Rough approximations based on bisimulations

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Rough approximations based on bisimulations Ping Zhu a,b,∗ , Huiyang Xie c , Qiaoyan Wen b a

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School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China State Key Laboratory of Networking and Switching, Beijing University of Posts and Telecommunications, Beijing 100876, China College of Science, Beijing Forestry University, Beijing 100083, China

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Article history: Received 29 March 2016 Received in revised form 29 October 2016 Accepted 9 November 2016 Available online xxxx

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Keywords: Rough set Generalized approximation space Bisimulation Bisimilarity Labeled transition system

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In recent years, rough set theory initiated by Pawlak has been intensively investigated. When the classical rough sets based on equivalence relations have been extended to generalized rough sets based on binary relations, the lower and upper rough approximations, which are the core concepts of rough set theory, have been generalized in several different ways. A common feature of these generalized approximations is that they use only “one step” information of the underlying relation to discern objects. By “one step” in a binary relation we mean that the ordered pair of the starting and end points of the step belongs to the relation. Motivated by a rich notion, bisimulation, appearing in various areas of computer science, we introduce a kind of lower and upper rough approximations for generalized rough sets in this paper. Our lower and upper approximations are based on bisimulations, in particular, bisimilarity, which is the largest bisimulation. Roughly speaking, bisimilar objects are regarded as indiscernible. We present some basic properties of the new lower and upper rough approximations and illustrate our motivation and the applicability of our results by examples. Moreover, we make a detailed comparison between the rough approximations based on the underlying relation and the rough approximations based on bisimilarity. In particular, we provide a necessary and sufficient condition for the consistency of the two kinds of rough approximations. © 2016 Elsevier Inc. All rights reserved.

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1. Introduction

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In order to cope with incomplete or inexact knowledge in information systems, Pawlak initiated the study of rough set theory in the early 1980s [33,34]. Since then, we have witnessed a world-wide growth of interest in the theory and its applications in many fields (see, for example, [35,36,52]). Nowadays, it becomes an excellent tool to handle granularity of data. The starting point of rough set theory in [33,34] is that every object of the universe of discourse is associated with some information like data and knowledge, and the objects having the same information are indiscernible with respect to the available information. Formally, indiscernibility is an equivalence relation and any equivalence class is interpreted as a granule of knowledge. According to Pawlak’s terminology, any subset X of the universe U is called a concept in U . If a concept X is a union of some equivalence classes, then X is precise; otherwise X is vague. The basic idea of rough set theory consists in approximating incomplete or inexact concepts with a pair of precise concepts—its lower and upper approximations.

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Corresponding author at: School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China. E-mail addresses: [email protected] (P. Zhu), [email protected] (H. Xie), [email protected] (Q. Wen).

http://dx.doi.org/10.1016/j.ijar.2016.11.007 0888-613X/© 2016 Elsevier Inc. All rights reserved.

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Fig. 1. Two black boxes and their behaviors. (For interpretation of the colors in this figure, the reader is referred to the web version of this article.)

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However, as pointed out by many authors, an equivalence relation could fail to provide a realistic view of relationships among objects in real-world application. In other words, the requirement of equivalence relation as indiscernibility relation is too restrictive. In light of this, equivalence relation has been replaced by characteristic relation [14,15,38], similarity relation [43], tolerance relation [5,21,24,42], dominance relation and preorder [12,13], and even arbitrary binary relation [48, 49,51] in some extensions of the classical rough sets. At the same time, the corresponding lower and upper approximations have been intensively investigated. Besides the constructive approach extensively adopted in the literature mentioned above, some scholars (see, for example, [7,10,45,46]) considered algebraic approach in the study of rough sets based on binary relations. In terms of rough sets based on general binary relations, there are a large number of lower and upper approximations (see, for example, [16, Section 2.3] and the references therein). Among others, Yao explored the lower and upper approximations that use successor or predecessor neighborhoods to replace equivalence classes in classical rough set theory [47, 49,50], in which some perfect constructive approaches and axiomatical characterizations have been provided. Taking a similarity relation into account, Slowinski and Vanderpooten [43] introduced a generalized definition of rough approximations based on the concept of ambiguity proposed by themselves. In [1], Abo-Tabl introduced three types of generalized rough approximations by using the intersection of right neighborhoods and compared these types with Yao’s lower and upper approximations. In addition, Abu-Donia generalized the classical rough approximations by utilizing a family of different types of relations [2]. It is worth noting that in essence, the aforementioned generalized rough approximations based on binary relations are only dependent on “one step” information of the underlying relations. By “one step” in a binary relation we mean that the ordered pair of the starting and end points of the step belongs to the relation. Formally, for a relation R ⊆ U × U and any s, t ∈ U , we say that there is one step from s to t if (s, t ) ∈ R. For instance, the elements that have one step from x give rise to the successor neighborhood R s (x) = { y ∈ U | (x, y ) ∈ R } of x, and the elements that have one step to x form the predecessor neighborhood R p (x) = { y ∈ U | ( y , x) ∈ R } of x. In general, the rough approximations via neighborhoods are based on the successor neighborhoods or predecessor neighborhoods. For example, the lower and upper approximations defined respectively by app X = {x ∈ U | R s (x) ⊆ X } and app X = {x ∈ U | R s (x) ∩ X = ∅} are typical ones which are based on successor neighborhoods [47,49]. This implies that the indiscernibility is characterized by “one step” information of the underlying relation. However, in some rich and complex data sets, “one step” information may not be sufficient for discerning objects. To illustrate it, let us consider an example, which is inspired by [29, Chapter 2].

