Some Remarks on the Concept of Approximations from the View of Knowledge Engineering

Some Remarks on the Concept of Approximations from the View of Knowledge Engineering

Tsau Young Lin (San Jose State University, USA), Rushin Barot (San Jose State University, USA) and Shusaku Tsumoto (Shimane Medical University, Japan)
DOI: 10.4018/978-1-4666-1743-8.ch020


The concepts of approximations in granular computing (GrC) vs. rough set theory (RS) are examined. Examples are constructed to contrast their differences in the Global GrC Model (2nd GrC Model), which, in pre-GrC term, is called partial coverings. Mathematically speaking, RS-approximations are “sub-base” based, while GrC-approximations are “base” based, where “sub-base” and “base” are two concepts in topological spaces. From the view of knowledge engineering, its meaning in RS-approximations is rather obscure, while in GrC, it is the concept of knowledge approximations.
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Rs-Approximations In (Infinite) Universe

The approximation theory of RS is well known. For preciseness, we will recall the notion here. Let U be a classical set, called the universe. Let 978-1-4666-1743-8.ch020.m01 be a partition, namely, a family of subsets, called equivalence classes, that are mutually disjoint and their union is the whole universe U. Then the pair (U, 978-1-4666-1743-8.ch020.m02) is called approximation space in RS. Pawlak introduced following two definitions. Observe that Pawlak focus on finite universe. However we allow U to be infinite.

Let X be an arbitrary subset of the universe U.

Definition (RS) 1 Let E be an arbitrary equivalence class of R.

  • 1.

    Upper approximation:978-1-4666-1743-8.ch020.m03

  • 2.

    Lower approximation:978-1-4666-1743-8.ch020.m04

This definition is the formal form of the intuitive upper and lower approximations

Definition (RS)2 Let p be an arbitrary element of U.

  • 1.

    Closure978-1-4666-1743-8.ch020.m05 = {978-1-4666-1743-8.ch020.m06, if 978-1-4666-1743-8.ch020.m07, then 978-1-4666-1743-8.ch020.m08}; note that C[X] is a closed set in the sense of topological spaces.

  • 2.

    InteriorI[X] = {978-1-4666-1743-8.ch020.m09 such that 978-1-4666-1743-8.ch020.m10 & 978-1-4666-1743-8.ch020.m11}.

In RS community, the previous definitions are directed generalized to Covering Cov by interpreting E as member of Cov .


Counter Intuitive Phenomena

In this section, we present some Counter Intuitive phenomena of approximations. The first example was generated to answer some questions raised in a conversation with Tian Yang, Guangming Lang, Jing Hao from Hunan University.

  • Example 1. Let the universe U be the real line. Let us consider the collection COV of all open half lines, namely, the sets of the following form {u | u < a} and {u | a < u} for a 978-1-4666-1743-8.ch020.m12 U. These half lines form a sub-base of the usual topology in real line; Here the “usual topology “ is a technical term referring to the topology of commonly known closure of the whole set.

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