The minimization of axiom sets characterizing generalized approximation operators

Information Sciences 176 (2006) 887–899 www.elsevier.com/locate/ins

The minimization of axiom sets characterizing generalized approximation operators Xiao-Ping Yang

a,*

, Tong-Jun Li

a,b

a

b

Information College, Zhejiang Ocean University, Zhoushan, Zhejiang 316004, PR China Institute for Information and System Sciences, Faculty of Science, Xi’an Jiaotong University, Xi’an, Shaanxi 710049, PR China Received 31 August 2004; accepted 18 January 2005

Abstract In the axiomatic approach of rough set theory, rough approximation operators are characterized by a set of axioms that guarantees the existence of certain types of binary relations reproducing the operators. Thus axiomatic characterization of rough approximation operators is an important aspect in the study of rough set theory. In this paper, the independence of axioms of generalized crisp approximation operators is investigated, and their minimal sets of axioms are presented.
2005 Elsevier Inc. All rights reserved. Keywords: Approximation operators; Axioms; Rough sets

*

Corresponding author. Tel.: +86 580 8180230. E-mail addresses: [email protected] (X.-P. Yang), [email protected] (T.-J. Li).

0020-0255/$ - see front matter
2005 Elsevier Inc. All rights reserved. doi:10.1016/j.ins.2005.01.012

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1. Introduction Rough set theory is a generalization of the classical set theory for modelling systems with uncertain or incomplete information, and it has recently aroused much interest in both theory and applications. For example, it may be used to unravel knowledge hidden in information systems and express the knowledge in the form of decision rules. There are two approaches to the development of rough set theory: constructive and axiomatic. In the constructive approach, primitive notions include binary relations on the universe, partitions of the universe, neighborhood systems and Boolean algebras, and based on these notions the lower and upper approximation operators are constructed [6,8,9,14–16,18,24,26,28–33,35]. On the other hand, the axiomatic approach takes the lower and upper approximation operators as primitive notions, and a set of axioms is used to characterize the approximation operators produced using the constructive approach. In terms of axiomatic approach, rough set theory may be interpreted as an extension of the classical set theory with two additional unary operators. The lower and upper approximation operators are related to the necessity (box) and possibility (diamond) operators in modal logic, and the interior and closure operators in topological space [2,3,7,9,10,19,23,29,30,32]. Under this approach, a set of axioms is used to characterize approximation operators that are the same as the ones produced by using constructive approach. Zakowski [34] studied a set of axioms on approximation operators, and Comer [4,5] investigated axioms on approximation operators in relation to cylindric algebras within the context of Pawlak information systems [13]. Lin and Liu [9] proposed six axioms on a pair of abstract operators on the power set of universe in the framework of topological space, under which there exists an equivalence relation reproducing the lower and upper approximation operators by the constructive approach. Similar result was reported by Wiweger [23]. However, these studies are restricted to Pawlak rough set algebra defined by equivalence relations. Wybraniec–Skardowska [28] examined many axioms on various classes of approximation operators and proposed several constructive methods to generate them. Mordeson [11] investigated the axiomatic characterization of approximation operators defined by covers, and Thiele [19] explored the axiomatic characterization within modal logic. The most important axiomatic studies for crisp rough sets are done by Yao et al. [29,30,32,33], where various crisp rough set algebras are characterized using different sets of axioms. The research of axiomatic approach has also been extended to approximation operators in fuzzy environment [1,10,12,17,20– 22,25,27]. However, the abovementioned studies have not solved the important problem of the independence and minimization of the axiom set for approximation operators. This paper attempts to solve this problem for generalized crisp

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rough approximation operators. Independence of axioms for rough approximation operators is investigated, and minimal axiom sets corresponding to various generalized approximation operators are presented. The paper is organized as follows: In Section 2, we are to give some basic notions related to the rough approximation operators, and review the existed axiom sets of the generalized approximation operators in Section 3. In Section 4, we are to summarize the research results in the form of theorems, and prove the theorems in Section 5. Finally, in Section 6, we are to conclude the paper with a summary and an outlook for further research.