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Example 1.1. Suppose that we want to discern reactive systems by interacting with them and observing the change of their states. Clearly, it is better to interact enough times instead of single time and then make a discrimination based on the observed sequence. Let us think of two reactive systems as two black boxes with one blue button each, as shown in Fig. 1. Following [29], we interact with the black boxes by trying to press the buttons. Sometimes the button goes down, which means that we succeed, and sometimes it does not. This is the only way that we can tell the difference between the two black boxes. For the first black box, we assume that its button can successively go down two times from its initial state and after that, it does not continue; for the second one, we assume that it can always go down. We describe their behaviors by the right state transition graph in Fig. 1. Formally, we are considering a generalized approximation space (U , R ), where U = {s, u , v , t } and R = {(s, u ), (u , v ), (t , t )}. Let us approximate the concept X = {s, t } of their initial states. If we chose the above lower and upper approximation operators based on successor neighborhoods, we would obtain that app X = {x ∈ U | R s (x) ⊆ X } = { v , t } and app X = {x ∈ U | R s (x) ∩ X = ∅} = {t }. Recall that the basic idea of rough set theory consists in describing incomplete or inexact concepts with a pair of precise concepts which are granules of knowledge. Obviously, the states s, u , v, and t are different from one another; for instance, in state s the left button can successively go down two times, while in state u the left button can only go down one time. Therefore, each state should be regarded as a granule of knowledge. In light of this, approximating the concept {s, t } by itself (namely, the union of two knowledge granules {s} and {t }) is much better than using { v , t } and {t }. However, the approximation operators in the literature cannot give rise to such an approximation. We will revisit it in detail by Example 3.1.

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The above example motivates us to characterize indiscernibility by exploiting multi-step information. Fortunately, there is an elegant concept in computer science, called bisimulation [28,32,40], that reflects our basic idea. A bisimulation is a

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binary relation on the states of a discrete event system such that associated states behave in the same way, in the sense that one state can simulate the other and vice-versa. Intuitively, two states are bisimilar if they match each other’s moves. The union of all bisimulations gives the largest bisimulation, which is referred to as bisimilarity. It turns out to be an equivalence relation. Following the basic idea of process algebra [28], we can regard the elements of the universe U as states and the steps like (s, t ) arising from a relation R as moves. In this way, a generalized approximation space (U , R ) can be viewed as a discrete event system. As a result, we can characterize indiscernibility by bisimilarity. The aim of the paper is to provide lower and upper approximations established by bisimulations, in particular, bisimilarity. After introducing the definition, we examine the basic properties of the new lower and upper approximations and further illustrate the applicability of our results by examples. Moreover, we make a comparison between the rough approximations based on the underlying relation and our new rough approximations based on bisimilarity. In particular, we present a characterization of when the two kinds of approximations produce the same result. The remainder of the paper is structured as follows. In Section 2, we briefly review some basics of Pawlak’s rough set theory, generalized rough sets based on relations, and bisimulations. We introduce rough approximations based on bisimulations and discuss related properties in Section 3. Section 4 is devoted to illustrating the applicability of rough approximations based on bisimilarity. We address the relationship between the two kinds of approximations in Section 5 and conclude the paper in Section 6 with a brief discussion on the future research.

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R

app X = {x ∈ U | [x] R ⊆ X } and app R X = {x ∈ U | [x] R ∩ X = ∅}, R

(1)

respectively. A rough set in (U , R ) is an equivalence class of subsets of U , modulo the two approximations. The ordered pair   app X , app R X is a particular representation of such an equivalence class, called increasing representation [31]. In light of R

this, the general notion of rough sets can be simply identified with the rough approximations of a given set. The ordered pair (U , R ) is said to be an approximation space. The lower and upper approximations possess the following properties. Proposition 2.1 ([33,34]). Let (U , R ) be an approximation space. For any X , Y ⊆ U , the following properties hold. (1) (2) (3) (4) (5)

app X ⊆ X ⊆ app R X . R app ∅ = app R ∅ = ∅, app U = app R U = U . R R app ( X ∩ Y ) = app X ∩ app Y and app R ( X ∩ Y ) ⊆ app R X ∩ app R Y . R R R app ( X ∪ Y ) ⊇ app X ∪ app Y and app R ( X ∪ Y ) = app R X ∪ app R Y . R R R app X ⊆ app Y and app R X ⊆ app R Y , if X ⊆ Y . R

R

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2.2. Generalized rough sets based on relations

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We begin with some concepts of Pawlak’s rough sets and their properties. Let U be a finite and nonempty universe, and let R ⊆ U × U be an equivalence relation on U , which is called an indiscernibility relation in rough set theory. We write U / R for the set of all equivalence classes induced by R and denote by [x] R the equivalence class that contains x. Such equivalence classes are also called elementary sets; every union (not necessarily nonempty) of elementary sets is called a definable set. For any X ⊆ U , one can approximate it by a pair of definable sets. The lower approximation app X and the upper approximation app R X of X are formally defined by

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2.1. Pawlak’s rough set theory

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2. Preliminaries

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The classical rough sets have been extended by replacing R with an arbitrary relation. In this subsection, we review the notion of generalized rough sets based on relations [47,49]. Thus, suppose that R ⊆ U × U is any binary relation on U . We refer to the ordered pair (U , R ) as a generalized approximation space. For any x, y ∈ U , if (x, y ) ∈ R, we say that y is R-related to x, x is a predecessor of y, and y is a successor of x. Denote by R s (x) the set of elements that are R-related to x, namely,

R s (x) = { y ∈ U | (x, y ) ∈ R },

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which is also called the successor neighborhood of x. The notion of a successor neighborhood is often called R-neighborhood and in mathematics it is known as a Pierce product. Accordingly, one can define the predecessor neighborhood of x. By substituting equivalence classes in Equation (1) with successor neighborhoods, we can approximate any subset X ⊆ U by a pair of subsets of U . Formally, the lower approximation app X and the upper approximation app R s X of X based on a

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binary relation R are defined by

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app

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Proposition 2.2 ([49]). Let (U , R ) be a generalized approximation space. For any X , Y ⊆ U , the generalized approximation operators have the following basic properties.

