- Email: [email protected]
A method of dissimilarity analysis
Omega 33 (2005) 431 – 434 www.elsevier.com/locate/omega
A method of dissimilarity analysis S.S. Alam, S. Ghosh∗ Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India Accepted 29 June 2004 Available online 3 September 2004
Abstract This paper finds the essential differences between objects in terms of available attributes in a knowledge representation system (KR-system) where neither condition nor decision attributes are distinguished. We find these differences by using the concepts of “almost indiscernibility relation” and “approximate decision logic (ADL)” language. Here, an information system has been considered where attribute values are not always quantitative, rather subjective having vague or imprecise meanings. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Almost indiscernibility relation; ADL-language; -reduct; -core
1. Introduction The issue of knowledge representation is of primary importance in current research in artificial intelligence (AI). The knowledge representation system (KR-system) or an information system can be perceived as a data table, columns of which are labeled by attributes, rows are labeled by objects and each row represents a piece of information about the corresponding object. In fact, data table can be viewed as a model for logic, here called decision logic [5,7,11], which is generally used to derive conclusions from data available in the KR-system. The fundamental notion of decision logic is decision algorithm which is a set of decision rules (implications) or a sequence of instructions. The decision logic is very useful for simplification of decision algorithm by elimination of unnecessary conditions in each decision rule of a decision algorithm separately. A decision table [1] is a special and important class of information system and is a kind of prescription which specifies what decision
∗ Corresponding author.
E-mail addresses: [email protected] (S.S. Alam), [email protected] (S. Ghosh). 0305-0483/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.omega.2004.07.008
should be undertaken when some conditions are satisfied. In a decision table [2,4,10], condition and decision attributes are distinguishable. But in this paper, we consider the information system where neither condition nor decision attributes are distinguished. Therefore we are basically not interested in dependencies among attributes [6,8], but in description of some objects in terms of available attributes, in order to find the essential differences between objects of interest. This type of problem of differentiation of various objects is often of crucial importance in decision making. In this paper, we have proposed a logic which is of inductive character and is intended as a tool for data analysis, i.e., our main concern is in discovering dependencies in data and data reduction. The data table may also be looked at from a different angle, namely as a set of propositions about reality and consequently can be treated by means of logical tools. One more important remark concerning the decision algorithm seems in order. Formulas can be true or false, but the decision algorithm, which is a set of formulas, cannot have attributes of truth or falsity. Instead, consistency and inconsistency are the basic features of decision algorithms. In other words, our account, in contrast to the philosophy of deduction, stresses on rather the consistency of data than their truth (or falsity) and our main interest is not in
432
S.S. Alam, S. Ghosh / Omega 33 (2005) 431 – 434
the investigation of theorem-proving mechanisms in the introduced logic, but in analysis, in computational terms (decision algorithms, condition-action rules), of how some facts are derived from data.
Definition 3.2. If is a formula, then the set ||S defined as ||S = {x ∈ U : x S } will be called the meaning of the formula in S.
2. Preliminaries The following definitions and preliminaries are required in the sequel of our work and hence presented in brief. Definition 2.1. Let S = (U, A) be an information system and P ⊆ A. For a chosen ∈ [0, 1], we define P () a binary relation over U by x, y ∈ P ()
iff fx (a)I fy (a)
for all
a ∈ P.
We notice that P () is not exactly the indiscernibility relation IND(P ) defined by Pawlak [9]; rather it can be viewed as an “almost indiscernibility relation” over U [3]. Definition 2.2. Let S = (U, A) be an information system. Consider a level value ∈ [0, 1]. The subset of attributes P is -independent, if for every Q ⊂ P , P () = Q(). Otherwise subset P is -dependent. A subset P ⊆ Q ⊆ A is a -reduct of Q, if P is -independent subset of Q and P () = Q(). Definition 2.3. Let S = (U, A) be an information system, P ⊆ A, and a ∈ P . Consider a level value ∈ [0, 1]. An attribute a is -superfluous in P if Q() = P () where Q=P −{a}. Otherwise the attribute a is -indispensible in P . 3. Semantics of approximate decision logic (ADL) Formulas are meant to be used as description of objects of the universe. Of course, some objects may have the almost same description, thus formulas may also describe subsets of objects obeying properties expressed by these formulas of ADL. In particular, almost atomic formula (a, v) or in short av is interpreted as a collection of all objects x such that (a(x), v) ∈ P (), for a choosen value of . Definition 3.1. An object x ∈ U satisfies the formula in S = (U, A) denoted by x S (or in short x , if S is understood) if and only if the following conditions are satisfied: 1. 2. 3. 4.
x(a, v) iff (a(x), v) ∈ P (), x ∼ iff non x , x ∨ iff x or x , x ∧ iff x and x , As a corollary of the above we get
1. x → iff x ∼ ∨ , 2. x ≡ iff x → and x → .