2. Rough approximation operators Let U be a finite and nonempty set called the universe of discourse. The class of all subsets of U will be denoted by PðU Þ. Let R be a binary relation on U, that is, R
U · U. "x 2 U, denote Rs(x) = {y 2 U:(x, y) 2 R}, Rs(x) is called the successor neighborhood of x with respect to R. The relation R is referred to as serial if Rs(x) 5 ; for all x 2 U; R is referred to as reflexive if x 2 Rs(x) for all x 2 U; R is referred to as symmetric if "(x, y) 2 U · U, y 2 Rs(x) implies x 2 Rs(y); R is referred to as transitive if "x, y, z 2 U, y 2 Rs(x) and z 2 Rs(y) imply z 2 Rs(x); R is referred to as Euclidean if "x, y, z 2 U, y 2 Rs(x) and z 2 Rs(x) imply z 2 Rs(y). If R is a binary relation on U, the pair (U, R) is called a generalized approximation space. For any set X
U, a pair of lower and upper approximations, R(X) and RðX Þ, are defined, respectively, as follows: RðX Þ ¼ fx 2 U : Rs ðxÞ
X g;

RðX Þ ¼ fx 2 U : Rs ðxÞ \ X 6¼ ;g:

The pair ðRðX Þ; RðX ÞÞ is referred to as a generalized crisp rough set, and R, R : PðU Þ ! PðU Þ are referred to as lower and upper generalized crisp approximation operators, respectively. Theorem 2.1 follows immediately from the definition [15,30,35]: Theorem 2.1. For any relation R on U, its lower and upper approximation operators satisfy the following properties: for all A; B 2 PðU Þ, ðL1Þ RðAÞ ¼ Rð AÞ;

ðU1Þ RðAÞ ¼ Rð AÞ;

ðL2Þ RðU Þ ¼ U ;

ðU2Þ Rð;Þ ¼ ;;

ðL3Þ RðA \ BÞ ¼ RðAÞ \ RðBÞ;

ðU3Þ RðA [ BÞ ¼ RðAÞ [ RðBÞ;

ðL4Þ A
B ) RðAÞ
RðBÞ; ðU4Þ A
B ) RðAÞ
RðBÞ; ðL5Þ RðA [ BÞ  RðAÞ [ RðBÞ;

ðU5Þ RðA \ BÞ
RðAÞ \ RðBÞ;

where A is the complement of A in U.

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Properties (L1) and (U1) show that R and R are dual approximation operators. Properties with the same number may be considered as dual properties. With respect to certain special types, e.g., serial, reflexive, symmetric, transitive and Euclidean binary relations on the universe of discourse U, the approximation operators have additional properties [30,35]. Theorem 2.2. Let R be an arbitrary crisp binary relation on U, and R and R are the lower and upper generalized crisp approximation operators, respectively. Then ð1Þ R is serial

ð2Þ R is reflexive ð3Þ R is symmetric ð4Þ R is transitive ð5Þ R is Euclidean

()

ðL0Þ

Rð;Þ ¼ ;;

()

ðU0Þ

RðU Þ ¼ U ;

()

ðLU0Þ RðAÞ
RðAÞ 8A 2 PðU Þ:

()

ðL6Þ

RðAÞ
A 8A 2 PðU Þ;

()

ðU6Þ

A
RðAÞ 8A 2 PðU Þ:

()

ðL7Þ

A
RðRðAÞÞ 8A 2 PðU Þ;

()

ðU7Þ

RðRðAÞÞ
A 8A 2 PðU Þ:

()

ðL8Þ

RðAÞ
RðRðAÞÞ 8A 2 PðU Þ;

()

ðU8Þ

RðRðAÞÞ
RðAÞ 8A 2 PðU Þ:

()

ðL9Þ

RðAÞ
RðRðAÞÞ 8A 2 PðU Þ;

()

ðU9Þ

RðRðAÞÞ
RðAÞ 8A 2 PðU Þ:

