Granular computing and dual Galois connection

doi:10.1016/j.ins.2007.07.008

Information Sciences

Volume 177, Issue 23, 1 December 2007, Pages 5365-5377

https://doi.org/10.1016/j.ins.2007.07.008 Get rights and content

Abstract

A covering model for granular computing in a set-theoretic setting is studied in this paper. Under this model, a zooming-in operator is redefined. Combinations of the zooming-in and zooming-out operators form two pairs of approximation operators of the original and the granulated universe of discourse. Their properties are examined in detail. For the two pairs of lower and upper approximation operators, it is proved that the duality is always true. For a generalized approximation space, the approximation representations are just the combination operators formed on the basis of the zooming-out and zooming-in operators. Relationships between these combination operators and the dual Galois connection are also analyzed.

Introduction

In the real world, information is often granular and elements within an information granule has to be dealt with as a whole rather than individually. The idea of information granularity has been explored in a number of fields such as rough sets, fuzzy sets, cluster analysis, database, machine learning, and data mining [1], [5], [6], [7], [15], [16], [19], [20], [23], [25], [26], [27]. In recent years, there is a renewed interest in granular computing [3], [8], [9], [12], [13], [14], [21], [22], [24], [28], [30], [31], [32], and it has become increasingly important in information processing, particularly in soft computing.

Granular computing is a collective term referring to theories, methodologies, techniques, and tools for the analysis of information granules, e.g., groups, classes, intervals or clusters, encountered in problem solving [19], [25], [27]. The purpose of granular computing is to seek for an approximation scheme which can effectively solve a complex problem, albeit not in the most precise way. It allows us to view a phenomenon with different levels of granularity (i.e., different grain sizes). The process of constructing information granule is called information granulation. It means that we divide objects into a series of granules, with each granule being an object set assembled via an indiscernibility relation, a similarity relation, etc.

Many models of granular computing have been proposed over the years [20], [27]. Rough set [10], [11] is perhaps the one that popularizes it. Rough set models enable us to precisely define and analyze many notions of granular computing. Though one can gain a better understanding of granular computing within the rough set framework, there are many fundamental issues which have not been thoroughly investigated. Granulation of the universe of discourse, description of granules, relationship between granules, and computing with granules, to name but a few examples, are issues that need further scrutiny. Granulation of a universe of discourse is one of the important aspects whose solution has significant bearing on granular computing.

Based on a simple granulation structure, namely a partition of the universe of discourse, Yao [21] proposed a model of granular computing. Results from rough sets, quotient space theory, belief functions, and power algebra are reformulated, re-interpreted, and combined for computation with granules. The model is constructed by granulating a finite universe of discourse through a family of pairwise disjoint subsets under an equivalence relation. For the universe of discourse and the granulated universe of discourse induced by an equivalence relation, two basic operations, called zooming-in and zooming-out, are introduced. The zooming-in operation expands an element of the granulated universe of discourse into a subset of the original universe of discourse, and hence reveals more information about it. The zooming-out operation, on the other hand, moves from the original universe of discourse into its granulated counterpart by ignoring some details. Therefore, computations in the two universes of discourse can be connected through the two operations.

The partition model is actually based on the Pawlak approximation space. In the real world, however, we seldom have a clear-cut partitioning of space. A certain degree of overlapping among the partitions is common in classification. To better model reality, Ma et al. [9] extended the space to a generalized approximation space, and formulated a covering model by granulating a finite set of a universe of discourse into a family of overlapping granules on the basis of a reflexive relation. With respect to the zooming-in and zooming-out operations, they gave a discussion on the covering model for granular computing. Under the covering model, however, relationships between subsets of the granulated universe of discourse do not hold in the original universe of discourse. Consequently, relationships between subsets of the two universes of discourse cannot be carried over to each other. Furthermore, although rough set approximations of an ordinary subset of a universe of discourse in the generalized approximation space [18], [29], [30] can be obtained by a combination of these operations, the duality may not hold. To overcome these shortcomings, it is necessary to have a new perspective on the zooming-in operation under the covering model.