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(1) app R s ∅ = ∅, app

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Y and app R s X ⊆ app R s Y , if X ⊆ Y .

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In this subsection, we recall the concept of bisimulations and their basic properties, including the largest bisimulation, called bisimilarity. The reader may refer to [28,32,40] for more details. Usually, we should introduce bisimulations on labeled transition systems (LTSs), because these are the most common structures on which bisimulation has been investigated. LTSs themselves are a kind of widely used formal model in computer science [22], and they are essentially labeled directed graphs. However, as a first step toward introducing bisimulation into rough set theory, we limit us to (unlabeled) transition systems, a special class of LTSs with a single label. In doing so, we can present our main concepts and ideas in a simpler, more consistent form. Recall that an (unlabeled) transition system is a pair (U , R ), where U is a set of states and R ⊆ U × U is a transition relation. Formally, (unlabeled) transition systems, generalized approximation spaces, and unlabeled directed graphs mean the same thing, namely, a set with a binary relation. Based on generalized approximation spaces, we can state the notion of bisimulations as follows. Definition 2.1. Let (U , R ) be a generalized approximation space. A binary relation B ⊆ U × U is called a bisimulation if for all (s, t ) ∈ B, (1) (s, s ) ∈ R implies that (t , t ) ∈ R for some (2) (t , t ) ∈ R implies that (s, s ) ∈ R for some

t ∈ U with (s , t ) ∈ B; s ∈ U with (s , t ) ∈ B.

Two states s and t are bisimilar, denoted s ∼ t, if there exists a bisimulation B such that (s, t ) ∈ B. The relation ∼ is referred to as bisimilarity.

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Intuitively, for a relation to be a bisimulation, related states must be able to “match” transitions of each other by moving to related states. The version of bisimulation for LTSs simply requires that matched transitions have the same label. It should be noted that in general a bisimulation is a partial match of transitions, since it requires matching only for those states that appear in the relation. Simple examples of bisimulations are the identity relation Id U = {(s, s) | s ∈ U } and the empty relation. Bisimulation has a number of appealing properties. For instance, it is preserved by various operations on relations. To state such properties, we need a few more notations. In general, we define the converse R −1 of a binary relation R and the composition R 1 ◦ R 2 of two binary relations R 1 and R 2 by

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R −1 = {(t , s) | (s, t ) ∈ R }, R 1 ◦ R 2 = {(r , t ) | there exists s ∈ U such that (r , s) ∈ R 1 and (s, t ) ∈ R 2 }, respectively. Proposition 2.3 ([28]). Let (U , R ) be a generalized approximation space. Suppose that each B i , i ∈ I , is a bisimulation. Then the 1 relations B − , B 1 ◦ B 2 , and ∪i ∈ I B i are all bisimulations. i

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2.3. Bisimulations

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The union of all bisimulations gives rise to the largest bisimulation, i.e., bisimilarity. Furthermore, it has been shown that the bisimilarity ∼ is an equivalence relation on U [28]. Remarkably, there are some efficient algorithms for bisimilarity in the literature. The reader is referred to the excellent survey [41, Chapter 3]; here we do not go into the detail of its algorithmic, complexity and decidability properties. To illustrate the above notions, we present two examples here. One is based on a generalized approximation space; the other is based on an LTS, i.e., labeled approximation space.

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Fig. 2. A generalized approximation space, where the relation depicted by dotted arrows is a bisimulation.

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Fig. 3. A labeled approximation space.

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Example 2.1. Fig. 2 presents a generalized approximation space, where U = {s1 , s2 , . . . , s5 , t 1 , t 2 , . . . , t 5 } and the relation R is indicated by solid arrows. The graph follows from [9], which shows a bisimulation between two nonisomorphic graphs. By definition, there are many bisimulations, such as Id U , {(s4 , t 4 ), (s5 , t 5 )}, {(s3 , s4 ), (s5 , s5 )}, and B = {(s1 , t 1 ), (s2 , t 2 ), (s2 , t 3 ), (s3 , t 4 ), (s4 , t 4 ), (s5 , t 5 )}. By a routine computation or using classical algorithms for bisimilarity (see, for example, [41]), we can obtain that

∼ = IdU ∪ B ∪ B −1 ∪ {(s3 , s4 ), (s4 , s3 ), (t 2 , t 3 ), (t 3 , t 2 )},   U /∼ = {s1 , t 1 }, {s2 , t 2 , t 3 }, {s3 , s4 , t 4 }, {s5 , t 5 } . As a result, we see that s1 and t 1 are bisimilar, s2 , t 2 , and t 3 are bisimilar, s3 , s4 , and t 4 are bisimilar, and s5 and t 5 are bisimilar.

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Although the relations in the above examples are acyclic, our model of generalized approximation spaces does not assume the condition, as subsequently shown in Figs. 4–6. Nevertheless, for a generalized approximation space with an acyclic relation, let us show that it is easy to decide whether or not two states are bisimilar. To this end, we need to introduce some notions. Let (U , R ) be a generalized approximation space. A state s in U is said to be steady, if there is no state t such that (s, t ) ∈ R, namely, R s (s) = ∅. For example, in Fig. 2, both s5 and t 5 are steady. A run of (U , R ) is a finite, nonempty sequence σ = s1 s2 · · · sn of states such that (si , si+1 ) ∈ R, for all 1 ≤ i ≤ n − 1; for the run σ , we say that it begins with s1 and define the length of σ as n − 1. For any state s, we denote by L (s) the set of all lengths of runs beginning with s. For instance, in Fig. 2, L (s1 ) = L (t 1 ) = {3} and L (s5 ) = L (t 5 ) = ∅. It should be noted that for any state s ∈ U , L (s) is always finite, when R is acyclic.