The following are important propositions which explain the meaning of a formula. Propositions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
|(a, v)|S = {x ∈ U : (a(x), v) ∈ P ()}, | ∼ |S = −||S , | ∨ |S = ||S ∪ ||S , | ∧ |S = ||S ∩ ||S , | → |S = −||S ∪ ||S , | ≡ |S = (−||S ∩ −||S ) ∪ (||S ∩ ||S ), S iff ||S = U , S ∼ iff ||S = , S → iff ||S ⊆ ||S , S ≡ iff ||S = ||S .
4. Decision rules and decision algorithms Now we are going to define two basic concepts in the ADL-language, namely, those of decision rule and decision algorithm. Any implication → will be called a decision rule in the KR-knowledge; and are referred to as the predecessor and successor of → , respectively. If a decision rule → is true in S for a given , we say that the decision rule is -consistent in S; otherwise, the decision rule is -inconsistent in S for that value of . If → is a decision rule and and are P -basic and Q-basic formulas, respectively, then the decision rule → will be called a P Q-basic decision rule, (in short P Q-rule) or basic rule when P Q is known. Any finite set of decision rules in an ADL-language is referred to as a decision algorithm in the ADL-language. Any finite set of basic decision rules will be called a basic decision algorithm. If all decision rules in a basic decision algorithm are P Q-decision rules, then the algorithm is said to be P Q-decision algorithm or in short P Q-algorithm. The P Q-algorithm is -consistent in S for a given , if and only if all its decision rules are -consistent (true) in S for that value of ; otherwise, the algorithm is -inconsistent in S. In order to check whether or not a decision algorithm is -consistent for given , we have to check whether or not all its decision rules are true. Now the following proposition gives a much simpler method to solve this problem. Proposition. A PQ-decision rule → in a P Q-decision algorithm is -consistent (true) for a given in S if and only if for any P Q-decision rule → in P Q-decision
S.S. Alam, S. Ghosh / Omega 33 (2005) 431 – 434
algorithm, = implies = . That is, in order to check whether or not a decision rule → is true, we have to show that the predecessor of the rule discerns the decision class from the remaining decision classes of the decision algorithm in question.
Fuzzy proximity relation Sa
5. Reduction of decision rules
Fuzzy proximity relation Sb
If is P -basic formula and Q ⊆ P , then by /Q we mean the Q-basic formula obtained from the formula by removing from all elementary formulas (a, v) such that a ∈ P − Q. Let → be a P Q-rule, and a ∈ P . An attribute a is -dispensible for a given value of in the rule → if and only if
A G VG
L M Sh VS
Example. Suppose we are going to choose an aircraft and several aircrafts are available as follows: U
a
b
c
d
e
1 2 3 4 5 6
A G VG G VG G
L M Sh VS Sh L
Lo H A Av H H
VS S S VS Q S
S U M U U S
For the sake of simplicity, we consider the attributes a,b,c,d, and e, which are characterized by speed, how long it can fly, height, how quick can it turn, and stability limit, respectively. Aircrafts in the approach can be treated as objects and their characteristics as attributes; however, condition and decision attributes are not distinguished in the table. Formally it is convenient to consider all the attributes both as condition and decision attributes, and carry out all computations under this assumption. Now let us consider the fuzzy proximity relations of the domain of attributes:
A
G
VG
1.0 0.4 0.2
0.4 1.0 0.8
0.2 0.8 1.0
L
M
Sh
VS
1.0 0.8 0.5 0.1
0.8 1.0 0.6 0.4
0.5 0.6 1.0 0.8
0.1 0.4 0.8 1.0
Fuzzy proximity relation Sc
S → implies S /(P − {a}) → . Otherwise, the attribute a is -indispensible in → for that value of . If all attributes a ∈ P are -indispensible in → , then → will be called -independent. The subset of attributes R ⊆ P will be called a -reduct of P Q-rule → if /R → is -independent and S → implies S /R → . If R is the -reduct of the P Q-rule → , then /R → is said to be reduced. The set of all -indispensible attributes in → will be called -core of → , and will be denoted by CORE( → ).