This paper extends on the covering model in [9] which granulates, on the basis of a reflexive relation, a finite universe of discourse into a family of overlapping granules. We define in Section 2 the zooming-in operation for a universe of discourse and its granulated counterpart, interchangeably called the granulated universe of discourse, induced by a reflexive relation. To be in line with the literature, however, we still call it the zooming-in operation. Connections between the elements of the original universe of discourse and its granulated counterpart are studied. In Section 3, we study two pairs of approximation operators formed by the combinations of the zooming-out and zooming-in operations. They are employed to study the connections between the computations in the two universes of discourse. It is shown that in a generalized approximation space, approximation representations are just the combination of the zooming-out and zooming-in operators, and the duality is always true for different combinations. In Section 4, we discuss relationships between an inner operator, a closure operator and operators of different combinations formed by the zooming-in and zooming-out operators, and study the dual Galois connection thus formed. We then conclude the paper with a summary in Section 5.

Section snippets

Preliminaries

Let U be a finite and nonempty set called a universe of discourse, and R ⊆ U × U a binary relation on U. For any x ∈ U, the set R_s(x) = {y ∈ U; (x, y) ∈ R} is called the R-neighborhood of x. The relation R is reflexive if for all x ∈ U, x ∈ R_s(x); R is symmetric if for all x, y ∈ U, x ∈ R_s(y) implies y ∈ R_s(x); R is transitive if for all x, y, z ∈ U, x ∈ R_s(y) and y ∈ R_s(z) implies x ∈ R_s(z); and R is Euclidean if for all x, y, z ∈ U, y ∈ R_s(x) and z ∈ R_s(x) implies z ∈ R_s(y). Furthermore, R is a similarity relation on U if it is

Approximation representations on U and U/(R)

It is well known that lower and upper approximations of a classical set are also subsets of the same universe of discourse. From the definitions of the zooming-in and zooming-out operations, we can combine them to derive some new operators on the same universe of discourse. In fact, not only can we obtain approximation operators of a reflexive approximation space, but we can also obtain approximation operators of a granulated universe of discourse.

Inner operator, closure operator and dual Galois connection

In this section, we examine the relationships of an inner operator, a closure operator and different combination operators formed by the zooming-in and zooming-out operations. Furthermore, we study the dual Galois connections formed by these combination operators.

The following definitions can be found in [4]:

Definition 4.1

Let $P = (P, ⩽)$ be an ordered set. The mapping c : P → P is called a closure operator, if

(1)
$x ⩽ c (x) (for any x \in P)$ ;
(2)
$x ⩽ y \Rightarrow c (x) ⩽ c (y) (for any x, y \in P)$ ;
(3)
$c (c (x)) = c (x) (for any x \in P)$ .

Definition 4.2

Let $P = (P, ⩽)$ be an ordered set. The

Conclusion

We have studied in this paper a covering model granulated by a reflexive binary relation. Under this covering model, the conventional zooming-in operator has been redefined so that it is theoretically more general and conceptually more realistic. Using the newly constructed zooming-in and zooming-out operators, we have investigated the connections between the elements of a universe of discourse and that of its granulated counterpart. Combinations of the redefined zooming-in operator and the

Acknowledgements

The authors would like to thank the anonymous referees for their constructive comments. This work was supported by the earmarked research grant CUHK4711/06H of the Hong Kong Research Council.

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Granular computing and dual Galois connection

Abstract

Introduction

Section snippets

Preliminaries

Approximation representations on U and U/(R)

Inner operator, closure operator and dual Galois connection

Conclusion

Acknowledgements

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Information Sciences

Fuzzy Sets and Systems

Information Sciences

Information Sciences

Granular Computing: An Introduction

Rough fuzzy sets and fuzzy rough sets

International Journal of General Systems

Approximations and rough sets based on tolerances

LNAI 2005

Granular computing on binary relations I: data mining and neighborhood systems, II: rough set representations and belief functions