If R is an equivalence relation on U, and the pair (U, R) is the Pawlak approximation space, more interesting properties of lower and upper approximation operators can be derived [15]. 3. Axiom sets of the generalized approximation operators In an axiomatic approach, the primitive notion is a system ðPðU Þ; \; [; ; L; H Þ, where ðPðU Þ; \; [; Þ is the set algebra, and L; H : PðU Þ ! PðU Þ are unary operators on the power set PðU Þ. We call L and H approximation operators, to indicate their intended physical interpretation. They are defined by axioms without direct reference to binary relations. Definition 3.1. Let L; H : PðU Þ ! PðU Þ be two unary operators on the power set PðU Þ. They are dual operators if ðL1 Þ LX ¼ H  X

8X
U

or

ðH1 Þ HX ¼ L  X

8X
U :

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By the duality of L and H, it is sufficient to introduce one operator and to define the other by using (L1) or (H1). For example, one may define the operator L and regard H as an abbreviation of  L . The use of both operators is for the sake of simplicity and clarity. We are interested in the following conditions (axioms) on L and H: ðL2 Þ LU ¼ U :

ðH2 Þ H ; ¼ ;:

ðL3 Þ LðX \ Y Þ ¼ LX \ LY 8X ; Y
U :

ðH3 Þ H ðX [ Y Þ ¼ HX [ HY 8X ; Y
U :

ðLH0 Þ LX
HX 8X
U :

ðL0 Þ L; ¼ ;:

ðL6 Þ LX
X 8X
U:

ðH6 Þ X
HX 8X
U :

ðL7 Þ X
LHX 8X
U :

ðH7 Þ HLX
X 8X
U :

ðL8 Þ LX
LLX 8X
U :

ðH8 Þ HHX
HX 8X
U :

ðL9 Þ HX
LHX 8X
U :

ðH9 Þ HLX
LX 8X
U :

ðH 0 Þ HU ¼ U :

Definition 3.2. Let L and H be dual operators, if there exists a serial (reflexive, symmetric, transitive, Euclidean, respectively) relation R on U such that LX ¼ RX ;

HX ¼ RX

8X
U ;

we call L and H serial (reflexive, symmetric, transitive, Euclidean, respectively) approximation operators, the corresponding algebra ðPðU Þ; \; [; ; L; H Þ is called a serial (reflexive, symmetric, transitive, Euclidean, respectively) rough set algebra. In this paper, we only consider dual operators. For completeness, we summarize the main results of axiomatic characterizations of the rough approximation operators corresponding to different binary relations in Theorems 3.3–3.8 [29,35]. Theorem 3.3. Let L; H : PðU Þ ! PðU Þ be dual operators. Then there exists a binary relation R on U such that "X
U, LX ¼ RX ; HX ¼ RX if and only if L satisfies axioms L2 and L3: ðL2 Þ LU ¼ U :

ðL3 Þ

LðX \ Y Þ ¼ LX \ LY

8X ; Y
U

or equivalently H satisfies axioms H2 and H3: ðH2 Þ

H ; ¼ ;:

ðH3 Þ H ðX [ Y Þ ¼ HX [ HY

8X ; Y
U :

Theorem 3.4. Let L; H : PðU Þ ! PðU Þ be dual operators. Then there exists a serial relation R on U such that "X
U, LX ¼ RX ; HX ¼ RX if and only if L satisfies axioms L2 and L3, or equivalently, H satisfies axioms H2 and H3, and meanwhile, L and H satisfy one of the following equivalent axioms: ðLH0 Þ LX
HX

8X
U :

ðL0 Þ L; ¼ ;:

ðH0 Þ

HU ¼ U :

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Theorem 3.5. Let L; H : PðU Þ ! PðU Þ be dual operators. Then there exists a reflexive relation R on U such that "X
U, LX ¼ RX ; HX ¼ RX if and only if L satisfies axioms L2, L3 and L6, or equivalently, H satisfies axioms H2, H3 and H6. Theorem 3.6. Let L; H : PðU Þ ! PðU Þ be dual operators. Then there exists a symmetric relation R on U such that "X
U, LX ¼ RX ; HX ¼ RX if and only if L satisfies axioms L2, L3 and L7, or equivalently, H satisfies axioms H2, H3 and H 7. Theorem 3.7. Let L; H : PðU Þ ! PðU Þ be dual operators. Then there exists a transitive relation R on U such that "X
U, LX ¼ RX ; HX ¼ RX if and only if L satisfies axioms L2, L3 and L8, or equivalently, H satisfies axioms H2, H3 and H8. Theorem 3.8. Let L; H : PðU Þ ! PðU Þ be dual operators. Then there exists an Euclidean relation R on U such that "X
U, LX ¼ RX ; HX ¼ RX if and only if L satisfies axioms L2, L3, and L9, or equivalently, H satisfies axioms H2, H3 and H 9.