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Example 2.2. Fig. 3 presents a labeled approximation space, where U = {s1 , s2 , . . . , s4 , t 1 , t 2 , . . . , t 5 } and the transition relation is indicated by solid arrows with labels. Assume that they are the state transition graphs of two black boxes S and T , respectively, and that each black box has three buttons labeled by a, b, or c. Although they generate the same set {a, ab, ac } of strings, their behaviors are different: For S, we can always choose between the button labeled by b and the button labeled by c after the button labeled by a, while for T , we can only successively press either the button labeled by b or the button labeled by c (but not both) after the button labeled by a. As a result, we see that s1 and t 1 are not bisimilar.

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The following is a classic example appearing in concurrency theory of computer science.

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Proposition 2.4. Suppose that (U , R ) is a generalized approximation space, where R is acyclic. Then for any states s, t ∈ U , s ∼ t if and only if L (s) = L (t ).

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Proof. By the definition of bisimilarity, the necessity is obvious for an acyclic relation. For the sufficiency, set B = {(s, t ) ∈ U × U | L (s) = L (t )}. There is no difficulty to check that B is a bisimulation, and thus, s ∼ t. 2

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In this section, we investigate the lower and upper approximations induced by bisimulations. For the consistency with rough set theory, we focus on generalized approximation spaces and leave the related study of labeled approximation spaces for future investigations. Let (U , R ) be a generalized approximation space. Due to the fact that the identity relation Id U is a bisimulation, there always exists a bisimulation. It should be noted that bisimulation itself is a binary relation. Therefore, for any bisimulation B ⊆ U × U , it follows that Equation (2) gives the lower approximation app X and the upper approximation app B s X of X ,

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Bs

X = {x ∈ U | B s (x) ⊆ X } and app B s X = {x ∈ U | B s (x) ∩ X = ∅}.

For simplicity, from now on we write app X and app B X for app B

Bs

(3)

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Example 3.1. Consider the generalized approximation space (U , R ) in Fig. 1, where U = {s, u , v , t } and R = {(s, u ), (u , v ), (t , t )}. Let us approximate the concept X = {s, t } of their initial states. (1) We first consider the approximations based on the underlying relation R. We recall that app X = { v , t } and app R X = {t }. R (2) We then consider the approximations based on bisimilarity. Observe that the bisimilarity ∼ is exactly Id U . It yields by Equation (2) that app X = app ∼ X = {s, t } = X . ∼

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Proposition 3.1. The conclusions in Proposition 2.1 hold for the bisimilarity ∼, and the conclusions in Proposition 2.2 hold for any bisimulation B.

(1) app (2) app (3) app

Id U

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B1◦B2

∪i ∈ I B i

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B1

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app

app

Bi

B2

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X and app B 1 ◦ B 2 X = app B 1 app B 2 X .

X and app ∪i∈ I B i X =



i∈ I

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app B i X .

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(2) Let us first prove that app

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app

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B1◦B2

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B1

app



B2



= x ∈ U | { y ∈ U | (x, y ) ∈ B 1 ◦ B 2 } ⊆ X

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Id U

X = {x ∈

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X . By definition, we have that

X = {x ∈ U | ( B 1 ◦ B 2 )s (x) ⊆ X }

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and

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U | {x} ⊆ X } = X and app IdU X = {x ∈ U | {x} ∩ X = ∅} = X .

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Proposition 3.2. Let (U , R ) be a generalized approximation space. Suppose that each B i , i ∈ I , is a bisimulation. For any X ⊆ U , we have the following.

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For some special forms of bisimulations, we have the following results on approximations.

Proof. (1) It follows by the definition of Id U that Id U s (x) = {x} for all x ∈ X . Therefore, we obtain that app

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Noting that bisimulation is a binary relation and bisimilarity is an equivalence relation, we have the following fact.

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Which one does follow our intuition? It should be noted that the basic idea of rough set theory consists in describing incomplete or inexact concepts with a pair of precise concepts. Obviously, the states s, u , v, and t are different from one another, and each one can be regarded as a precise concept. In light of this, it is obvious that the second approach, which approximates {s, t } by itself, follows our intuition.

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Before presenting the properties of such approximations, we would like to revisit Example 1.1.

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X and app B s X , respectively.

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app

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namely,

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app

B1

app

 B2

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= {x ∈ U | B 1s (x) ⊆ app B X } 2  = x ∈ U | { z ∈ U | (x, z) ∈ B 1 } ⊆ { z ∈ U | B 2s ( z) ⊆ X } .

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X ⊆ app



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X , we assume by contradiction that there is x ∈ app

X \app



1

To show that app

X , where

1

2

the symbol A \ B denotes the set of all elements which are members of A but not members of B. It means that there is some z ∈ U such that (x , z ) ∈ B 1 and B 2 s ( z )  X . The latter implies that there exists y ∈ U such that ( z , y ) ∈ B 2 and y ∈ / X . As a result, we find that (x , y ) ∈ B 1 ◦ B 2 with y ∈ / X , which contradicts the assumption x ∈ app X . Conversely, to verify

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3 4 5

B1◦B2





⊆ app B

that app

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It follows from x ∈ / app

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that (x , z )

B2

B1◦B2

app





X \app

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B2

B1◦B2

B 2 with y

∈





app B 1 app B 2 X

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10 11 12

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and

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= x ∈ U | { y ∈ U | (x, y ) ∈ B 1 ◦ B 2 } ∩ X = ∅

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B 1 and ( z , y ) 

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app B 1 ◦ B 2 X = {x ∈ U | ( B 1 ◦ B 2 )s (x) ∩ X = ∅}

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X that there is y ∈ U \ X , such that (x , y ) ∈ B 1 ◦ B 2 . Therefore, there exists z ∈ U such

B1

app

B2

X.

