Dynamic Analysis, Learning, and Robust Control of Complex SystemsView this Special Issue
New Rough Approximations Based on E-Neighborhoods
This paper puts forward some rough approximations which are motivated from topology. Given a subset , we can use 8 types of -neighborhoods to construct approximations of an arbitrary on the one hand. On the other hand, we can also construct approximations relying on a topology which is induced by an -neighborhood. Properties of these approximations and relationships between them are studied. For convenience of use, we also give some useful and easy-to-understand examples and make a comparison between our approximations and those in the published literature.
1. Introduction and Preliminaries
The problem of imperfect knowledge became a crucial issue for computer scientists, especially in the area of information system and artificial intelligence [1, 2]. There are various approaches to manipulate and understand imperfect knowledge, among which is rough set theory. Rough set was proposed by Pawlak [3, 4] in 1982 which has been generalized in many ways [5–13]. What we are concerned about are those methods whose ideas are motivated from topology, for example, methods constructing the lower and upper approximations by using different kinds of neighborhoods, such as right and left neighborhoods [10, 14], minimal right neighborhoods , and intersection and union neighborhoods (see [16, 17]). In fact, a combination of rough set theory and topological theory became the main goal of many studies (see [18–25]).
The present paper is a continuation to these works where we initiate new types of neighborhoods (namely, -neighborhoods) to give lower and upper approximations of an arbitrary set directly or indirectly. In Section 2, we apply -neighborhoods to establish the concepts of -neighborhoods and discuss the main properties. With the help of examples, we show the relationships between them as well as with -neighborhoods and -adhesion neighborhoods. In Section 3, we formulate and study the concepts of -lower and -upper approximations, -boundary region, -positive and -negative regions, and -accuracy measure of a subset and make comparisons between them with respect to different types of . In Section 4, we study the previous concepts from a topological view and explore their main properties. In Section 5, we give some conclusions and make a plan for future works.
Now, we recall some basic properties and results of rough set theory, particularly those related to some types of neighborhood systems.
Definition 1 (see [3, 5, 16]). (1)A subset (also called a binary relation on ) is said to be an equivalence relation if it is reflexive (i.e., for each ), symmetric (i.e., if ), and transitive (i.e., whenever and ). It is said to be a preorder (or quasi-order) if it is reflexive and transitive. It is said to be a partial order if it is an antisymmetric (i.e., whenever and ) preorder. It is said to be a diagonal if . It said to be serial if for every , such that (also written as ).(2)(See [4, 26] for a special case). For an equivalence relation on and a subset , the two related sets and are called the Pawlak lower approximation and upper approximation of , respectively (where is the set of all equivalence classes).
Proposition 1 (see [3, 4] for a special case). The lower approximations and the upper approximations have the following properties (, the power set of ):(1) and .(2) and .(3) and . Particularly, and if .(4) and .
Definition 2. A subset is called a topology on (and is called a topological space) if it is closed under arbitrary union and finite intersection. A topology satisfying that every open set is also closed is called a clopen topology. We will use to denote the interior of (i.e., the union of all open sets that are contained in ) and to denote the closure of (i.e., the intersection of all closed sets containing ) in this paper.
Let . Then, is a topological space, , , and is the smallest open neighborhood of . This inspires many people to define the lower approximation and the upper approximation by neighborhoods; actually, our approach in this paper is also motivated from topology. In the following, we write .
Definition 3 (see [10, 14, 16]). Let and .(1)The 8 kinds of -neighborhoods are defined as follows: , , (if for some or , otherwise), (if for some or , otherwise), , , , and (2)The triple is called a -neighborhood space (in brief, -NS), where is a mapping from to which associated each with a -neighborhood
Definition 4 (see ). Let . The -adhesion neighborhoods are defined as follows:(1)(2)(3)(4)(5)(6)(7)(8)
In this section, we introduce the notions of -neighborhoods using -neighborhoods and study their properties.
Definition 5. Let . The -neighborhoods are defined as follows:(1)(2)(3)(4)(5)(6)(7)(8)We give the following example to illustrate how we calculate different types of neighborhoods. Also, we will benefit from this example to clarify some obtained results.
Example 1. Let and . Then, the -neighborhoods and -neighborhoods of a point are as in Table 1.
