Generalized Ideals and Co-granular Rough Sets
03 Jul 2017-pp 23-42
TL;DR: This research will be of relevance for a number of logico-algebraic approaches to rough sets that proceed from point-wise definitions of approximations and also for using alternative approximation in spatial mereological contexts involving actual contact relations.
Abstract: Lattice-theoretic ideals have been used to define and generate non granular rough approximations over general approximation spaces over the last few years by few authors. The goal of these studies, in relation based rough sets, have been to obtain nice properties comparable to those of classical rough approximations. In this research paper, these ideas are generalized in a severe way by the present author and associated semantic features are investigated by her. Granules are used in the construction of approximations in implicit ways and so a concept of co-granularity is introduced. Knowledge interpretation associable with the approaches is also investigated. This research will be of relevance for a number of logico-algebraic approaches to rough sets that proceed from point-wise definitions of approximations and also for using alternative approximations in spatial mereological contexts involving actual contact relations. The antichain based semantics invented in earlier papers by the present author also applies to the contexts considered.
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TL;DR: This paper defines several measurements to compare the granularity of neighborhood granulations, and generates “OR” and “AND” decision rules based on multigranulation fusion strategies that are employed to make decisions in the presence of disease diagnosis problems.
Abstract: Multigranulation rough set over two universes provides a new perspective to combine multiple granulation knowledge in a multigranulation space in practical reality. Note that there are always non-essential neighborhood granulations, which would affect the efficiency and quality of decision making. Therefore, selecting valuable granulations and reducing worthless ones are necessary for the application of multigranulation rough set in decision process. In this paper, we first define several measurements to compare the granularity of neighborhood granulations, using which the granulation selection with multigranulation rough set is characterized. Then, the selection algorithms in the multigranulation space are developed. Third, we generate “OR” and “AND” decision rules based on multigranulation fusion strategies. As an application, these decision rules are employed to make decisions in the presence of disease diagnosis problems. In the end, the effectiveness and efficiency of the proposed algorithms are examined with numerical experiments on selective data sets.
19 citations
29 Jun 2020
TL;DR: This expository paper is intended to explain basic aspects of granular computing from a critical perspective, their range of applications and provide directions relative to general rough sets and related formal approaches to vagueness.
Abstract: A number of nonequivalent perspectives on granular computing are known in the literature, and many are in states of continuous development. Further related concepts of granules and granulations may be incompatible in many senses. This expository paper is intended to explain basic aspects of these from a critical perspective, their range of applications and provide directions relative to general rough sets and related formal approaches to vagueness. General granular principles related to knowledge are also mentioned.
9 citations
Cites background from "Generalized Ideals and Co-granular ..."
...In the present author’s classification, a rough approximation operator may be granular (in the axiomatic sense), co-granular, pointwise, abstract or empirical [31]....
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Posted Content•
TL;DR: Higher order versions of granular operator spaces and variants are presented uniformly as partial algebraic systems for generalizing Skowron-Polkowski style of rough mereology and further algorithms grounded in mereological nearness, suited for decision-making in human-machine interaction contexts, are proposed.
Abstract: Granular operator spaces and variants had been introduced and used in theoretical investigations on the foundations of general rough sets by the present author over the last few years. In this research, higher order versions of these are presented uniformly as partial algebraic systems. They are also adapted for practical applications when the data is representable by data table-like structures according to a minimalist schema for avoiding contamination. Issues relating to valuations used in information systems or tables are also addressed. The concept of contamination introduced and studied by the present author across a number of her papers, concerns mixing up of information across semantic domains (or domains of discourse). Rough inclusion functions (\textsf{RIF}s), variants, and numeric functions often have a direct or indirect role in contaminating algorithms. Some solutions that seek to replace or avoid them have been proposed and investigated by the present author in some of her earlier papers. Because multiple kinds of solution are of interest to the contamination problem, granular generalizations of RIFs are proposed, and investigated. Interesting representation results are proved and a core algebraic strategy for generalizing Skowron-Polkowski style of rough mereology (though for a very different purpose) is formulated. A number of examples have been added to illustrate key parts of the proposal in higher order variants of granular operator spaces. Further algorithms grounded in mereological nearness, suited for decision-making in human-machine interaction contexts, are proposed by the present author. Applications of granular \textsf{RIF}s to partial/soft solutions of the inverse problem are also invented in this paper.
5 citations
Cites background from "Generalized Ideals and Co-granular ..."