1◦B2

B1◦B2

X , we assume, by contradiction, that there exists some x ∈ app

B1

∈ / X . It yields that { z ∈ U | (x, z) ∈ B 1 }  { z ∈ U | B 2s (z) ⊆ X }, and thus  

x ∈ / app B app B X , which is absurd. This completes the proof of app B ◦ B X = app B app B X . 1 2 1 2 1 2   We now show that app B 1 ◦ B 2 X = app B 1 app B 2 X . It follows from definition that 

∈

B2

X

B1

B1◦B2

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B1

app

app



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= {x ∈ U | B 1s (x) ∩ app B 2 X = ∅}  = x ∈ U | { z ∈ U | (x, z) ∈ B 1 } ∩ { z ∈ U | B 2s ( z) ∩ X = ∅} = ∅ . 

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We first verify that app B 1 ◦ B 2 X ⊆ app B 1 app B 2 X . For any x ∈ app B 1 ◦ B 2 X , we see that { y ∈ U | (x , y ) ∈ B 1 ◦ B 2 } ∩ X = ∅, which implies that there is some y ∈ X such that (x , y ) ∈ B 1 ◦ B 2 . Therefore, there exists z ∈ U such that (x , z ) ∈ B 1 and ( z , y ) ∈ B 2 . The former means that z ∈ B 1 s (x ), and the latter, together with y ∈ X , means that z ∈ app B 2 X . We thus









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obtain that z ∈ B 1s (x ) ∩ app B 2 X . It implies that x ∈ app B 1 app B 2 X , and thus app B 1 ◦ B 2 X ⊆ app B 1 app B 2 X . Conversely,

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for any given x

= ∅. It follows from the latter

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that there is y ∈ X such that ( z , y ) ∈ B 2 . As a result, we obtain that y ∈ { y  ∈ U | (x ,y ) ∈ B 1 ◦ B 2 } ∩ X , and thus ( B 1 ◦ B 2 )s (x ) ∩ X = ∅, which yields that x ∈ app B 1 ◦ B 2 X . Hence, we have that app B 1 app B 2 X ⊆ app B 1 ◦ B 2 X , which finishes the

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proof of app B 1 ◦ B 2 X = app B 1 app B 2 X .

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27

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∈ app B 1





app B 2 X , there exists z



∈

U such that (x , z ) ∈ B 1 and B 2 s ( z ) ∩ X



(3) To prove the two equalities, it is convenient to have the following observation: for any x ∈ U , (∪i ∈ I B i )s (x) = ∪i ∈ I B i s (x), which can be easily checked. Using this equality, we obtain that

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app

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∪i ∈ I B i

X = {x ∈ U | (∪i ∈ I B i )s (x) ⊆ X }

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= {x ∈ U | ∪i ∈ I B i s (x) ⊆ X }

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= ∩i ∈ I {x ∈ U | B i s (x) ⊆ X }

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= ∩i ∈ I app B X

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i

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and

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app ∪i∈ I B i X = {x ∈ U | (∪i ∈ I B i )s (x) ∩ X = ∅}

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 = {x ∈ U | ∪i ∈ I B i s (x) ∩ X = ∅}

 = {x ∈ U | ∪i ∈ I B i s (x) ∩ X = ∅}

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= ∪i ∈ I {x ∈ U | B i s (x) ∩ X = ∅}

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= ∪i ∈ I app B i X ,

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which completes the proof of (3).

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Remark 3.1.

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(1) For the convenience of the reader, we have provided the proof of Proposition 3.2. In fact, the assertions in the proposition are axioms of Propositional Dynamic Logic, or PDL, in [37], and the proposition can be viewed as a corollary of the relational interpretation of PDL.1

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The authors would like to thank one of the anonymous reviewers for pointing out it.

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Fig. 4. A fraction of a multi-agent recommendation network.

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(2) Although we stated the above proposition for bisimulations in generalized approximation spaces, we have not used any specific properties of bisimulations in the proof of the theorem. Indeed, the assertions of Proposition 3.2 hold for general binary relations and labeled approximation spaces.

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∼

i app B i

i app B i

X and app ∼ X =

X , where B i runs over all bisimulations.

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In this section, we apply the previous results to two examples arising from multi-agent recommendation networks and web mining, respectively. Let us begin with the first one.

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Example 4.1. Fig. 4 shows an example of graph pattern matching discussed in [11], which is a fraction of a multi-agent recommendation network. In the graph, each node represents a music shop agent (M S A), a book server agent (B S A), a customer (C ), or a facilitator agent (F A) assisting customers to find B S As and M S As; each edge indicates a recommendation. It is not difficult to check that

  U /∼ = { M S A 1 , M S A 2 }, { B S A 1 , B S A 2 }, { F A 1 , F A 2 , C 1 , C 2 }, { F A 3 , F A 4 }, {C 3 , C 4 , . . . , C k } .

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(4)

Assume that Fig. 4 specifies the condition attributes of agents, and a decision attribute shows that only the agents M S A 1 , M S A 2 , and B S A 1 , which consist of all music shop agents and one book server agent, are satisfactory. Let us consider the approximations of the concept X = { M S A 1 , M S A 2 , B S A 1 } of satisfactory agents based on the underlying relation and bisimilarity, respectively. For the approximations based on the underlying relation R shown in Fig. 4, it follows from Equation (2) that the lower approximation app X and the upper approximation app R X of X are R

app X = { M S A 1 , C 3 , C 4 , . . . , C k }, R

app R X = { M S A 1 , B S A 1 , B S A 2 }, respectively. However, it seems difficult to give an intuitive explanation of such approximations. For the approximations based on the bisimilarity ∼, we have that

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4. Illustrative examples

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Proof. Because ∼ is the union of all bisimulations, the theorem follows immediately from the assertion (3) of Proposition 3.2. 2

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Theorem 3.1. Let (U , R ) be a generalized approximation space. For any X ⊆ U , we have that app X =



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As an immediate application of Proposition 3.2, we have the following result.