433
Lo H Av
Lo
H
Av
1.0 0.4 0.5
0.4 1.0 0.8
0.5 0.8 1.0
VS
S
Q
1.0 0.8 0.3
0.8 1.0 0.5
0.3 0.5 1.0
N
Ne
P
1.0 0.5 0.0
0.5 1.0 0.6
0.0 0.6 1.0
Fuzzy proximity relation Sd
VS S Q
Fuzzy proximity relation Se
N Ne P
As all the rows in the table are distinct, that means the table is -consistent (or that the corresponding decision rules are true), which means that each aircraft has unique characterization in terms of the given features. Now in order to eliminate the unnecessary features from our consideration, we have to find whether the features of these aircrafts are -dispensible or not. For this we will drop the attributes one by one and check that the table is -consistent or not. Dropping the attribute d we get U
a
b
c
e
1 2 3 4 5 6
A G VG G VG G
L M Sh VS Sh L
Lo H Av Av H H
S U M U U S
434
S.S. Alam, S. Ghosh / Omega 33 (2005) 431 – 434
which is an -inconsistent table due to rules aG bV S cAv eU → aG bV S cAv dV S eU (rule 4), aV G bSh cH eU → aV G bSh cH dQ eU (rule 5). Similarly, dropping the attribute b, e we get the inconsistent tables but dropping the attributes a and c we get the -consistent tables. Thus the core attribute is the set {b, d, e} and there are two reducts in the table {a, b, d, e} and {b, c, d, e}. Now we have to compute decision algorithm with a smaller number of decision rules which will give a description of essential differences between the aircrafts. Computing core values of attributes we obtain the following table (for the reduct {a, b, d, e}) U
a
b
d
e
1 2 3 4 5 6
A — — — — G
— M Sh VS — L
— — — VS Q —
— U M U — S.
It is easy to see that the core values are reducts of the decision rules, which can be presented in the form aA → 1 b M eU → 2 bSh eU → 3 b V S dV S eU → 4 dQ → 5 a G bL eS → 6 Thus each aircraft is uniquely characterized by a proper decision rule and this characterization can serve as a basis for aircraft evaluation. Similar characteristic can be obtained from another reduct. The method presented is obviously of a general nature and can be applied to various problems, such as marketing analysis, performance evaluation of a student in an examination, pattern recognition, etc., and lies in fact within the scope of the group decision making area.
6. Conclusion Here we replaced the concept of truth by the concept of almost indiscernibility. The relationship between the truth
and almost indiscernibility is of primary importance from the algorithmic point of view, because it allows us to replace checking whether some implications (decision rules) are true or not—by the investigation of almost indiscernibility of some elements of the universe. So, what we have obtained finally is the minimal set of decision rules which is equivalent to the original table, as far as the decisions are concerned. This means that in the simplified table only the minimal set of conditions, necessary to make decisions specified in the table, are included. That is, as a result, we obtained much simpler computation algorithm for knowledge reduction. The rough set approach seems to be a useful tool to trace the dissimilarities between objects, states, opinions, processes, etc.
References [1] Pawlak Z. Rough sets, theoretical aspects of reasoning about data. Dordrecht, Kluwer: Academic Publishers; 1991. [2] Boryczka M. Optimization of decision tables using rough sets. Bulletin of the Polish Academy of Science and Technology 1989;37:321–32. [3] De SK, Biswas R, Roy AR. Finding dependency of attributes in an information system. The Journal of Fuzzy Mathematics 1999;7:335–43. [4] Kowalczyk A, Szymanski J. Rough simplification of decision tables. Bulletin of the Polish Academy of Science and Technology 1989;37:359–74. [5] Krynicki M. A note on rough concept logic. Fundamenta Informaticae 1990;13:227–35. [6] Mrozek A. Rough sets and dependency analysis among attributes in computer implementation of expert inference models. International Journal of Man-Machine Studies 1989;30:457–73. [7] Orlowska E. Logic aspects of learning concepts. International Journal of Approximating Reasoning 1988;2:349–64. [8] Orlowska E. Semantics analysis of inductive reasoning. Theoretical Computer Science 1986;43:81–9. [9] Pawlak Z. Rough sets. International Journal of Information and Computer Sciences 1982;11:341–56. [10] Pawlak, Z. Rough sets and decision tables. Lecture notes in computer sciences, vol. 208. Berlin: Springer; 1985. p. 186–96. [11] Pawlak Z. Rough logic. Bulletin of the Polish Academy of Science and Technology 1987;35:253–8.
