S-Approximation: A New Approach to Algebraic Approximation

Abstract

We intend to study a new class of algebraic approximations, called -approximations, and their properties. We have shown that -approximations can be used for applied problems which cannot be modeled by inclusion based approximations. Also, in this work, we studied a subclass of -approximations, called M -approximations, and showed that this subclass preserves most of the properties of inclusion based approximations but is not necessarily inclusionbased. The paper concludes by studying some basic operations on -approximations and counting the number of -min functions. IntroductionUncertainty is often present in real-life applications. Uncertainty in noncrisp sets is characterized by nonempty boundary regions, in which nothing can be said about their elements with certainty. In classical set theory, a subset of a universe induces a partition { , − } over that universe. This partition can be interpreted as a knowledge on elements of ; that is, elements in are indiscernible to each other and also the same thing holds for items in − . This knowledge may be improved to another partition, for example, P, whose items in each partition are indiscernible to each other. In consequence, for a subset of , the problem of whether belongs to or not, with respect to knowledge P, may become undecidable; that is, some elements indiscernible to with respect to knowledge P may be in , whereas some other indiscernible elements to with knowledge P may not belong to . To cope with such uncertainty, some tools were invented such as the Dempster-Shafer theory of evidence [1], theory of fuzzy sets [2][3][4][5], and theory of rough sets [6][7][8]. Rough set theory and fuzzy set theory are two independent approaches for uncertainty. There is a connection between rough set theory and Dempster-Shafer theory. Strictly speaking, lower and upper approximations of rough set theory correspond to the inner and outer reductions from Dempster-Shafer theory [9].Rough set theory and its generalizations are all based on the inclusion relation [7,8,[10][11][12][13][14][15], which is a limitation in approximations. In this work, we introduce a new concept named -approximation set. This concept is independent of inclusion relation and contains rough sets and their generalizations as special cases. We provide some examples of approximations using this new concept, which cannot be obtained by rough set theory. This paper is organized as follows. The notion ofapproximation sets is proposed in Section 2, followed by considering some operations on them. The definition of M conditioned rough sets is proposed in Section 3 and the number of such sets is counted. Then we conclude the paper. S-ApproximationIn this section, with regard to Dempster's multivalued mappings [16], we propose a new mathematical approach to study approximation spaces and we will show that this concept can be independent of inclusion relations and the rough set and its generalizations are all special cases of this concept. where is the complement of with respect to .Example 2. Let and be none...

Neighborhood system S-approximation spaces and applications

“…In this section, we remind some preliminary facts and definitions for S-approximation spaces from [10,33]. An S-approximation space is a quadruple G = (U, W, T, S), where U and W are finite non-empty sets, T is a mapping of the form T : U → P(W ) and S is a mapping of the form S : P(W ) × P(W ) → {0, 1}.…”

“…In addition, the mapping S denotes a (large) class of measures of which, the set inclusion is a special case. For some examples of different S mappings, one can refer to [10,33]. We would like to emphasize that the mapping S is just restricted to have a form as S : P(W ) × P(W ) → {0, 1} and nothing more.…”

“…There are several tools to deal with uncertainty in these applications like the theory of rough sets and its extensions [6,22,23,25,26,43,46], fuzzy set theory [47][48][49], Dempster-Shafer theory of evidence or belief functions [7,32] and S-approximation spaces [10,33]. The concept of S-approximation spaces was introduced in [10] as a generalization of Dempster-Shafer theory of evidence and rough set theory. Then, it was studied from a three-way decision [45] viewpoint in [33].…”

On twelve types of covering-based rough sets

“… 1988 ; Pawlak 1991 , 1995 ), and its extensions (Yao 1996 , 1998 ; Skowron and Stepaniuk 1996 ; Pei and Xu 2004 ), S-approximation spaces (Hooshmandasl et al. 2014 ; Shakiba and Hooshmandasl 2015a , b ; Shakiba et al. 2016 ), granular computing (Bargiela and Pedrycz 2003 ; Lin 2002 , 2003 ) and fuzzy set theory (Zadeh 1965 , 1983 ).…”

A combinatorial approach to certain topological spaces based on minimum complement S-approximation spaces

“…Rough set theory and its generalizations are all based on the inclusion relation [24,22,18,19,41,40,39,42], which can be considered as a limitation. In this work, we use a new concept named S-approximation set which is proposed in [6]. This concept is independent from the inclusion relation and contains rough sets and their generalizations as special cases.…”