Fundamenta Informaticae - Volume 142, issue 1-4

Article Type: Other

DOI: 10.3233/FI-2015-1281

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. v-viii, 2015

Authors: Ciucci, Davide | Dubois, Didier | Prade, Henri

Article Type: Research Article

Abstract: The square of opposition is as old as logic. There has been a recent renewal of interest on this topic, due to the emergence of new structures (hexagonal and cubic) extending the square. They apply to a large variety of representation frameworks, all based on the notions of sets and relations. After a reminder about the structures of opposition, and an introduction to their gradual extensions (exemplified on fuzzy sets), the paper more particularly studies fuzzy rough sets and rough fuzzy sets in the setting of gradual structures of opposition.

Keywords: square of opposition, fuzzy set, fuzzy relation, rough set

DOI: 10.3233/FI-2015-1282

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 1-19, 2015

Authors: Inuiguchi, Masahiro

Article Type: Research Article

Abstract: In this paper we focus on generalizations of the classical rough set approach to fuzzy environments. There are two aspects of rough set approaches: classification and approximation. In the classification aspect, by rough set approaches we can classify objects into positive and negative examples of a class. On the other hand, in the approximation aspect, by rough set approaches we obtain the lower and upper approximations of a class. The former model works better in the attribute reduction while the latter model works better in the rule induction. In the setting of the classical rough set approach, the lower approximation …is nothing but the set of positive examples and the upper approximation is the complementary set of negative examples. However, these equalities do not always hold in the generalized settings. Most of fuzzy rough set models proposed earlier are defined in the classification aspect. The approaches based on those models do not always work well in approximating fuzzy subsets. In this paper we define the fuzzy rough set models in the approximation aspect. We investigate their fundamental properties and demonstrate the advantages of fuzzy set approximation. Finally we consider attribute reduction based on the proposed fuzzy rough set models. Show more

Keywords: rough sets, fuzzy sets, inclusion degree, certainty qualification, implication function

DOI: 10.3233/FI-2015-1283

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 21-51, 2015

Authors: Vluymans, Sarah | D’eer, Lynn | Saeys, Yvan | Cornelis, Chris

Article Type: Research Article

Abstract: Data used in machine learning applications is prone to contain both vague and incomplete information. Many authors have proposed to use fuzzy rough set theory in the development of new techniques tackling these characteristics. Fuzzy sets deal with vague data, while rough sets allow to model incomplete information. As such, the hybrid setting of the two paradigms is an ideal candidate tool to confront the separate challenges. In this paper, we present a thorough review on the use of fuzzy rough sets in machine learning applications. We recall their integration in preprocessing methods and consider learning algorithms in the supervised, …unsupervised and semi-supervised domains and outline future challenges. Throughout the paper, we highlight the interaction between theoretical advances on fuzzy rough sets and practical machine learning tools that take advantage of them. Show more

Keywords: fuzzy sets, rough sets, fuzzy rough sets, machine learning

DOI: 10.3233/FI-2015-1284

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 53-86, 2015

Authors: Wu, Wei-Zhi | Li, Tong-Jun | Gu, Shen-Ming

Article Type: Research Article

Abstract: Axiomatic characterizations of approximation operators are important in the study of rough set theory. In this paper, axiomatic characterizations of relation-based fuzzy rough approximation operators determined by a fuzzy implication operator ℐ are investigated. We first review the constructive definitions and properties of lower and upper ℐ-fuzzy rough approximation operators. We then propose an operator-oriented characterization of ℐ-fuzzy rough sets. We show that the lower and upper ℐ-fuzzy rough approximation operators generated by an arbitrary fuzzy relation can be described by single axioms. We further examine that ℐ-fuzzy rough approximation operators corresponding to some special types of fuzzy relations, such …as serial, reflexive, and 𝒯-transitive ones, can also be characterized by single axioms. Show more

Keywords: Approximation operators, fuzzy implication operators, fuzzy rough sets, rough sets, triangular norms

DOI: 10.3233/FI-2015-1285

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 87-104, 2015

Authors: Liu, Guilong

Article Type: Research Article

Abstract: This paper studies quasi-discrete closure spaces and fuzzy closure spaces. We show that any topological closure cT induced by a closure c is the smallest extension from a closure space to a topological closure space in both crisp and fuzzy environment, in addition, a characterization of the continuous mappings in quasi-discrete closure spaces is obtained. We propose the concept of quasi-discrete fuzzy closure spaces in the context of fuzzy sets and establish a one to one correspondence between quasi-discrete fuzzy closure spaces and reflexive fuzzy relations. We also discuss the relationship between topological closure cT …and closure c in quasi-discrete fuzzy closure spaces and show that the process from closure c to topological closure cT can be realized via the process from a reflexive fuzzy relation to its transitive closure. Show more

