Approximation Properties on a Set Based on Equivalence Relations and Dominance Relations

Equivalence and dominance relations used earlier in fault diagnosis procedures are defined as relations between faults, similar to the relations used for fault collapsing. Since the basic entity of diagnostic fault simulation and test generation is a fault pair, and not a single fault, we introduce a framework where equivalence and dominance relations are defined for fault pairs. Using equivalence and dominance relations between fault pairs, we define a fault pair collapsing process, where fault pairs are removed from consideration under diagnostic fault simulation and test generation since they are guaranteed to be distinguished when other fault pairs are distinguished. Another concept, which was used earlier to enhance fault collapsing, is the level of similarity between faults. We extend this definition into a level of similarity between fault pairs and discuss its use for fault pair collapsing. The level of similarity encompasses equivalence and dominance relations between fault pairs, and extends them to allow additional fault pair collapsing.

Optimal multiprocessor task scheduling using dominance and equivalence relations

Rough set theory is a new mathematical tool to deal with vagueness and uncertainty. The classical rough set theory based on equivalence relation has made a great progress, while the equivalence relation is too harsh to meet and is extended to dominance relation in real world. It is important to investigate rough computational methods for rough set theory, which is one of the bottleneck problems in the development of rough set theory. In this article, rough computational approach to upper ap-proximation reduction (UAR) is discussed based on dominance matrix in inconsistent ordered information systems (IOIS). The algorithm of upper approximation reduction is obtained, from which we can provide approach to upper approximation reduction operated simply in inconsistent systems based on dominance relations. Finally, an example illustrates the validity of this method, and shows the method is excellent to a complicated information system.

This model for fuzzy rough sets is one of the most important parts in rough set theory. Moreover, it is based on an equivalence relation (indiscernibility relation). However, many systems are not only concerned with fuzzy sets, but also based on a dominance relation because of various factors in practice. To acquire knowledge from the systems, construction of model for fuzzy rough sets based on dominance relations is very necessary. The main aim to this paper is to study this issue. Concepts of the lower and the upper approximations of fuzzy rough sets based on dominance relations are proposed. Furthermore, model for fuzzy rough sets based on dominance relations is constructed, and some properties are discussed.

The attribute set of some information systems is composed of both regular attributes and criteria. In order to obtain information reduction of this type of information systems, equivalence relation should be defined on the regular attributes and dominance relation on the criteria. Firstly, suppose condition attributes are criteria and decision attributes are regular attributes, dominance-equivalence relation is introduced,and the Discernibility-Matrix (DM) method of reduct generation is developed and compared with the attribute significance method. Secondly,when condition attributes are the hybrid of regular attributes and criteria, equivalence-dominance relation is then defined and Discernibility-Matrix approach of reduction generation is also provided.The effectiveness of this method is shown by both theoretical proof and illustrative example.

Equivalence and dominance relations used earlier in fault diagnosis procedures are defined as relations between faults, similar to the relations used for fault collapsing. Since the basic entity of diagnostic fault simulation and test generation is a fault pair, and not a single fault, we introduce a framework where equivalence and dominance relations are defined for fault pairs. Using equivalence and dominance relations between fault pairs, we describe a fault pair collapsing process, where fault pairs are removed from consideration under diagnostic fault simulation and test generation since they are guaranteed to be distinguished when other fault pairs are distinguished. We demonstrate the full extent of fault pair collapsing by considering circuits with small numbers of inputs. We also describe an efficient fault pair collapsing procedure for larger circuits based on structural analysis

Rough set theory is an effective mathematical tool for dealing with inconsistencies in information systems. Dominance based rough set (DBRS) is an extension to the original rough set, in which the equivalence relation is replaced by a dominance relation. However, in some condition, the lower approximation of DBRS can be emptied by only one "malicious" object. The variable consistency dominance based rough set (VC-DBRS) is proposed to avoid situations like this. The core conception of VC-DBRS is introducing parameters to control the consistency of objects including in lower approximations. There are several kinds of consistency measures have been proposed, but it is difficult to compute them by manual, especially for a large data set. It is necessary to find out an approach to calculate these measures automatically. This paper proposes a new algorithm based on dominance matrices to calculate the rough membership for VC-DBRS, and then presents how to use this measure to get the lower and upper approximation.