4. Minimal axiom sets of operators We now discuss independence of axiom sets characterizing serial, reflexive, symmetric, transitive, and Euclidean approximation operators. The results are presented in the form of theorems as follows: Theorem 4.1. Axioms L2 and L3 are independent. Theorem 4.2. Axioms L2, L3 and L0 in Theorem 3.4 are independent, i.e., any two of the axioms cannot derive the third one. Theorem 4.3. Axioms L2, L3 and L6 in Theorem 3.5 are independent, i.e., any two of the axioms cannot derive the third one. Theorem 4.4. Axioms L2, L3 and L7 in Theorem 3.6 are dependent. L3, L7 can derive L2. {L3, L7} is the minimal axiom set to characterize symmetric approximation operators. Theorem 4.5. Axioms L2, L3 and L8 in Theorem 3.7 are independent. Theorem 4.6. Axioms L2, L3 and L9 in Theorem 3.8 are dependent. L3, L9 can derive L2. {L3, L9} is the minimal axiom set to characterize Euclidean approximation operators.

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5. Proof of theorems By the dual properties of L and H, we are only to prove the theorems for L. Proof of Theorem 4.1. Firstly, we show that L2 ; L3. Let U = {a, b, c} and Y0 = {a, c}. X and LX for all X
U are respectively listed in columns 1 and 2 of Table 1. HX, LHX and LLX calculated by the duality of L and H are listed in columns 3–5, L(X \ Y0) and LX \ LY0 in column 6. Let X0 = {a, b}, then L(X0 \ Y0) = {a}, LX0 \ LY0 = ; (see the items underlined in Table 1). Hence, L(X0 \ Y0) 5 LX0 \ LY0. It is easy to see that L2 is satisfied. This implies L2 ; L3. Secondly, we show that L3 ; L2. Let U = {a, b, c}. LX for all X
U is listed in column 2 of Table 2. HX calculated by the duality of L and H is listed in colunm 3. L(X \ Y) and LX \ LY for all X, Y
U are listed in columns 4–11. It is easy to see that L(X \ Y) = LX \ LY for all X,Y
U, i.e., L3 is satisfied, and LU = {a, b} 5 U (see the item underlined in Table 2), i.e., L2 is not satisfied. This implies L3 ; L2. Thus, we have proved that L2 and L3 are independent. h Remark. This theorem implies that {L2, L3}, or equivalently, {H2, H3}, is the minimal axiom set to characterize the generalized rough approximation operators generated from an arbitrary binary relation. Proof of Theorem 4.2. Firstly, we show that L2, L3 ; L0. Let U = {a, b}. X and LX for all X
U are respectively listed in columns 1 and 2 of Table 3. HX, LHX, LLX calculated by the duality of L and H are listed in columns 3–5, and L(X \ Y), LX \ LY in columns 6–9. It is easy to see that L2 and L3 are satisfied, and L0 is not satisfied. This implies L2, L3 ; L0. Secondly, we show that L2, L0 ; L3. Let U = {a, b, c}. X and LX for all X
U are respectively listed in columns 1 and 2 of Table 1. HX calculated by Table 1 Example to check that L2 ; L3, L2, L0 ; L3, L2, L6 ; L3, L7 ; L3, L2, L8 ; L3, L9 ; L3 X

LX

HX

LHX

LLX

L(X \ Y0), LX \ LY0 Y0 = {a, c}

U {a, b} {a, c} {b, c} {a} {b} {c} ;

U {a, b} {c} {b, c} {a} ; {c} ;

U {a, b} U {b, c} {a} {a, b} {c} ;

U {a, b} U {b, c} {a} {a, b} {c} ;

U {a, b} {c} {b, c} {a} ; {c} ;

{c}, {c} fag; ; {c}, {c} {c}, {c} {a}, ; ;, ; {c}, {c} ;, ;