Keywords: Fuzzy relation, Topology, Fuzzy topological closure, Quasi-discreteness, Fuzzy set

DOI: 10.3233/FI-2015-1286

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 105-115, 2015

Authors: Lang, Guangming | Yang, Tian

Article Type: Research Article

Abstract: In practical situations, interval-valued fuzzy sets are of interest because fuzzy sets of this kind are frequently encountered. In this paper, motivated by the needs for solving imprecise problems, we generalize the concept of shadowed sets for understanding interval-valued fuzzy sets and provide a solution to compute a pair of thresholds by searching for a balance of uncertainty. Then we present three-way approximations of interval-valued fuzzy sets and a formulation for calculating the pair of thresholds using single-valued loss functions. We also compute three-way approximations of interval-valued fuzzy sets using interval-valued loss functions. Afterwards, we employ several examples to illustrate …that how to take an action for an object with an interval-valued membership grade using an interval-valued loss function. Show more

Keywords: Decision-theoretic rough sets, Interval-valued fuzzy sets, Interval-valued loss function, Shadowed sets

DOI: 10.3233/FI-2015-1287

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 117-143, 2015

Authors: Ma, Jian-Min | Yao, Yiyu

Article Type: Research Article

Abstract: Pawlak’s rough set model considers the rough approximations based on an equivalence relation. Multi-granulation rough set models concern rough approximations based on multiple equivalence relations. In this paper, we examine six types of rough set approximations in multi-granulation fuzzy approximation spaces (MGFASs). We construct a partition of the given universe based on a fuzzy binary relation in a fuzzy approximation space. Based on the partition, we introduce a pair of rough set approximations. In a multi-granulation fuzzy approximation space, by a family of fuzzy binary relations, we introduce two kinds of rough set approximations in terms of the union and …intersection of fuzzy relations, respectively. A pair of rough set approximations based on the family of fuzzy binary relations is also discussed. Furthermore, the optimistic and pessimistic multi-granulation rough set approximations are investigated due to the fuzzy binary relations in aMGFAS. Properties of these rough set approximations are demonstrated. Finally, we examine relationships of them. It is proved that the lower and upper approximations generated by a family of fuzzy binary relations are the pair nearest to the undefinable set, and the pessimistic multi-granulation lower and upper approximations are the pair farthest to the undefinable set. Show more

Keywords: multi-granulation, multi-granulation fuzzy approximation space, optimisticmulti-granulation approximation, pessimistic multi-granulation approximation

DOI: 10.3233/FI-2015-1288

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 145-160, 2015

Authors: Ma, Zhou-Ming | Mi, Ju-Sheng

Article Type: Research Article

Abstract: Multi-granulation rough set(MGRS), as a kind of fusion mechanism of different information or data, is an useful development of Pawlak rough set theory. Firstly, this paper gives an introduction for various types of MGRS, their properties and axiomatization characterizations are studied. We show that, except for the optimistic one, each of the existing MGRS means a single granulation rough set. Then, we made a comparative analysis on the different uncertainty measures among the various multi-granulation approximation spaces. At the basis of investigating for the existing uncertainty measures, we discuss their limitations via some examples, and propose a total ordered relation …among approximation spaces, even in the more general covering ones. It will be better than the original partial relation in revealing uncertainty, which conceal in the approximation space or covering one. Finally, based on the total ordered relation, we present improved information entropy, rough entropy, knowledge granulation and axiomatic definition of the knowledge granulation measures. It is proved that they are more reasonable than the original ones. Then, some novel uncertainty measures and improved fusion uncertainty measures about various granulations are also proposed. By employing these measures, granulation measures of various MGRSs are defined and studied. Show more

Keywords: Rough set, Multi-granulation, Total ordered relation, Uncertainty measure

DOI: 10.3233/FI-2015-1289

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 161-181, 2015

Authors: Qin, Keyun | Luo, Junfang | Pei, Zheng

Article Type: Research Article

Abstract: Rough set approach for knowledge discovery in incomplete information systems has been extensively studied. This paper conduct a further study of valued tolerance relation based rough approximations. We make an analysis of the existing rough approximabilities and propose a new approach for lower (upper) approximability, which is a generalization of Pawlak approximation operators for complete information system. The approach has also been generalized to fuzzy cases. Some basic properties of the approximation operators are examined.

Keywords: Rough set, tolerance relation, valued tolerance relation, lower (upper) approximability

DOI: 10.3233/FI-2015-1290

Citation: Fundamenta Informaticae, vol. 142, no. 1-4, pp. 183-194, 2015