Communication between information systems with covering based rough sets
Introduction
The theory of rough sets [28], [29], [30], [31], proposed by Pawlak, is a mathematical tool to deal with inexact or uncertain knowledge in information systems. This theory can approximate subsets of universes by two definable subsets called lower and upper approximations and unravel knowledge hidden in information systems [8], [9], [10], [14], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [35], [36], [45], [55], [56], [57], [58], [59], [60], [61]. Another application of rough set theory is to reduce the number of attributes in databases. Given a dataset with discrete attribute values, we can find a subset of attributes that are the most informative and has the same discernible capability as the original attributes [15], [16], [18], [19], [37], [38], [39], [40], [41], [42], [43], [50], [51], [52], [53], [54].
Information system, as a mathematical model in artificial intelligence, is an important application area of rough sets. Over the last decades, there has been much work on information systems with rough sets, including some successful applications in machine learning, pattern recognition, decision analysis, and knowledge discovery in databases. These topics may be divided broadly into two classes: One is to investigate internally the information mining and processing in information systems. The other is to study information transmission or mapping between information systems, which is so called the communication between information systems [1], [11], [12], [13], [17], [20], [33], [34], [47], [48], [49], [62].
Communication is directly related to the issue of mappings of information systems while preserving their basic functions. The original motivation to study communication between information systems is to find a relatively small database which has the same results on reduction as the original database [1], [12], [13], [20]. This topic can be extended to many important application fields such as data fusion, data compression [49], and information tranmision [11], [34]. According to ideas in [11], [33], [34], communication can be explained as translating the information contained in one granular world into the granularity of another granular world and thus providing a mechanism for exchanging information with other granular worlds. From mathematical viewpoints, these communications can be considered as to compare some structures and properties of different information systems via mappings, which are useful tools to study the relationship between information systems.
The theory of rough sets combined with homomorphic mapping concepts in algebras is a new strategy to study communication between information systems. According to the viewpoint in rough sets, a rough approximation space is actually a granular information world and an information system can be seen as a combination of some approximation spaces on the same universe. Thus a mapping between approximation spaces can induce a mapping between information systems and communication between information systems can be explained as a mapping between two information systems [12], [13], [20], [33], [34]. The notion of homomorphism based on rough sets, as a special mapping between information systems, was firstly introduced by Grzymala-Busse in [12], [13]. Later, Li and Ma discussed some features of redundancy and reductions of complete information systems under some homomorphisms [20]. As showed in these works, the notion of homomorphism on information systems is useful in aggregating sets of objects, attributes, and descriptors of original systems. However, these two studies are mainly concentrated on the problem about attribute reductions under homomorphisms. They did not discuss the issue related to set approximations. Furthermore, their researches are both limited in the framework of traditional rough sets.
As we know, traditional rough set model works in case that the values of attributes are only symbolic. In reality, the values of attributes could be both symbolic and real-valued in information systems. Thus the notions of homomorphisms based on traditional rough sets cannot be applied into studying commication between information systems containing real-valued data. To deal with this problem, several consistent functions based on binary relations were introduced and investigated in [47], [49]. In [48], Wang further defined the notions of consistent functions based on fuzzy relations for constructing attribute reducts and examining some invariant properties of homomorphisms in fuzzy environments.
The theory of covering based rough sets [60] is another important generalization of classical rough sets to deal with information systems based on coverings which is so called covering information system. This theory has been amply demonstrated to be useful by successful applications in a variety of problems [2], [3], [4], [5], [6], [7], [9], [26], [35], [36], [44], [46], [63], [64], [65], [66]. Recently, Wang et al. introduced the concepts of hommorphisms with covering based rough sets in order to construct attribute reducts of covering information systems in communications[51]. A problem with Wang’s study is that many important issues were not discussed. For example, sets approximations and attribute reductions in covering decision systems have not been considered and the necessary conditions of some hommorphic properties of covering information systems were also not presented.
The article makes some new contributions to the development of the theory of communications between information systems and between decision systems. We firstly introduce the concepts of consistent functions in covering information systems. Later, we respectively define the concepts of homomorphisms between covering information systems and between covering decision systems. Under the condition of homomorphisms, we discuss some properties of covering information systems and decision systems, and present some relationships of structural features of original systems and their image systems. We find that some basic properties of original systems, such as set approximations, attribute reductions, can be reserved under the condition of homomorphisms in both covering information systems and covering decision systems.
The remainder of this paper is organized as follows. In Section 2, we review the relevant concepts in rough set theory. In Section 3, we present the definitions of consistent functions related to coverings and investigate their properties. In Section 4, we define the concepts of covering mappings between two universes and investigate their properties. In Section 5, we introduce the concepts of homomorphisms between covering information systems and study their properties. In Section 6, we present the concepts of homomorphisms between covering decision systems and study their properties. Section 7 presents conclusions.
Section snippets
Preliminaries
This section mainly reviews some basic notions related to this paper.
Consistent functions and their properties
Mapping is a basic mathematical tool to study the relationship between two sets. Similarly, we study communication between covering information systems by mapping. In order to study invariant properties of covering information systems in communications, we firstly introduce the concepts of consistent functions with respect to coverings and investigate their basic properties in this section. Let U be a nonempty set called a universe. We denote by C(U) the set of all coverings on U. Definition 3.1 Let U and V be
Covering mappings and their properties
In order to develop tools for studying communication between information systems with covering based rough sets, this section is devoted to introduce covering mappings and explore their properties. Let us first review the extension principle of classical sets.
Let f be a mapping from U to V, f: U → V, u∣ → f(u) = v ∈ V, ∀u ∈ U. The extension principle of classical sets shows that f can induce a mapping from the power set P(U) to the power set P(V) and an inverse mapping from P(V) to P(U), that is,
Homomorphisms between covering information systems and their properties
As mentioned in Introduction, communication from one information system to another one can be regarded as homomorphism between information systems from mathematical viewpoints. By means of the results of the above sections, we introduce the notions of homomorphisms as tools to study communication between covering information systems, and examine its some invariant properties.
Let f: U → V be a surjective mapping from U to V, Δ = {Ci:i = 1, 2, … , m} a family of coverings on U and Γ = {E1, E2, … , En} a family of
Homomorphisms between covering decision systems and their properties
In this section, we investigate some invariant properties of communication between differrent covering decision systems under the condition of homomorphisms. Covering decision systems include consistent covering decision systems and inconsistent decision systems. Next, we start our study on consistent covering decision systems by defining some related notions first. Definition 6.1 Let Δ = {Ci:i = 1, … , m} be a family of coverings on U, D a decision equivalence relation on U and U/D = {[x]D:x ∈ U} the decision partition
Conclusions
In this paper, we study the communications between covering information systems and between covering decision systems. We show that a mapping between two universes can induce a covering on one universe according to a given covering on the other universe. A covering information system or a covering decision system can be regarded as a combination of some covering approximation spaces on the same universe. A mapping between covering approximation spaces can be seen as a mapping between covering
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Nos. 61070242 and 71171080), the Fundamental Research Funds for the Central Universities (No. HIT.NSRIF. 2010078), the natural science foundation of Hebei Province (F2012201023), Shenzhen Key Laboratory for High Performance Data Mining with Shenzhen New Industry Development Fund under grant No. CXB201005250021A, and Scientific Research Project of Hebei University (09265631D-2).
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