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app X = { M S A 1 , M S A 2 }, ∼

app ∼ X = { M S A 1 , M S A 2 , B S A 1 , B S A 2 }, by using the above partition U / ∼. Recall that in classical rough sets, the lower approximation is also referred to as the positive region of a concept. Therefore, we see that all music shop agents can be positively (i.e., unambiguously) classified as belonging to the target set of satisfactory agents. The upper approximation { M S A 1 , M S A 2 , B S A 1 , B S A 2 } includes all objects that might be members of the target set of satisfactory agents. The negative region U \app ∼ X consists of the objects that

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can be definitely ruled out as members of the target set of satisfactory agents. The boundary region { B S A 1 , B S A 2 }, given by app ∼ X \app X , consists of those objects that can neither be ruled in nor ruled out as members of the target set. In this ∼ way, we can obtain some rules, such as, all music shop agents are satisfactory and all facilitator agents are not satisfactory. In contrast with the rough approximations based on the underlying relation, it shows the advantage of representing a target set by the rough approximations based on bisimilarity. As another application, let us consider deadlock detection in Fig. 4. It is well known that bisimilarity has a strong ability to distinguish deadlock (i.e., inability to proceed). This is based upon the intuition that we cannot consider two states “the same” when one, and only one of them, can cause a deadlock, that is, states that will terminate (deadlock) should not be equivalent to states that may not terminate (deadlock). For example, we cannot consider C 2 and C 3 “the same”. Furthermore, let us distinguish the customer set C = {C 1 , C 2 , . . . , C k } and the facilitator agent set F A = { F A 1 , F A 2 , F A 3 , F A 4 } by the rough approximations based on the bisimilarity ∼, respectively. Using Equations (2) and (4), it is easy to obtain that

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32

The lower approximation app C consists of all customers C 3 , C 4 , . . . , C k that have no ability to proceed, that is, ∼ C 3 , C 4 , . . . , C k are deadlocks. The upper approximation app ∼ C includes all objects F A 1 , F A 2 , C 1 , C 2 , . . . , C k that might be members of the target set C . This means that some customers in the upper approximation may not be unable; indeed, customers C 1 and C 2 are capable of proceeding. The lower approximation app F A consists of all facilitator agents F A 3 , F A 4 ∼ that can cause a deadlock, while the upper approximation app ∼ F A consists of all objects F A 1 , F A 2 , F A 3 , F A 4 , C 1 , C 2 that might be members of the target set of facilitator agents. On the other hand, if we consider the rough approximations based on the underlying relation, we can obtain that R R

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Again, there is not an intuitive explanation of such approximations.

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The second example is concerned with web mining. By the rough approximations based on the bisimilarity, it presents the nodes that may be infected by a virus.

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∼

app R F A = { B S A 1 , B S A 2 , C 1 , C 2 }.

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app ∼ F A = { F A 1 , F A 2 , F A 3 , F A 4 , C 1 , C 2 }.

app F A = {C 1 , C 2 , . . . , C k },

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app F A = { F A 3 , F A 4 },

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app R C = { F A 1 , F A 2 , F A 3 , F A 4 };

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app ∼ C = { F A 1 , F A 2 , C 1 , C 2 , . . . , C k };

∼

app C = { F A 1 , F A 2 , F A 3 , F A 4 , C 3 , C 4 , . . . , C k },

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app C = {C 3 , C 4 , . . . , C k },

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Example 4.2. Let us consider a fraction of a web graph with 11 nodes in Fig. 5. The largest bisimulation, bisimilarity, on the web graph is the relation













∼= (si , s j ) | i , j ∈ {1, 2, 3} ∪ (t i , t j ) | i , j ∈ {1, 2, . . . , 5} ∪ (u i , u j ) | i , j ∈ {1, 2, 3} . Assume that we find that the nodes in V = {s1 , s2 , s3 , u 2 } are infected by a virus. Using the approximations based on the bisimilarity ∼, we see that

app V = {s1 , s2 , s3 } and app ∼ V = {s1 , s2 , s3 , u 1 , u 2 , u 3 }, ∼

where U is the set of all nodes in the web graph. It is a common knowledge that similar nodes are infected by a virus with near probabilities. Therefore, we can believe that all s nodes (i.e., {s1 , s2 , s3 }) in the web graph can be positively (i.e., unambiguously) classified as belonging to the target set of infected nodes. The upper approximation {s1 , s2 , s3 , u 1 , u 2 , u 3 } includes all nodes that may be infected. It suggests us check for u 1 and u 3 to determine whether they are infected.

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5. Relationships between the two kinds of approximations

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Broadly speaking, for a generalized approximation space (U , R ) we are concerned with the two kinds of approximations. One is based on the underlying relation R, and the other is based on the derived bisimilarity ∼. In the previous examples in the last section, we have not paid attention to the relationship between the approximations based on R and those based on ∼. This section is devoted to disclosing the relationships between the two kinds of approximations. We begin the discussion of the relationships by an example, which shows that unfortunately, in general there is no desired relationship between them.

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Example 5.1. Let us revisit Example 2.1. Recall that the relation R has been shown in Fig. 2, and U /∼ = {s1 , t 1 }, {s2 , t 2 , t 3 },  {s3 , s4 , t 4 }, {s5 , t 5 } . We first compare the lower approximations and then consider the upper approximations.