Rough set approach is a very useful tool to handle the unclear and ambiguous data. In this approach, a set's boundary region is used to express the incorrectness. Based on an equivalence relation, we can have upper and lower approximations for rough set. As rough sets make use of the equivalence relation property, they remain rigid. Rough set theory becomes a time consuming process because we need to find all the equivalence classes. It is unreliable and inefficient for real time applications where the data sets may be very large. In this paper, we provide a solution to this problem with covering based rough set approach. Covering based rough set is an extension of rough set approach in which the equivalence relation has been relaxed. This method is based on coverings rather than partitions. This makes it more flexible than rough sets and it is more convenient for dealing with complex applications. The purpose of covering based rough set is to get more number of overlapping. We uses covering based similarity measure which gives better results as compared to rough set which uses set and sequence similarity measure.

Rough sets were originally defined in the presence of an equivalence relation. This concept is applied to several cases where the equivalence relation, or equivalently, the partition, is replaced with several relations or with a covering. Slowinski and Vanderpooten [1] extended the rough set to a case in which an equivalence relation is replaced with a similarity relation (see also Greco et al. [2]). Yao and Lin [3] and Yao [4, 5] discussed rough sets under various kinds of extended equivalence relations. Bonikowski et al. [6] investigated rough sets under a covering which is an extension of a partition. Inuiguchi and Tanino [7] have also considered rough sets under a similarity relation and a covering. Greco et al. [2] defined rough sets under an ordering relation and showed the importance of their approach in multicriteria decision making problems.

The rough set (RS) and multi-granulation rough set (MGRS) theories have been successfully extended to accommodate preference analysis by substituting the equivalence relation (ER) with the dominance relation (DR). On the other hand, bipolarity refers to the explicit handling of positive and negative aspects of data. In this paper, with the help of bipolar fuzzy preference relation (BFPR) and bipolar fuzzy preference δ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}-covering (BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C), we put forward the idea of BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C based optimistic multi-granulation bipolar fuzzy rough set (BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C-OMG-BFRS) model and BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C based pessimistic multi-granulation bipolar fuzzy rough set (BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C-PMG-BFRS) model. We examine several significant structural properties of BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C-OMG-BFRS and BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C-PMG-BFRS models in detail. Moreover, we discuss the relationship between BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C-OMG-BFRS and BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C-PMG-BFRS models. Eventually, we apply the BFPδ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\delta $$\\end{document}C-OMG-BFRS model for solving multi-criteria decision-making (MCDM). Furthermore, we demonstrate the effectiveness and feasibility of our designed approach by solving a numerical example. We further conduct a detailed comparison with certain existing methods. Last but not least, theoretical studies and practical examples reveals that our suggested approach dramatically enriches the MGRS theory and offers a novel strategy for knowledge discovery, which is practical in real-world circumstances.

Clustering is categorised as hard or soft in nature. Soft clusters may have fuzzy or rough boundaries. Rough clustering can help researchers to discover overlapping clusters in many applications such as web mining and text mining. Rough set approach is a very useful tool to handle the unclear and ambiguous data. As rough sets make use of the equivalence relation property, they remain rigid and it is unreliable and inefficient for real time applications where the datasets may be very large. In this paper, we provide a solution to this problem with covering-based rough set approach. Covering-based rough set is an extension of rough set approach in which the equivalence relation has been relaxed. This method is based on coverings rather than partitions. This makes it more flexible than rough sets and it is more convenient for dealing with complex applications. Clustering sequential data is one of the vital research tasks. We uses covering-based similarity measure which gives better results as compared to rough set which uses set and sequence similarity measure. In this paper, covering-based rough fuzzy set clustering approach is proposed to resolve the uncertainty of sequence data.