A method of dissimilarity analysis S.S. Alam, S. Ghosh∗ Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur-721302, India Accepted 29 June 2004 Available online 3 September 2004
Abstract This paper finds the essential differences between objects in terms of available attributes in a knowledge representation system (KR-system) where neither condition nor decision attributes are distinguished. We find these differences by using the concepts of “almost indiscernibility relation” and “approximate decision logic (ADL)” language. Here, an information system has been considered where attribute values are not always quantitative, rather subjective having vague or imprecise meanings. 䉷 2004 Elsevier Ltd. All rights reserved. Keywords: Almost indiscernibility relation; ADL-language; -reduct; -core
1. Introduction The issue of knowledge representation is of primary importance in current research in artificial intelligence (AI). The knowledge representation system (KR-system) or an information system can be perceived as a data table, columns of which are labeled by attributes, rows are labeled by objects and each row represents a piece of information about the corresponding object. In fact, data table can be viewed as a model for logic, here called decision logic [5,7,11], which is generally used to derive conclusions from data available in the KR-system. The fundamental notion of decision logic is decision algorithm which is a set of decision rules (implications) or a sequence of instructions. The decision logic is very useful for simplification of decision algorithm by elimination of unnecessary conditions in each decision rule of a decision algorithm separately. A decision table [1] is a special and important class of information system and is a kind of prescription which specifies what decision
∗ Corresponding author.
E-mail addresses: [email protected] (S.S. Alam), [email protected] (S. Ghosh). 0305-0483/$ - see front matter 䉷 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.omega.2004.07.008
should be undertaken when some conditions are satisfied. In a decision table [2,4,10], condition and decision attributes are distinguishable. But in this paper, we consider the information system where neither condition nor decision attributes are distinguished. Therefore we are basically not interested in dependencies among attributes [6,8], but in description of some objects in terms of available attributes, in order to find the essential differences between objects of interest. This type of problem of differentiation of various objects is often of crucial importance in decision making. In this paper, we have proposed a logic which is of inductive character and is intended as a tool for data analysis, i.e., our main concern is in discovering dependencies in data and data reduction. The data table may also be looked at from a different angle, namely as a set of propositions about reality and consequently can be treated by means of logical tools. One more important remark concerning the decision algorithm seems in order. Formulas can be true or false, but the decision algorithm, which is a set of formulas, cannot have attributes of truth or falsity. Instead, consistency and inconsistency are the basic features of decision algorithms. In other words, our account, in contrast to the philosophy of deduction, stresses on rather the consistency of data than their truth (or falsity) and our main interest is not in
432
S.S. Alam, S. Ghosh / Omega 33 (2005) 431 – 434
the investigation of theorem-proving mechanisms in the introduced logic, but in analysis, in computational terms (decision algorithms, condition-action rules), of how some facts are derived from data.
Definition 3.2. If is a formula, then the set ||S defined as ||S = {x ∈ U : x S } will be called the meaning of the formula in S.
2. Preliminaries The following definitions and preliminaries are required in the sequel of our work and hence presented in brief. Definition 2.1. Let S = (U, A) be an information system and P ⊆ A. For a chosen ∈ [0, 1], we define P () a binary relation over U by x, y ∈ P ()
iff fx (a)I fy (a)
for all
a ∈ P.
We notice that P () is not exactly the indiscernibility relation IND(P ) defined by Pawlak [9]; rather it can be viewed as an “almost indiscernibility relation” over U [3]. Definition 2.2. Let S = (U, A) be an information system. Consider a level value ∈ [0, 1]. The subset of attributes P is -independent, if for every Q ⊂ P , P () = Q(). Otherwise subset P is -dependent. A subset P ⊆ Q ⊆ A is a -reduct of Q, if P is -independent subset of Q and P () = Q(). Definition 2.3. Let S = (U, A) be an information system, P ⊆ A, and a ∈ P . Consider a level value ∈ [0, 1]. An attribute a is -superfluous in P if Q() = P () where Q=P −{a}. Otherwise the attribute a is -indispensible in P . 3. Semantics of approximate decision logic (ADL) Formulas are meant to be used as description of objects of the universe. Of course, some objects may have the almost same description, thus formulas may also describe subsets of objects obeying properties expressed by these formulas of ADL. In particular, almost atomic formula (a, v) or in short av is interpreted as a collection of all objects x such that (a(x), v) ∈ P (), for a choosen value of . Definition 3.1. An object x ∈ U satisfies the formula in S = (U, A) denoted by x S (or in short x , if S is understood) if and only if the following conditions are satisfied: 1. 2. 3. 4.
x(a, v) iff (a(x), v) ∈ P (), x ∼ iff non x , x ∨ iff x or x , x ∧ iff x and x , As a corollary of the above we get
1. x → iff x ∼ ∨ , 2. x ≡ iff x → and x → .
