The two sides of the theory of rough sets

doi:10.1016/j.knosys.2015.01.004

Knowledge-Based Systems

Volume 80, May 2015, Pages 67-77

https://doi.org/10.1016/j.knosys.2015.01.004 Get rights and content

Abstract

There exist two formulations of the theory of rough sets. A conceptual formulation emphasizes on the meaning and interpretation of the concepts and notions of the theory, whereas a computational formulation focuses on procedures and algorithms for constructing these notions. Except for a few earlier studies, computational formulations dominate research in rough sets. In this paper, we argue that an oversight of conceptual formulations makes an in-depth understanding of rough set theory very difficult. The conceptual and computational formulations are the two sides of the same coin; it is essential to pay equal, if not more, attention to conceptual formulations. As a demonstration, we examine and compare conceptual and computational formulations of two fundamental concepts of rough sets, namely, approximations and reducts.

Introduction

To develop and study a theory related to computation, one must precisely define its basic concepts and notions. A conceptual definition or formulation of a concept focuses on the meaning, interpretation and inherent properties of the concept, whereas a computational definition or formulation focuses on algorithms and methods for testing if something is an instance of the concept or constructing an instance of the concept. A conceptual definition, although essential for understanding, may not directly give a computationally efficient method. On the other hand, a computational definition, although suitable for computation, may push the meaning of a concept to the background and make an understanding difficult. Conceptual and computational definitions are the two sides of the same coin; it is necessary to consider both of them.

An important application of the theory of rough sets is analyzing data and reasoning about data [40], with a very strong computational orientation. It is therefore not surprising that the majority of studies of rough sets concentrate on computational definitions and formulations. A review of literature shows that there are very limited studies on conceptual formulations of rough sets, except for a few earlier studies that motivate the introduction of rough set theory [28], [29], [30], [36], [37], [56]. Many theoretical studies of rough sets introduce new concepts and produce new results without a clear interpretation of their meaning.

An overlook of studies on conceptual definitions and formulations may hinder further development of rough set theory. Therefore, we advocate a change of attention to conceptual definitions and formulations of the basic concepts and notions of rough set theory. Our goal is not to introduce new concepts per se, but to recast existing concepts and results in a setting of conceptual formulations. To demonstrate the power and value of conceptual definitions and formulations of rough set theory, we examine and compare both conceptual and computational formulations of rough set approximations and reducts.

In Section 2, we provide a brief discussion on conceptual and computational formulations across many disciplines. Section 3 examines a conceptual definition of rough set approximations in terms of the definability of sets. More specifically, the definability of sets in an information table is a primitive notion, from which rough set approximations are naturally derived. Section 4 reviews computational definitions and shows the equivalence of conceptual definitions and computational definitions. Section 5 gives a conceptual definition of a reduct of a set. Section 6 shows that attribute reducts, relative attribute reducts, attribute–value reducts, and rule reducts [12], [40] can be unified under the conceptual definition introduced in Section 5. More importantly, we have a simple explanation of Pawlak rough set analysis [40] as a three-step process [60]: (1) finding an attribute reduct to simplify an information table, (2) finding an attribute–value pair reduct to simplify the left-hand-side of a classification or decision rule, and (3) finding a rule reduct to simplify a set of classification rules. Section 7 looks at the construction of a reduct. We present three classes of algorithms by using deletion, addition-deletion, and addition strategies [61], respectively.

Section snippets

Conceptual versus computational formulations

According to Watt and van den Berg [51], concepts are the building blocks of scientific theories. A scientific concept consists of a label, a theoretical definition and an operational definition. The theoretical definition of a concept concerns the meaning of the concept. An operational definition provides ways so that the concept can be objectively observed or measured. A more detailed discussion on definitions, their purposes, and various kinds of definitions (including theoretical

A conceptual understanding of approximations

From a conceptual point of view, rough sets approximations are introduced due to the undefinability of some subsets of objects in an information table. In other words, a basic idea of rough set theory is the approximations of undefinable sets by definable sets.

Computational definitions of approximations

The conceptual Definition 3 provides a semantically sound interpretation of rough set approximations. However, it is computationally difficult to construct the approximations of a set by using the definition directly. In this section, we present a computational formulation for efficient construction of approximations and show its equivalence to the conceptual formulation.

A conceptual understanding of reducts

We give a general definition of reducts of a set [60] and examine three types of reducts in rough set theory.

Reducts and Pawlak rough set analysis

Yao and Fu [60] re-interpret Pawlak rough set analysis in terms of constructing three types of reducts in three steps. As another demonstration of the power of conceptual definitions and formulations, we review the main results in this section.

Reduct construction algorithms

If a property $P$ is not monotonic with respect to set inclusion, it is impossible to have a computationally efficient method. In this section, we will consider the class of properties with monotonicity. A conceptual understanding of a reduct offers a unified framework for defining and interpreting a reduct. To further demonstrate the value of the unified framework, we describe generic algorithms for reduct construction based on three search strategies [61].

Conclusion

When studying and applying rough set theory, it is important to realize that there exist two sides of the theory. One is a sound conceptual understanding and the other is an efficient computation. With limited studies on conceptual understanding, we are struggling with a search for a simple and elegant formulation and explanation of rough set theory. We are confused by different expressions of the same notions. It may be necessary to revisit some of the earlier studies that motivated the

Acknowledgements

This work was supported in part by a Discovery Grant from NSERC, Canada. The author thanks Professor Hamido Fujita for encouragement and constructive discussions. The author is very grateful to reviewers for their valuable and insightful comments.