For a formal context, we define an equivalence relation on the set of attributes. Through this equivalence relation, we define the lower and upper approximation operators relative to the family of semiconcepts of the formal context. We study on the two operators with the further properties that are interesting and valuable in the theory of rough set. In addition, we research on the lattice properties of all of semiconcepts in a formal context. Using this lattice, we set up two operators, and find their approximation properties in the theory of rough set. The two ways for giving approximations generalize the idea of Pawlak rough set approximations from one universal set to two non-related universal sets. We provide examples to exam the correct of the two approximation ways in this paper.

Normally, in some complex information systems, the binary relation on domain of any attribute is just a kind of ordinary binary, which does not meet some common properties such as reflexivity, transitivity or symmetry. In view of the above-mentioned facts this paper attempts to employ FCA(Formal Concept Analysis), proposes a rough set model based on FCA, in which equivalence relations, dominance relations, similarity relations(or tolerance relations) and neighborhood relations on universe are expanded to general binary relations and problems in rough set theory are discussed based on FCA. Particularly, from the above description of complex information systems, we can see that the relation in domain of any attribute may be extremely complex, which often leads to high time complexity and space complexity in the process of knowledge acquisition. For above reason this paper introduces granular computing(GrC), which can effectively reduce the complexity to a certain extent.

The definition of basic rough sets depends upon a single equivalence relation defined on the universe or several equivalence relations taken one each taken at a time. In the view of granular computing, classical rough set theory is based upon single granulation. The basic rough set model was extended to rough set model based on multi-granulations (MGRS) in [6], where the set approximations are defined by using multi-equivalences on the universe and their properties were investigated. Using the hybridized rough fuzzy set model introduced by Dubois and Prade [2], rough fuzzy set model based on multi-granulation is introduced and studied by Wu and Kou [15]. Topological properties of rough sets introduced by Pawlak in terms of their types were recently studied by Tripathy and Mitra [11]. These were extended to the context of incomplete multi granulation by Tripathy and Raghavan [12]. Recently, the concept of multi-granulations based on rough fuzzy sets by Tripathy and Nagaraju [13]. In a recent paper, Tripathy et al [14] introduced the concept of incomplete multigranulation on rough intuitionistic fuzzy sets (MGRIFS) and studied some of its topological properties. In this paper we continue further by introducing the concept of accuracy measures on MGRIFS and prove some of their properties. Our findings are true for both complete and incomplete intuitionistic fuzzy rough set models based upon multi granulation. The concepts and results established in [13] and [14] open a new direction in the study of multigranulation for further study.KeywordsRough SetsFuzzy rough setsIntuitionistic Fuzzy Rough Setsmulti granular fuzzy rough sets and multigranular intuitionistic fuzzy rough setsaccuracy measure

Theories of fuzzy sets and rough sets are powerful mathematical tools for modelling various types of uncertainty. Molodtsov (Comput Math Appl 37:19–31, 1999 [6]) initiated a novel concept called soft sets, a new mathematical tool for dealing with uncertainties. It has been found that fuzzy sets, rough sets, and soft sets are closely related concepts (Aktas and Cagman in Inf Sci 1(77):2726–2735, 2007 [1]). Research works on soft sets are very active and progressing rapidly in these years. In 2001, Maji et al. (J Fuzzy Math 9(3):589–602, 2001 [5]) proposed the idea of intuitionistic fuzzy soft set theory and established some results on them. Based on an equivalence relation on the universe of discourse, Dubois and Prade (Int J Gen Syst 17:191–209, 1990 [3]) introduced the lower and upper approximation of fuzzy sets in a Pawlak approximation space and obtained a new notion called rough fuzzy sets. Feng et al. (Soft Compt 14:899–911, 2009 [4]) introduced lower and upper soft rough approximation of fuzzy sets in a soft approximation space and obtained a new hybrid model called soft rough fuzzy sets which is the extension of Dubois and Prade’s rough fuzzy sets. The aim of this chapter is to consider lower and upper soft rough intuitionistic fuzzy approximation of intuitionistic fuzzy sets in intuitionistic fuzzy soft approximation space (IF soft approximation space) and obtain a new hybrid model called soft rough intuitionistic fuzzy sets which can be seen as extension of both the previous work by Dubois and Prade and Feng et al.