The following are important propositions which explain the meaning of a formula. Propositions. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
|(a, v)|S = {x ∈ U : (a(x), v) ∈ P ()}, | ∼ |S = −||S , | ∨ |S = ||S ∪ ||S , | ∧ |S = ||S ∩ ||S , | → |S = −||S ∪ ||S , | ≡ |S = (−||S ∩ −||S ) ∪ (||S ∩ ||S ), S iff ||S = U , S ∼ iff ||S = , S → iff ||S ⊆ ||S , S ≡ iff ||S = ||S .
4. Decision rules and decision algorithms Now we are going to define two basic concepts in the ADL-language, namely, those of decision rule and decision algorithm. Any implication → will be called a decision rule in the KR-knowledge; and are referred to as the predecessor and successor of → , respectively. If a decision rule → is true in S for a given , we say that the decision rule is -consistent in S; otherwise, the decision rule is -inconsistent in S for that value of . If → is a decision rule and and are P -basic and Q-basic formulas, respectively, then the decision rule → will be called a P Q-basic decision rule, (in short P Q-rule) or basic rule when P Q is known. Any finite set of decision rules in an ADL-language is referred to as a decision algorithm in the ADL-language. Any finite set of basic decision rules will be called a basic decision algorithm. If all decision rules in a basic decision algorithm are P Q-decision rules, then the algorithm is said to be P Q-decision algorithm or in short P Q-algorithm. The P Q-algorithm is -consistent in S for a given , if and only if all its decision rules are -consistent (true) in S for that value of ; otherwise, the algorithm is -inconsistent in S. In order to check whether or not a decision algorithm is -consistent for given , we have to check whether or not all its decision rules are true. Now the following proposition gives a much simpler method to solve this problem. Proposition. A PQ-decision rule → in a P Q-decision algorithm is -consistent (true) for a given in S if and only if for any P Q-decision rule → in P Q-decision
S.S. Alam, S. Ghosh / Omega 33 (2005) 431 – 434
algorithm, = implies = . That is, in order to check whether or not a decision rule → is true, we have to show that the predecessor of the rule discerns the decision class from the remaining decision classes of the decision algorithm in question.
Fuzzy proximity relation Sa
5. Reduction of decision rules
Fuzzy proximity relation Sb
If is P -basic formula and Q ⊆ P , then by /Q we mean the Q-basic formula obtained from the formula by removing from all elementary formulas (a, v) such that a ∈ P − Q. Let → be a P Q-rule, and a ∈ P . An attribute a is -dispensible for a given value of in the rule → if and only if
A G VG
L M Sh VS
Example. Suppose we are going to choose an aircraft and several aircrafts are available as follows: U
a
b
c
d
e
1 2 3 4 5 6
A G VG G VG G
L M Sh VS Sh L
Lo H A Av H H
VS S S VS Q S
S U M U U S
For the sake of simplicity, we consider the attributes a,b,c,d, and e, which are characterized by speed, how long it can fly, height, how quick can it turn, and stability limit, respectively. Aircrafts in the approach can be treated as objects and their characteristics as attributes; however, condition and decision attributes are not distinguished in the table. Formally it is convenient to consider all the attributes both as condition and decision attributes, and carry out all computations under this assumption. Now let us consider the fuzzy proximity relations of the domain of attributes:
A
G
VG
1.0 0.4 0.2
0.4 1.0 0.8
0.2 0.8 1.0
L
M
Sh
VS
1.0 0.8 0.5 0.1
0.8 1.0 0.6 0.4
0.5 0.6 1.0 0.8
0.1 0.4 0.8 1.0
Fuzzy proximity relation Sc
S → implies S /(P − {a}) → . Otherwise, the attribute a is -indispensible in → for that value of . If all attributes a ∈ P are -indispensible in → , then → will be called -independent. The subset of attributes R ⊆ P will be called a -reduct of P Q-rule → if /R → is -independent and S → implies S /R → . If R is the -reduct of the P Q-rule → , then /R → is said to be reduced. The set of all -indispensible attributes in → will be called -core of → , and will be denoted by CORE( → ).
