The minimization of axiom sets characterizing generalized approximation operators
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], [15], [16], [18], [24], [26], [28], [29], [30], [31], [32], [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], [21], [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 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.
Section snippets
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 . Let R be a binary relation on U, that is, R ⊆ U × U. ∀x ∈ U, denote Rs(x) = {y ∈ U:(x, y) ∈ 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) ≠ ∅ for all x ∈ U; R is referred to as reflexive if x ∈ Rs(x) for all x ∈ U; R is referred to as symmetric if ∀(x, y) ∈ U × U, y ∈ Rs(x) implies x ∈ Rs(y); R is referred to as transitive if ∀x, y
Axiom sets of the generalized approximation operators
In an axiomatic approach, the primitive notion is a system , where is the set algebra, and are unary operators on the power set . 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 be two unary operators on the power set . They are dual operators if By the duality of L and H, it is
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.
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) ≠ LX0 ∩ LY0. It is easy to see that L2 is satisfied. This implies L2 ⇏ L3. Secondly, we
Conclusion
This paper has investigated the independence of axioms characterizing generalized approximation operators. The minimal sets of axioms characterizing generalized crisp rough approximation operators have been obtained, which complements the results of Yao [29]. The minimal set of axioms characterizing approximation operators in fuzzy environment is to be found and verified.
Acknowledgments
The authors would like to thank the anonymous referees for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (No. 60373078) and the Scientific Research Project of the Education Department of Zhejiang Province in China (No. 20040538).
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