References (66)

N. Azam et al.
Analyzing uncertainties of probabilistic rough set regions with game-theoretic rough sets
Int. J. Approx. Reason.
(2014)
N. Azam et al.
Game-theoretic rough sets for recommender systems
Knowl.-Based Syst.
(2014)
D. Ciucci et al.
A map of dependencies among three-valued logics
Inform. Sci.
(2013)
X.F. Deng et al.
Decision-theoretic three-way approximations of fuzzy sets
Inform. Sci.
(2014)
B.Q. Hu
Three-way decisions space and three-way decisions
Inform. Sci.
(2014)
X.Y. Jia et al.
Minimum cost attribute reduction in decision-theoretic rough set models
Inform. Sci.
(2013)
X.Y. Jia et al.
On an optimization representation of decision-theoretic rough set model
Int. J. Approx. Reason.
(2014)
D.C. Liang et al.
Systematic studies on three-way decisions with interval-valued decision-theoretic rough sets
Inform. Sci.
(2014)
D. Liu et al.
Incorporating logistic regression to decision-theoretic rough sets for classifications
Int. J. Approx. Reason.
(2014)
D. Liu et al.
Probabilistic model criteria with decision-theoretic rough sets
Inform. Sci.
(2011)

X.A. Ma et al.

Decision region distribution preservation reduction in decision-theoretic rough set model

Inform. Sci.

(2014)

E. Orłowska et al.

Expressive power of knowledge representation systems

Int. J. Man–Machine Stud.

(1984)

Z. Pawlak

Information systems, theoretical foundations

Inform. Syst.

(1981)

Z. Pawlak

Rough classification

Int. J. Man–Machine Stud.

(1984)

Y.H. Qian et al.

Multigranulation decision-theoretic rough sets

Int. J. Approx. Reason.

(2014)

J.T. Yao et al.

Decision-theoretic rough sets and beyond

Int. J. Approx. Reason.

(2014)

Y.Y. Yao

Relational interpretations of neighborhood operators and rough set approximation operators

Inform. Sci.

(1998)

Y.Y. Yao

Three-way decisions with probabilistic rough sets

Inform. Sci.

(2010)

H. Yu et al.

An automatic method to determine the number of clusters using decision-theoretic rough set

Int. J. Approx. Reason.

(2014)

X.Y. Zhang et al.

Reduction target structure-based hierarchical attribute reduction for two-category decision-theoretic rough sets

Inform. Sci.

(2014)

B. Zhou

Multi-class decision-theoretic rough sets

Int. J. Approx. Reason.

(2014)

J.R. Billif et al.

Conceptual Physics: Matter in Motion

(1969)

Z. Bonikowski

A certain conception of the calculus of rough sets

Notre Dame J. Formal Logic

(1992)

M.K. Chakraborty et al.

Rough sets: some foundational issues

Fundam. Inform.

(2013)

D. Ciucci, Orthopairs in the 1960s: historical remarks and new ideas, in: Proceedings of RSCTC 2014, LNCS(LNAI) 8536,...

H. Fujita, Research directions in subjective decision support system: case study on medical diagnosis, in: Proceedings...

H. Fujita et al.

Evidential probabilities for rough set in a case of competitiveness

J.W. Grzymala-Busse

Knowledge acquisition under uncertainty – a rough set approach

J. Intell. Robot. Syst.

(1988)

J.W. Grzymala-Busse

A new version of the rule induction system LERS

Fundam. Inform.

(1997)

K. Guttmannova et al.

Promoting conceptual understanding of statistics: definitional versus computational formulas

Teaching Psychol.

(2005)

P.G. Hewett

Conceptual Physics

(1998)

P.J. Hurley

A Concise Introduction to Logic

(2000)

H.X. Li, X.Z. Zhou, B. Huang, D. Liu, Cost-sensitive three-way decision: a sequential strategy, in: Proceedings of RSKT...

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View full text

The two sides of the theory of rough sets

Abstract

Introduction

Section snippets

Conceptual versus computational formulations

A conceptual understanding of approximations

Computational definitions of approximations

A conceptual understanding of reducts

Reducts and Pawlak rough set analysis

Reduct construction algorithms

Conclusion

Acknowledgements

Int. J. Approx. Reason.

Knowl.-Based Syst.

Inform. Sci.

Inform. Sci.

Inform. Sci.

Inform. Sci.

Int. J. Approx. Reason.

Inform. Sci.

Int. J. Approx. Reason.

Inform. Sci.

Inform. Sci.

Int. J. Man–Machine Stud.

Inform. Syst.

Int. J. Man–Machine Stud.

Int. J. Approx. Reason.

Int. J. Approx. Reason.

Inform. Sci.

Inform. Sci.

Int. J. Approx. Reason.

Inform. Sci.

Int. J. Approx. Reason.

Conceptual Physics: Matter in Motion

A certain conception of the calculus of rough sets

Notre Dame J. Formal Logic

Rough sets: some foundational issues

Fundam. Inform.

Evidential probabilities for rough set in a case of competitiveness

Knowledge acquisition under uncertainty – a rough set approach

J. Intell. Robot. Syst.

A new version of the rule induction system LERS

Fundam. Inform.

Promoting conceptual understanding of statistics: definitional versus computational formulas

Teaching Psychol.

Conceptual Physics

A Concise Introduction to Logic