources, among others. Typically, the problem area is complex enough that a more simple traditional algorithm cannot provide a proper solution. The foundation of a successful expert system depends on a series of technical procedures and development that may be designed by technicians and related experts. As such, expert systems do not typically provide a definitive answer, but provide probabilistic recommendations [12]. Expert systems provide an advantage when dealing with uncertainty as compared to decision trees. With decision trees, all the facts must be known to arrive at an outcome. Probability theory is devoted to dealing with theories of uncertainty. There are many theories of probability – each with advantages and disadvantages. 12
  • 2. National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012) Proceedings published in International Journal of Computer Applications® (IJCA)  For any X  U and an equivalence relation R  IND( K ) , there associate two subsets:  Lower approximation of X in A is the set RX  {Y  U / R : Y  X } U which can of X with the The elements of RX are those elements of be certainly classified knowledge of R .  as elements Upper of approximation X in A is the set RX  {Y  U / R : Y  X  }  fig: (2) Development of an Expert system The elements of RX are those elements of U, which can certainly be classified as elements of X, and the elements of RX are those elements of U, which can possibly be classified as elements of X, employing knowledge of R. 3. ROUGH SETS 3.1 Definitions and Notations Rough set theory is a technique deals with uncertainty. In this section we reintroduce some basic notations of Rough set theory [1, 2, 13, 18].    U (  ) is the universe and be a finite set of objects. R is the indiscernibility relation, or equivalence relation over U .  Indiscernibility is the inability to distinguish between two or more values. A= (U, R) an ordered pair is called an approximation space.  [ x] R denotes the equivalence class or R containing an element x  U .  For any subset (  )   , the intersection of all equivalence relations in    is denoted by IND() and is called the indiscernibility relation over  . Elementary sets in A – the equivalence classes of R. Definable set in A–Any finite union of elementary sets in A. RX is the set of elements of X which can be possibly classified as elements of X employing knowledge of R . The boundary of X is, RX  RX .  The borderline region is the undecidable area of the universe. We say that X is rough with respect to R if and only if RX  RX , equivalently BN R ( X )   . X is said to be Rdefinable if and only if RX  RX , or BN R ( X )   . 3.2 Application of Rough Set We briefly highlight few applications of RST [5, 9, 14]. We again feel discussing on various applications will increase the length of paper. a. Representation of uncertain or imprecise knowledge. b. Empirical learning and knowledge acquisition from experience. c. Knowledge analysis. d. Analysis of conflicts. e. Evaluation of the quality of the available information with respect to its consistency and the presence or absence of repetitive data patterns. f. Identification and evaluation of data dependencies. g. Approximate pattern classification. h. Reasoning with uncertainty. i. Information-preserving data reduction. 3.3 Equivalence relations of Rough Sets and Classification of Expert Systems As discussed through the definitions and notations, the rough set philosophy is based upon the notion of equivalence relations. This describes the partitions made of classes of indiscernible objects with common lower and upper approximations. Incomplete information can be analyzed by means of classical approximation spaces. Such spaces consist of a universe and an indiscernibility relation, which is always equivalence relation. Knowledge granules determined by these spaces are equivalence classes. The equivalence relation divides the universe U into pair-wise disjoint subsets, called equivalence classes of objects. That is, for an object x∈.U, the equivalence class of objects containing x is given by: [x] = {x’|xE(V)x’} In expert system technology, the knowledge base is expressed with natural language rules IF ... THEN ... For examples :  "IF it is living THEN it is mortal" fig: (3) Rough Set Approach 13
  • 3. National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012) Proceedings published in International Journal of Computer Applications® (IJCA)  certainity with which the information is known. There are different ways in which these numbers can be defined, and how they are combined during the inference process [15, 21]. An important features of expdrt systems is their ability to explain themselves. Given that the system knows which rules were used during the inference process, the system can provide those rules to the user as means for explaaining the results. By looking explanations, the knowledge engineer can seen how the system is behaving, and how the rules and data are inteacting. This is very valueable diagnositc tool during development. 4. UNCERTAINITY WITH EXPERT SYSTEMS AND CLASSIFICATIONS 4.1 What is Uncertainty? 5. CONCLUSIONS "IF his age = known THEN his year of birth = date of today - his age in years"  "IF the identity of the germ is not known with certainty AND the germ is gram-positive AND the morphology of the organism is "rod" AND the germ is aerobic THEN there is a strong probability (0.8) that the germ is of type enterobacteriacae". This formulation has the advantage of speaking in everyday language which is very rare in computer science (a classic program is coded). Rules express the knowledge to be exploited by the expert system [20].     