433
Lo H Av
Lo
H
Av
1.0 0.4 0.5
0.4 1.0 0.8
0.5 0.8 1.0
VS
S
Q
1.0 0.8 0.3
0.8 1.0 0.5
0.3 0.5 1.0
N
Ne
P
1.0 0.5 0.0
0.5 1.0 0.6
0.0 0.6 1.0
Fuzzy proximity relation Sd
VS S Q
Fuzzy proximity relation Se
N Ne P
As all the rows in the table are distinct, that means the table is -consistent (or that the corresponding decision rules are true), which means that each aircraft has unique characterization in terms of the given features. Now in order to eliminate the unnecessary features from our consideration, we have to find whether the features of these aircrafts are -dispensible or not. For this we will drop the attributes one by one and check that the table is -consistent or not. Dropping the attribute d we get U
a
b
c
e
1 2 3 4 5 6
A G VG G VG G
L M Sh VS Sh L
Lo H Av Av H H
S U M U U S
434
S.S. Alam, S. Ghosh / Omega 33 (2005) 431 – 434
which is an -inconsistent table due to rules aG bV S cAv eU → aG bV S cAv dV S eU (rule 4), aV G bSh cH eU → aV G bSh cH dQ eU (rule 5). Similarly, dropping the attribute b, e we get the inconsistent tables but dropping the attributes a and c we get the -consistent tables. Thus the core attribute is the set {b, d, e} and there are two reducts in the table {a, b, d, e} and {b, c, d, e}. Now we have to compute decision algorithm with a smaller number of decision rules which will give a description of essential differences between the aircrafts. Computing core values of attributes we obtain the following table (for the reduct {a, b, d, e}) U
a
b
d
e
1 2 3 4 5 6
A — — — — G
— M Sh VS — L
— — — VS Q —
— U M U — S.
It is easy to see that the core values are reducts of the decision rules, which can be presented in the form aA → 1 b M eU → 2 bSh eU → 3 b V S dV S eU → 4 dQ → 5 a G bL eS → 6 Thus each aircraft is uniquely characterized by a proper decision rule and this characterization can serve as a basis for aircraft evaluation. Similar characteristic can be obtained from another reduct. The method presented is obviously of a general nature and can be applied to various problems, such as marketing analysis, performance evaluation of a student in an examination, pattern recognition, etc., and lies in fact within the scope of the group decision making area.
6. Conclusion Here we replaced the concept of truth by the concept of almost indiscernibility. The relationship between the truth
and almost indiscernibility is of primary importance from the algorithmic point of view, because it allows us to replace checking whether some implications (decision rules) are true or not—by the investigation of almost indiscernibility of some elements of the universe. So, what we have obtained finally is the minimal set of decision rules which is equivalent to the original table, as far as the decisions are concerned. This means that in the simplified table only the minimal set of conditions, necessary to make decisions specified in the table, are included. That is, as a result, we obtained much simpler computation algorithm for knowledge reduction. The rough set approach seems to be a useful tool to trace the dissimilarities between objects, states, opinions, processes, etc.
References [1] Pawlak Z. Rough sets, theoretical aspects of reasoning about data. Dordrecht, Kluwer: Academic Publishers; 1991. [2] Boryczka M. Optimization of decision tables using rough sets. Bulletin of the Polish Academy of Science and Technology 1989;37:321–32. [3] De SK, Biswas R, Roy AR. Finding dependency of attributes in an information system. The Journal of Fuzzy Mathematics 1999;7:335–43. [4] Kowalczyk A, Szymanski J. Rough simplification of decision tables. Bulletin of the Polish Academy of Science and Technology 1989;37:359–74. [5] Krynicki M. A note on rough concept logic. Fundamenta Informaticae 1990;13:227–35. [6] Mrozek A. Rough sets and dependency analysis among attributes in computer implementation of expert inference models. International Journal of Man-Machine Studies 1989;30:457–73. [7] Orlowska E. Logic aspects of learning concepts. International Journal of Approximating Reasoning 1988;2:349–64. [8] Orlowska E. Semantics analysis of inductive reasoning. Theoretical Computer Science 1986;43:81–9. [9] Pawlak Z. Rough sets. International Journal of Information and Computer Sciences 1982;11:341–56. [10] Pawlak, Z. Rough sets and decision tables. Lecture notes in computer sciences, vol. 208. Berlin: Springer; 1985. p. 186–96. [11] Pawlak Z. Rough logic. Bulletin of the Polish Academy of Science and Technology 1987;35:253–8.