Theorem 1. -neighborhoods have the following properties ():(1)(2)(3) iff (4)If is reflexive, then and (5)If is symmetric, then and (6)If is transitive, then (7)If is serial, then (8)If is symmetric and transitive, then and (if ) for each (9)If is preorder, then (10)If is a equivalence relation, then for each , all are identical, , and iff
Proof. (3) Obviously, for each . Then, for each . (4) Step 1: since is reflexive, and . This implies that and . Consequently, and . Step 2: let . By reflexivity of , we have . Therefore, ; thus, . Also, let . Then, . Since is reflexive, . Therefore, ; thus, . Hence, we obtain the desired result. (5) Since is symmetric, . Therefore, ; thus, . Consequently, . Similarly, . (6) Let . Then, . For each containing , we have . Since is transitive, . Therefore, ; thus, . Hence, ; consequently, . Similarly, . This implies that and . (7) It follows from (4) and the fact . (8) Step 1: we prove the case . Let . Then, , i.e., there exists , , and . Since is symmetric and transitive, . Therefore, ; thus, . Step 2: since is symmetric, by (5). We only prove the case . Let . Then, ; therefore, there is such that and . Now, let . Then, , i.e., there exists a such that and . Note that ; this means that ; consequently, ; also, ; thus, . Hence, , as required. (10) Step 1: by (4) and (8), and . It remains to prove . Since is symmetric, by (5). Let . Then, . Since is equivalence, . Therefore, ; consequently, . Thus, is proved. Step 2: we only prove the case . Let . Then, . Since is equivalence, . Therefore, . Conversely, . Since is reflexive, . Hence, .
3. Rough Approximations Using -Neighborhoods Directly
Let be -NS and . We devote this section to formulate the following concepts using -neighborhoods directly: -lower approximation, -upper approximation, -boundary region, -positive region, -negative region, and -accuracy measure of a subset . We will also illustrate the relationships between them and reveal the main properties with the help of examples.
Definition 6. Let and . Then, and are called -lower approximation and -upper approximation, respectively; , , and are called -boundary, -positive, and -negative regions of , respectively; is called the -accuracy measure of (if ), where denotes the cardinality of a set and denotes the cardinality of the set of natural numbers.
Theorem 2. -approximations have the following properties (, , ):(1).(2).(3) and . Particularly, and if .(4) and .(5)In general, and .
Proof. We only prove (3) and (4). (3) Step 1: obviously, . Conversely, for each (i.e., for each ), we have , and thus, . Hence, , as required. Step 2: obviously, . Conversely, for each , we have , i.e., for some . Thus, , as required. (4) . The second equality immediately comes by putting in the first.
Example 2. For -NS in Example 1, the -lower approximations and -upper approximations (), the -accuracy measure, the lower approximations and upper approximations in the sense of  for P<r>, and the lower approximations and upper approximations in the sense of  for Nr are given in Table 2 (where and ). From Table 2, we can see the approximations in this paper and those in  (resp., ) are incomparable.
4. Rough Approximations Induced by -Topologies
We will construct rough approximations using -neighborhoods indirectly in this section. Let be -NS . We first employ -neighborhoods to generate a topology (called an -topology) and then call the interior and the closure of a subset the -lower approximation and -upper approximation of , respectively. These kinds of approximations are compared with those in Section 3.
Theorem 3. (1) is a topology on satisfying whenever (2)Both and are topologies on (3)(4)(5)If is reflexive, then (6)If is serial, then (7)If is a equivalence relation, then is constant for all
Proof. We only prove (1). Obviously, is a topology on . For each and each , we need to prove . Without loss of generality, we assume . Suppose . Then, (by Theorem 1 (3)) and (as ), and thus, . This is a contradiction. Therefore, , i.e., .
Definition 7. Let and . Then, (the interior of in ) and (the closure of in ) are called the -lower approximation and -upper approximation, respectively; , , and are called -boundary, -positive, and -negative regions of , respectively; is called the -accuracy measure of (if ).
The relation between and is given by the following.
Theorem 4. for each (if ).
Proof. (i)Step 1: for each , we have (because and ), and thus, , which implies(ii)Step 2: let . If , then . If , then , and thus, . This means there exists such that and . Consequently, for any containing , we have . Therefore, , and thus, (because ). It follows that , and thus,From (1) and (2), we can see that .
Example 3. (1)Now, we exemplify an application of rough approximations introduced in this paper. Let be a group of people who have just reached Xian Yang Airport (but are not allowed to outbound station) from two countries (by two planes involving 200 people denoted by a set ) and will attend a meeting holding in Xi’an. Assume that and are actually infected of asymptomatic infection of new coronavirus, consists of and all who contacted after had contacted one of and , and and . To insure the safety of this meeting, each person contacting directly after contacts one of and is in can be looked to be the set of all people who must run a nucleic acid test, and some person contacting directly after contacts one of and is in can be looked to be the set of all people who should run a nucleic acid test.(2)For -NS in Example 1, the -lower approximations and -upper approximations () and the -accuracy measure are given in Table 3.
5. Concluding Remarks
Motivated by topology, this article has initiated two new rough approximations by introducing a new class of neighborhood systems (called -neighborhoods) using -neighborhoods. We have probed the main features and formulated the concepts of -lower and -upper approximations and -accuracy measure which are had contacted one of induced from different types of and compared them. We complete this work by studying these concepts from a topological view and comparing them. In all comparisons, we obtain higher accurate approximations in the case of .
In the upcoming works, we will study new types of neighborhoods in rough set theory and use them to define a topological structure. Also, we will investigate the -neighborhoods and approximations on the fuzzy rough set motivated from fuzzy control problems and fractional-order nonlinear systems.
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China under Grant nos. 11771263 and 61967001.
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