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app X 1 = {s5 , t 5 } and app X 1 = {s1 , t 1 }. ∼

As a result, we see that app X 1 ∩ app X 1 = ∅. R ∼ (2) For X 2 = {s1 , t 1 , s2 , s5 , t 5 }, we obtain that

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app X 2 = {s1 , s5 , t 5 } and app X 2 = {s1 , t 1 , s5 , t 5 }. R

∼

It gives rise to that app X 2  app X 2 . The notation X  Y means that X ⊆ Y and X = Y . R ∼ (3) For X 3 = {s3 , s5 , t 5 }, we find that

app X 3 = {s3 , s4 , t 4 , s5 , t 5 } and app X 3 = {s5 , t 5 }. ∼

R

Thus, app X 3  app X 3 . ∼ R (4) One may wonder whether or not there is a set X with app X = app X . In fact, it is possible. For example, for U = {s, t } R ∼ and R = {(s, t )}, we see that ∼= Id U , and thus app U = app U = U . R ∼ (5) We see by the cases (2)–(4) that there exists X such that app X ∩ app X = ∅. Is there a nontrivial intersection? Again, R ∼ it is possible. For X 4 = {s1 , t 1 , s2 , t 2 } in Example 2.1,

app X 4 = {s1 , s5 , t 5 } and app X 4 = {s1 , t 1 }, R

∼

which gives that app X 4 ∩ app X 4 = {s1 }. R ∼ (6) For X 5 = {s1 , t 5 }, we obtain that

app R X 5 = {t 4 } and app ∼ X 5 = {s1 , t 1 , s5 , t 5 }. It yields that app R X 5 ∩ app ∼ X 5 = ∅. (7) For X 6 = {s1 , s2 }, we see that

app R X 6 = {s1 } and app ∼ X 6 = {s1 , t 1 , s2 , t 2 , t 3 }. It gives rise to that app R X 6  app ∼ X 6 . (8) To give an instance to show that app ∼ X  app R X , we need to revisit Example 4.1. For X 7 = { F A 1 , F A 2 , C 1 , C 2 }, we obtain that

app R X 7 = { B S A 1 , B S A 2 , F A 1 , F A 2 , C 1 , C 2 } and app ∼ X 7 = { F A 1 , F A 2 , C 1 , C 2 }. This gives that app ∼ X 7  app R X 7 . (9) Consider the generalized approximation space (U , R ) with U = {s, t , u } and R = {(s, t ), (s, u ), (t , t )}. So ∼= Id U . For X = {s, t }, we obtain that app R X = app ∼ X = X . (10) For X 3 = {s3 , s5 , t 5 } in the case (3),

app R X 3 = {s2 , s3 , s4 , t 4 } and app ∼ X 3 = {s3 , s4 , t 4 , s5 , t 5 }. Thus app R X 3 ∩ app ∼ X 3 = {s3 , s4 , t 4 }, which is a nontrivial intersection of two upper approximations.

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(1) For X 1 = {s1 , t 1 }, we have that R

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It follows that the relationships between app X and app X as well as app R X and app ∼ X are random. Things change for R ∼ particular relations, some of which are widely used in rough set analysis. To this end, let us review some concepts. Let R be a binary relation on U . Recall that R is called serial, if for any s ∈ U , there exists a t ∈ U such that (s, t ) ∈ R, and R is called inverse serial, if for any s ∈ U , there exists a t ∈ U such that (t , s) ∈ R. The relation R is called reflexive, if (s, s) ∈ R for every s ∈ U . We say that R is symmetric, if (s, t ) ∈ R implies that

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(t , s) ∈ R. The relation R is called transitive, if (r , s) ∈ R and (s, t ) ∈ R imply that (r , t ) ∈ R. The relation R is called Euclidean, if (r , s) ∈ R and (r , t ) ∈ R imply that (s, t ) ∈ R. To state a simple fact for later use, we need one more concept. Let (U , R ) be a generalized approximation space. Recall that a state s in U is called steady, if there is no state t such that (s, t ) ∈ R. A state s is said to be unsteady if it cannot go into any steady state, that is, there is no path s0 s1 s2 · · · si ending with a steady state si , where s0 = s and (sk , sk+1 ) ∈ R for every 0 ≤ k ≤ i − 1. For instance, in Fig. 4, F A 1 , F A 2 , C 1 , C 2 are unsteady states, and the states M S A 1 , M S A 2 , B S A 1 , B S A 2 , F A 3 , F A 4 are neither steady nor unsteady.

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Lemma 5.1. Let (U , R ) be a generalized approximation space and set I = {(s, t ) | s and t are unsteady}. Then I is a bisimulation, and thus, all unsteady states are bisimilar. Proof. It follows immediately from the definitions of bisimulation and unsteady state.

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app X =

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∼

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U , if X = U ∅, otherwise

and

app ∼ X =

∅, if X = ∅ U , otherwise.

∼

R

Proof. When R is a serial or reflexive relation, we know that all states in U are unsteady, and thus they are bisimilar by Lemma 5.1. Therefore, we obtain that the bisimilarity ∼, the largest bisimulation, is exactly U × U . It means that [x]∼ = U , for any x ∈ U . By the definition of approximations, we obtain that app U = U and app X = ∅ for any X  U , and also ∼ ∼ app ∼ ∅ = ∅ and app ∼ X = U for any X = ∅. This proves the first part of the proposition. For the second part, observe that app U = U and app R ∅ = ∅. Hence, it always holds that app X ⊆ app X and app R X ⊆ R ∼ R app ∼ X , for any X ⊆ U . In addition, it follows from [49, Theorem 2] that app X ⊆ app R X , when R is a serial or reflexive R relation. Whence, we have that app X ⊆ app X ⊆ app R X ⊆ app ∼ X , as desired. 2 ∼

R

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Proof. We first verify that app X ⊆ app X . For any x ∈ app X , we have that [x]∼ ⊆ X . If R s (x) = ∅, then it is obvious by ∼ R ∼ definition that x ∈ app X . Otherwise, we may assume that R s (x) = {xi | i ∈ I } for some index set I . Because R is a symmetric R relation, we find that (xi , x) ∈ R for all i ∈ I , and moreover, we obtain by Lemma 5.1 that x ∼ xi for all i ∈ I . It follows that R s (x) ⊆ X . Hence, x ∈ app X , which leads to that app X ⊆ app X . R ∼ R We now check app R X ⊆ app ∼ X . Given any x ∈ app R X , we have that R s (x) ∩ X = ∅. Assume that x ∈ R s (x) ∩ X . Since



R is symmetric, we see that (x , x) ∈ R, and x and x are unsteady. It follows by Lemma 5.1 that x ∼ x , which means that x ∈ [x]∼ ∩ X . Hence, we obtain that x ∈ app ∼ X , which completes the proof of app R X ⊆ app ∼ X . 2

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∼

R

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For inverse serial, transitive, or Euclidean relations, in general there is no desired inclusion relationship among the approximations. The following are two instances.