894

Table 2 Example to check that L3 ; L2, L3, L0 ; L2, L3, L6 ; L2 X

HX

fa; bg ; ; ; ; ; ; ;

U U U U U U U {c}

L(X \ Y), LX \ LY U

Y = {a, b}

Y = {a, c}

Y = {b, c}

Y = {a}

Y = {b}

Y = {c}

Y=;

{a, b}, {a, b} ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

Table 3 Example to check that L2, L3 ; L0, L2, L3 ; L6, L3 ; L7, L3 ; L9 X

U fag {b} ;

LX

U fbg U fbg

HX

fag ; {a} ;

LHX

fbg fbg {b} {b}

LLX

U U U U

L(X \ Y), LX \ LY Y=U

Y = {a}

Y = {b}

Y=;

U, U {b}, {b} U, U {b}, {b}

{b}, {b}, {b}, {b},

U, U {b}, {b} U, U {b}, {b}

{b}, {b}, {b}, {b},

{b} {b} {b} {b}

{b} {b} {b} {b}

X.-P. Yang, T.-J. Li / Information Sciences 176 (2006) 887–899

U {a, b} {a, c} {b, c} {a} {b} {c} ;

LX

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895

the duality of L and H is listed in column 3. It is easy to see that LU = U, L; = ; for all X
U, i.e., L2 and L0 are satisfied, but L3 is not satisfied. This implies L2, L0 ; L3. Finally, we show that L3, L0 ; L2. Let U = {a, b, c}. LX for all X
U is listed in column 2 of Table 2. HX calculated by the duality of L and H is listed in column 3. It is easy to see that L3 and L0 are satisfied, and L2 is not satisfied. This implies L3, L0 ; L2. Thus, we have proved that L2, L3, and L0 are independent. h Remark. This theorem implies that {L2, L3, L0}, or equivalently, {H2, H3, H0}, is the minimal axiom set to characterize the serial approximation operators. Proof of Theorem 4.3. Firstly, we show that L2, L3 ; L6. Let U = {a, b}. LX for all X
U is listed in column 2 of Table 3. HX calculated by the duality of L and H is listed in column 3. It is easy to see that L2 and L3 are satisfied, and L6 is not satisfied. This implies L2, L3 ; L6. Secondly, we show that L3, L6 ; L2. Let U = {a, b, c}. LX for all X
U is listed in column 2 of Table 2. HX calculated by the duality of L and H is listed in column 3. It is easy to see that LU = {a, b} 5 U, i.e., L2 is not satisfied. This implies L3, L6 ; L2. Finally, we show that L2, L6 ; L3. Let U = {a, b, c}. LX for all X
U is listed in column 2 of Table 1. HX calculated by the duality of L and H is listed in column 3. It is easy to see that L2, L6 are satisfied, but L3 is not satisfied. This implies L2, L6 ; L3. Thus, we have proved that L2, L3, L6 are independent. h Remark. This theorem implies that {L2, L3, L6}, or equivalently, {H2, H3, H6}, is the minimal axiom set to characterize the reflexive approximation operators. Proof of Theorem 4.4. Firstly, we show that L3, L7 ) L2. In fact, if L7 is satisfied, i.e., X
LHX for all X
U, thus U
LHU. Since L3 implies that L satisfies monotonicity, we have LU  LHU  U, hence, LU = U. This implies L3, L7 ) L2. Secondly, we show that L3 ; L7. Let U = {a, b}. LX for all X
U is listed in column 2 of Table 3. HX calculated by the duality of L and H is listed in column 3. It is easy to see that L3 is satisfied, but L7 is not satisfied. This implies that L3 ; L7. Finally, we show that L7 ; L3. Let U = {a, b}. LX for all X
U is listed in column 2 of Table 1. HX calculated by the duality of L and H is listed in column 3. It is easy to see that L3 is not satisfied. This implies that L7 ; L3. Thus, we have proved that axioms L3 and L7 are independent, and they imply axiom L2. h