...The reader may refer to [40, 12] for more on the following definition....
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01 Jan 2018
TL;DR: In this research chapter, dualities and representations of various kinds associated with the semantics of rough sets are explained, critically reviewed, new proofs have been proposed, open problems are specified and new directions are suggested.
Abstract: In this research chapter, dualities and representations of various kinds associated with the semantics of rough sets are explained, critically reviewed, new proofs have been proposed, open problems are specified and new directions are suggested. Some recent duality results in the literature are also adapted for use in rough contexts. New results are also proved on granular connections between generalized rough and L-fuzzy sets by the present author. Philosophical aspects of the concepts have also been considered by her in this research chapter.
3 citations
01 Jan 2022
1 citations
References
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01 Jan 1998
TL;DR: A survey of results on relationships between the algebraic systems derived from the approximation spaces induced by information systems and various classes of algebras of relations is presented in this article.
Abstract: A survey of results is presented on relationships between the algebraic systems derived from the approximation spaces induced by information systems and various classes of algebras of relations. Rough relation algebras are presented and it is shown that they form a discriminator variety. A characterisation of the class of representable rough relation algebras is given. The family of closure operators derived from an approximation space is abstractly characterised as certain type of Boolean algebra with operators. A representation theorem is given which says that every such an algebra is isomorphic with a similar algebra that is derived from an information system.
40 citations
"Generalized Ideals and Co-granular ..." refers background in this paper
...This is also true of the representation results in [15,16,17]....
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01 Jan 2012
TL;DR: In this paper, the authors introduce rough natural number systems from both formal and less formal perspectives, which are used to improve most rough set-theoretical measures in general Rough Set theory and to represent rough semantics.
Abstract: New concepts of rough natural number systems are introduced in this research paper from both formal and less formal perspectives. These are used to improve most rough set-theoretical measures in general Rough Set theory (RST) and to represent rough semantics. The foundations of the theory also rely upon the axiomatic approach to granularity for all types of general RST recently developed by the present author. The latter theory is expanded upon in this paper. It is also shown that algebraic semantics of classical RST can be obtained from the developed dialectical counting procedures. Fuzzy set theory is also shown to be representable in purely granule-theoretic terms in the general perspective of solving the contamination problem that pervades this research paper. All this constitutes a radically different approach to the mathematics of vague phenomena and suggests new directions for a more realistic extension of the foundations of mathematics of vagueness from both foundational and application points of view. Algebras corresponding to a concept of rough naturals are also studied and variants are characterised in the penultimate section.
39 citations
TL;DR: Two algebraic semantics for bitten rough set theory are developed over similarity spaces and their abstract granular versions of rough set theories and connections with choice based generalized rough semantics developed by the present author are considered.
Abstract: We develop two algebraic semantics for bitten rough set theory ([19]) over similarity spaces and their abstract granular versions. Connections with choice based generalized rough semantics developed in [15] by the present author and general cover based rough set theories are also considered.
26 citations
"Generalized Ideals and Co-granular ..." refers background in this paper
...Ideals and filters have been used by the present author in algebraic semantics of general rough sets in some of her earlier papers like [9,1,10,11]....
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TL;DR: In this paper, the structure of rough objects is characterized and a theory of dependence for general rough sets is developed and used to internalize the Nelson-algebra based approximate semantics developed earlier by the present author.
Abstract: Rough Sets over generalized transitive relations like proto-transitive ones have been initiated recently by the present author. In a recent paper, approximation of proto-transitive relations by other relations was investigated and the relation with rough approximations was developed towards constructing semantics that can handle fragments of structure. It was also proved that difference of approximations induced by some approximate relations need not induce rough structures. In this research, the structure of rough objects is characterized and a theory of dependence for general rough sets is developed and used to internalize the Nelson-algebra based approximate semantics developed earlier by the present author. This is part of the different semantics of PRAX developed in this paper by her. The theory of rough dependence initiated in earlier papers is extended in the process. This paper is reasonably self-contained and includes proofs and extensions of representation of objects that have not been published earlier.
26 citations
TL;DR: The special properties of the rough sets which can be constructed by means of the congruences determined by ideals of lattice are studied and an example of their application in formal concept analysis is given.
23 citations
"Generalized Ideals and Co-granular ..." refers background in this paper
...Concepts of rough ideals have also been studied by different authors in specific algebras (see for example [12,13])- these studies involve the use of rough concepts within algebras....
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