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Example 5.2. Let us consider the generalized approximation space (U , R ) described by Fig. 6, which is just slightly more specific than that of Example 2.1. Obviously, R is an inverse serial relation. Nevertheless, we see that the bisimilarity ∼ is the same as the one in Example 2.1. Taking X = {s1 , t 1 , s2 , t 2 } and X = {s4 , t 4 }, we obtain that

app X = {s1 , s5 , t 5 },

app X = {s1 , t 1 },

app R X = {s2 , t 2 , t 3 },

app ∼ X = {s3 , s4 , t 4 }.

R

∼

This shows that there is no inclusion relation between app X and app X as well as between app R X and app ∼ X . R ∼

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Proposition 5.2. Let (U , R ) be a generalized approximation space. If R is a symmetric relation, then for any X ⊆ U , we have that app X ⊆ app X and app R X ⊆ app ∼ X .

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Proposition 5.1. Let (U , R ) be a generalized approximation space. If R is serial or reflexive, then for any X ⊆ U , we have that

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We see by the above lemma that F A 1 , F A 2 , C 1 , C 2 in Fig. 4 are bisimilar, as stated in Example 4.1.

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Example 5.3. Consider the generalized approximation space (U , R ) depicted in Fig. 7. Clearly, the relation R is both transitive and Euclidean. It is easy to check that ∼= Id U ∪ {(s, u ), (u , s), (t , v ), ( v , t )}. Taking X = {s, t , u }, we obtain that

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app X = {r , s, v }, R

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We end the section with a characterization of when the two kinds of approximations produce the same result.

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Theorem 5.1. Let (U , R ) be a generalized approximation space. Then both app X = app X and app R X = app ∼ X hold for all X ⊆ U R ∼ if and only if R = U × U . Proof. If R = U × U , then it is serial (and reflexive). Therefore, by Lemma 5.1 all the sates are bisimilar, and thus we have that ∼= U × U = R, which implies that app X = app X and app R X = app ∼ X for all X ⊆ U . This proves the sufficiency. R ∼ For the necessity, suppose that for all X ⊆ U , app X = app X and app R X = app ∼ X . Noting that app X ⊆ X ⊆ app ∼ X by R ∼ ∼ Proposition 3.1, we obtain that app X ⊆ X ⊆ app R X . It follows by [26, Proposition 2] that R is a serial relation. Furthermore, R in the proof of Proposition 5.1, we have obtained that ∼= U × U . If U has only an element, say s, then R = {(s, s)} because R is a serial relation. In this case, the necessity holds. For the case that U has at least 2 elements, by contradiction, we assume that R = U × U . Therefore, there is some (s, t ) ∈ / R. Consider the set X = U \{t }. Clearly, ∅ = X  U . Whence, we have that app X = app X = ∅. This means that for every x ∈ U , R s (x)  X . In particular, we see that R s (s)  X = U \{t }. R ∼ It forces that t ∈ R s (s), namely, (s, t ) ∈ R, which is a contradiction. This finishes the proof of the necessity. 2

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In some sense, the above theorem shows that there is a significant difference between the two kinds of approximations, because their consistency of approximations requires a quite strong condition on the underlying relation R.

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In the paper, we have proposed a kind of lower and upper rough approximations for generalized rough sets based on general binary relations. There are several problems which are worth further studying. Firstly, it should be noted that Janicki [19] investigated how to approximate an arbitrary relation by a relation (in particular, equivalence relation) with desired properties characterized by first-order predicate. Broadly speaking, along the same lines our approach is to approximate an arbitrary relation by a special equivalence relation with bisimilarity property. Is there any special role of the rough approximations based on bisimulations in generalized rough set theory? In particular, it should be pointed out that there is a modal logic, called Hennessy–Milner logic [17,18], for characterizing bisimilarity, and moreover, Yao [46] and Pagliani [30] introduced modal logic into the study of rough set theory. Therefore, a natural extension of our work would be approximating an arbitrary relation by a relation with desired properties characterized by modal logic. Secondly, the present work focuses on generalized approximation spaces. It would be interesting to consider the lower and upper approximations for labeled approximation spaces, where bisimilarity is really brought into play. Bringing labels into the study of rough sets is interesting itself; for example, Wu and Leung developed a block-labeled rough set model in [44]. Thirdly, it is also interesting to

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develop the lower and upper approximations by simulations [41], instead of bisimulations. Roughly speaking, simulation is a “one-sided” bisimulation, which only requires that one state is matched by the other, but does not require the converse. Fourthly, the illustrative examples in Section 4 focus on the lower and upper approximations. Noting that both simulation and bisimulation play an important role in graph pattern matching and graph compression (see, for example, [8,27]), it is desirable to apply the rough approximations based on simulations or bisimulations to graph data processing. Possible applications of the rough approximations to decision [25,39,44] and optimal approximations [20] are yet to be addressed. Finally, it remains to extend the present work to fuzzy rough sets, rough fuzzy sets, or formal fuzzy concept analysis [23] by using fuzzy bisimulations (see, for example, [3,4,6] and the references therein).

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The authors are grateful to the anonymous reviewers for pointing out some typos and suggesting a better way to organize the materials of the paper. This work was supported by the National Natural Science Foundation of China under Grants 61370053, 61370193, 61572081, and 61672107.

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