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Remark. This theorem implies that {L3, L7}, or equivalently, {H3, H7}, is the minimal axiom set to characterize symmetric approximation operators, and {L2, L3, L7} in Yao [29,35] is not minimal. Proof of Theorem 4.5. Firstly, we show that L2, L3 ; L8. Let U = {a, b, c}. LX for all X
U is listed in column 2 of Table 4. LLX calculated by the definition of L is listed in column 3. It is easy to see that L2 and L3 are satisfied, but L8 is not. (See the items underlined in Table 4.) This implies that L2, L3 ; L8. Secondly, we show that L2, L8 ; L3. Let U = {a, b, c}. LX for all X
U is in column 2 of Table 1, HX calculated by the duality of L and H is listed in column 3. It is easy to see that L2, L8 hold but L3 does not. This implies that L2, L8 ; L3. Finally, we show that L3, L8 ; L2. In fact, let U = {a, b, c}. Define LX = ; for all X
U, then LLX = ;, LX
LLX, i.e., L8 is satisfied, and L(X \ Y) = LX \ LY( = ;), i.e., L3 is also satisfied, and LU = ; 5 U, this means that L2 is not satisfied. This implies that L3, L8 ; L2. Thus, we have proved that axioms L2, L3, and L8 are independent. h Remark. This theorem implies that {L2, L3, L8}, or equivalently, {H2, H3, H8}, is the minimal axiom set to characterize the transitive approximation operators. Proof of Theorem 4.6. Firstly, we prove that L3, L9 ) L2. If not, we suppose that L obeys L3 and L9, but LU 5 U, then there exists x0 2 U such that x0 62 LU. It is easy to see that HX
U for all X
U. Obviously, X
U. Since L3 implies that L satisfies monotonicity, we have LU  LX, LU  L(X), LU  LHX. Notice that x0 62 LU, of course, x0 62 LX, for the same reason, x0 62 L(X), x0 62 LHX, thus x0 2  L(X) = HX. This means HX 6
LHX, i.e., L9 is not satisfied, which contradicts that L obeys L9. Thus, L3, L9 ) L2. Secondly, we show that L3 ; L9. Let U = {a, b}. LX for all X
U is listed in column 2 of Table 3. HX calculated by the duality of L and H is listed in column 3. It is easy to see that L3 holds, and L9 does not. This implies that L3 ; L9. Finally, we show that L9 ; L3. Let U = {a, b, c}, LX for all X
U is listed in column 2 of Table 1. HX calculated by the duality of L and H is listed in column 3. It is easy to see that L9 ; L3. Thus, we have proved that L3 and L9 are independent and they imply L2. Therefore, {L3, L9} is the minimal axiom set to characterize Euclidean approximation operators. h Remark. In Refs. [29,35], {L2, L3, L9} is the axiom set to characterize Euclidean approximation operators, and Theorem 4.6 implies that the axiom L2 can be omitted.

X

LX

LLX

L(X \ Y), LX \ LY Y=U

Y = {a, b}

Y = {b, c}

Y = {a, c}

Y = {a}

Y = {b}

Y = {c}

Y=;

U {a, b} {b, c} {c, a} {a} {b} {c} ;

U fa; cg {a, b} {b, c} {c} {a} {b} ;

U fb; cg {a, c} {a, b} {b} {c} {a} ;

U,U {a, c}, {a, c} {a, b}, {a, b} {b, c}, {b, c} {c}, {c} {a}, {a} {b}, {b} ;, ;

{a, c}, {a, c} {a, c}, {a, c} {a}, {a} {c}, {c} {c}, {c} {a}, {a} ;, ; ;, ;

{a, b}, {a, b} {a}, {a} {a, b}, {a, b} {b}, {b} ;, ; {a}, {a} {b}, {b} ;, ;

{b, c}, {b, c} {c}, {c} {b}, {b} {b, c}, {b, c} {c}, {c} ;, ; {b}, {b} ;, ;

{c}, {c}, ;, ; {c}, {c}, ;, ; ;, ; ;, ;

{a, a} {a}, {a} {a}, {a} ;, ; ;, ; {a}, {a} ;, ; ;, ;

{b}, ;, ; {b}, {b}, ;, ; ;, ; {b}, ;, ;

;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ; ;, ;

{c} {c} {c} {c}

{b} {b} {b}

{b}

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Table 4 Example to check that L2, L3 ; L8

897

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