Uncertainty is essentially lack of information to formulate a decision. Uncertainty may result in making poor or bad decisions. As living creatures, we are accustomed to dealing with uncertainty – that’s how we survive. Dealing with uncertainty requires reasoning under uncertainty along with possessing a lot of common sense [6, 10, 11]. 4.2 Theories to Deal with Uncertainty       Bayesian Probability Hartley Theory Shannon Theory Dempster-Shafer Theory Markov Models Zadeh’s Fuzzy Theory 4.3 Dealing with Uncertainty     Deductive reasoning – deals with exact facts and exact conclusions Inductive reasoning – not as strong as deductive premises support the conclusion but do not guarantee it. There are a number of methods to pick the best solution in light of uncertainty. When dealing with uncertainty, we may have to settle for just a good solution. 4.4 Uncertainty and Rules    There are several sources of uncertainty in rules: Uncertainty related to individual rules Uncertainty due to conflict resolution Uncertainty due to incompatibility of rules In this paper, we have reviewed Expert Systems and rough set theory. We discussed the classifications and uncertainity in Expert Systems data, and took an initiative to combat new approach is roughsets. It is observed that some theories previously dealing with uncertainity classifications. We focused a new technique is rough set theory to clasify the Expert Systems systematically and developed a practically feasible approach. For future work, we believe that the discussed approach will become suitability for classify the expert systems. 6. REFERENCES [1] Cheeseman, P. (1986) :“Probabilistic vs. Fuzzy Reasoning”, in Uncertainity in AI, L.N. Kanal and J.F. Lemmer, eds., Elsevier Scince Publichers, New York, N.Y., pp. 85-102. [2] Chelsea, MI.(1989), Dynamic Modelling and Expert Systems in Wastewater Engineering. Lewis Publishers, Inc., pp. 167-192. [3] Finn, G.A. (1989). Applications of Expert Systems in the Process Industry. In: G.G. Patry and D. Chapman (eds.). [4] Grzymala-Busse, J.(1988): LERS - A system for learning from examples based on rough sets, J. Intelligent and Robotics Systems, 1, pp. 3-16. [5] Grzymala-Busse, J. (1988): Knowledge Acquisition Under Uncertainty – A Rough Set Approach, J. Intelligent and Robotics Systems, 1, pp. 3-16. [6] Hayes-Roth, F., D.A. Watchman, and D.B. Lenat, eds. (1983), Building Expert Systems, Addison-Wesley, Reading, Mass. [7] Hyde, Andrew Dean (Sept 28, 2010), “The future of Artificial Intelligence”. [8] Ishizuka, M., K.S.Fu, and J.T.P. Ya, (1982) “A Rule-Based Inference with Fuzzy Set for Structural Damage Assessment”, in Approximate Reasonin in Decision Analysis, M.M. Gupta and E. Sanchez, eds., Elsevier North-Holland, New York, N.Y, pp. 261-268. [9] Jackson, Peter (1998), Introduction to Expert Systems (3rd Ed.), Addison Wesley, New-Delhi. [10] Nikolopoulos, Chris (1997), “Expert Systems: Introduction To First And Second Generation And Hybrid Knowledge Based Systems”, Mercell Dekker INC. fig: (4)Uncertainity and Rules Often the Knowledge is imperfect which causes uncertainty. To work in the real world, Expert systems must be able to deal with uncertainity[16]. One simple way is to associate anumeric value with each piece of information in the system. The numeric value represents the [11] Panda, G. K., Mitra, A. (2008): Rough Set Application in Social Network using Rule Induction. In: Proceedings of NCETIT, India, pp. 59-64. [12] Panda, G. K., Panda, B. S.(2009), “Preserving privacy in social networks – A Rough Set Approach”, In: Proceedings of ICONCT-09, MEPCO Shlenck Eng. College, Sivakasi, India, pp. 315-320, 14
  • 4. National Conference on Advancement of Technologies – Information Systems & Computer Networks (ISCON – 2012) Proceedings published in International Journal of Computer Applications® (IJCA) [13] Patterson, Dan W, (2007), “Introduction to Artificial Intelligence & Expert Systems”, Prentice-Hall India, New Delhi. [14] Pawlak, Z. (1982): Rough Sets. J. Inf. & Comp. Sc. II, 341356. [15] Pawlak, Z. (1991): Rough Sets-Theoretical Aspects of Reasoning about Data. Kluwer Acad Publ. [16] Pawlak, Z., Skowron, A.: Rough sets- Some Extensions. J. Information Sciences. 177(1), 28-40, 2007. [19] Wise, B. P. and M. Henrion, (1986), “A Framework for Comparing Uncertain Inference Systems to Probability”, in Uncertainty in AI, L.N. Kanal and J.F. Lemmer, eds., Elsevier Science Publishers, New York, N.Y., pp. 69-83. [20] Zadeh, L.A, , (1986) “Is probability Theory Sufficient for Dealing with Uncertainty in AI: A Negative View,” in Uncertainty in AI, Elsevier Science Publishers, New York, NY, pp. 103-116. [17] Rich, Elaine, and Kevin Knight, (2006), “Artificial Intelligence”, McGraw Hills Inc. [21] Zadeh, L.A., “Is Probability Theory Sufficient for Dealing with Uncertainty in AI: A Negative View” in Uncertainty in AI, L.N. Kanal and J.F. Lemmer, eds., Elsevier Science Publishers, New York, N.Y., 1986, pp. 103-116. [18] Sweeney, L. (2002): K-anonymity: A model for protecting privacy. Int. J. Uncertainty, Fuzziness and Knowledgebased System, vol. 10, no. 5, pp. 557-570. [22] Zhu, W (2007): Topological Approaches to Covering Rough Sets., J. Information Sciences.(USA), 177, 14991508. 15