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1

Generalization of Pawlak’s Approximations in hyper modules by set-valued homomorphism

Mirvakili S., Anvariyeh S. M., Davvaz B.

Foundations of Computing and Decision Sciences

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2017

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Vol. 42, No. 1

59--81

EN

The initiation and majority on rough sets for algebraic hyperstructures such as hypermodules over a hyperring have been concentrated on a congruence relation. The congruence relation, however, seems to restrict the application of the generalized rough set model for algebraic sets. In this paper, in order to solve this problem, we consider the concept of set-valued homomorphism for hypermodules and we give some examples of set-valued homomorphism. In this respect, we show that every homomorphism of the hypermodules is a set-valued homomorphism. The notions of generalized lower and upper approximation operators, constructed by means of a set-valued mapping, which is a generalization of the notion of lower and upper approximations of a hypermodule, are provided. We also propose the notion of generalized lower and upper approximations with respect to a subhypermodule of a hypermodule discuss some significant properties of them.

2

Weak Dependencies in Approximation Spaces

Ziarko W., Chen X.

Fundamenta Informaticae

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2013

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Vol. 127, nr 1-4

193--207

EN

The article reviews the basics of the variable precision rough set and the Bayesian approaches to data dependencies detection and analysis. The variable precision rough set and the Bayesian rough set theories are extensions of the rough set theory. They are focused on the recognition and modelling of set overlap-based, also referred to as probabilistic, relationships between sets. The set-overlap relationships are used to construct approximations of undefinable sets. The primary application of the approach is to analysis of weak data co-occurrence-based dependencies in probabilistic decision tables learned from data. The probabilistic decision tables are derived from data to represent the inter-data item connections, typically for the purposes of their analysis or data value prediction. The theory is illustrated with a comprehensive application example illustrating utilization of probabilistic decision tables to face image classification.

3

Soft Nearness Approximation Spaces

Öztürk M. A., İnan E.

Fundamenta Informaticae

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2013

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Vol. 124, nr 1/2

231--250

EN

In 1999, Molodtsov introduced the theory of soft sets, which can be seen as a new mathematical approach to vagueness. In 2002, near set theory was initiated by J. F. Peters as a generalization of Pawlak's rough set theory. In the near set approach, every perceptual granule is a set of objects that have their origin in the physical world. Objects that have, in some degree, affinities are considered perceptually near each other, i.e., objects with similar descriptions. Also, the concept of near groups has been investigated by İnan and Öztürk [30]. The present paper aims to combine the soft sets approach with near set theory, which gives rise to the new concepts of soft nearness approximation spaces (SNAS), soft lower and upper approximations. Moreover, we give some examples and properties of these soft nearness approximations.

4

Nearness of Visual Objects. Application of Rough Sets in Proximity Spaces

Peters J.F., Skowron A., Stepaniuk J.

Fundamenta Informaticae

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2013

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Vol. 128, nr 1-2

156--176

EN

The problem considered in this paper is how to describe and compare visual objects. The solution to this problem stems from a consideration of nearness relations in two different forms of Efremovič proximity spaces. In this paper, the visual objects are picture elements in digital images. In particular, this problem is solved in terms of the application of rough sets in proximity spaces. The basic approach is to consider the nearness of the upper and lower approximation of a set introduced by Z. Pawlak during the early 1980s as a foundation for rough sets. Two forms of nearness relations are considered, namely, a spatial EF- and a descriptive EF-relation. This leads to a study of the nearness of objects either spatially or descriptively in the approximation of a set. The nearness approximation space model developed in 2007 is refined and extended in this paper, leading to new forms of nearness approximation spaces. There is a natural transition from the two forms of nearness relations introduced in this article to the study of nearness granules.

5

Rough Set Based Reasoning About Changes

Skowron A., Stepaniuk J., Jankowski A., Bazan J.G., Swiniarski R.

Fundamenta Informaticae

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2012

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Vol. 119, nr 3/4

421-437

EN

We consider several issues related to reasoning about changes in systems interacting with the environment by sensors. In particular, we discuss challenging problems of reasoning about changes in hierarchical modeling and approximation of transition functions or trajectories. This paper can also be treated as a step toward developing rough calculus.

6

Development of Near Sets Within the Framework of Axiomatic Fuzzy Sets

Wang G., Ren Y., Liu X.

Fundamenta Informaticae

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2012

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Vol. 118, nr 3

291-304

EN

Near sets result from a generalization of rough sets, which introduced by Peters in 2006, and later formally defined in 2007. Near set theory provides a new framework for representation of objects characterized by the features that describe them. AFS (Axiomatic Fuzzy Set) theory was proposed by Liu (1998), which is a semantic methodology relating to the fuzzy theory. In this paper, a new version of near sets based on AFS theory is established, in which every object has an AFS fuzzy description with definitely semantics. The proposed approach to assessing the nearness (closeness) of objects is not defined directly using a distance metric, but depend on similarity of their fuzzy descriptions. It is also a natural linguistic description that is similar to humans perception. Moreover, an approach to set approximation based on the union of families of objects with similar fuzzy descriptions is given. The near sets based on AFS theory can be viewed as a new development of near sets within the fuzzy context.

7

Approximation Spaces in Rough–Granular Computing

Skowron A., Stepaniuk J.

Fundamenta Informaticae

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2010

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Vol. 100, nr 1/4

141-157

EN

We discuss some generalizations of the approximation space definition introduced in 1994 [24, 25]. These generalizations are motivated by real-life applications. Rough set based strategies for extension of such generalized approximation spaces from samples of objects onto their extensions are discussed. This enables us to present the uniform foundations for inducing approximations of different kinds of granules such as concepts, classifications, or functions. In particular, we emphasize the fundamental role of approximation spaces for inducing diverse kinds of classifiers used in machine learning or data mining.

8

Optimization in Discovery of Compound Granules

Jankowski A., Peters J.F., Skowron A., Stepaniuk J.

Fundamenta Informaticae

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2008

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Vol. 85, nr 1-4

249-265

EN

The problem considered in this paper is the evaluation of perception as a means of optimizing various tasks. The solution to this problem hearkens back to early research on rough set theory and approximation. For example, in 1982, Ewa Orowska observed that approximation spaces serve as a formal counterpart of perception. In this paper, the evaluation of perception is at the level of approximation spaces. The quality of an approximation space relative to a given approximated set of objects is a function of the description length of an approximation of the set of objects and the approximation quality of this set. In granular computing (GC), the focus is on discovering granules satisfying selected criteria. These criteria take inspiration from the minimal description length (MDL) principle proposed by Jorma Rissanen in 1983. In this paper, the role of approximation spaces in modeling compound granules satisfying such criteria is discussed. For example, in terms of approximation itself, this paper introduces an approach to function approximation in the context of a reinterpretation of the rough integral originally proposed by Zdzisaw Pawlak in 1993. We also discuss some other examples of compound granule discovery problems that are related to compound granules representing process models and models of interaction between processes or approximation of trajectories of processes. All such granules should be discovered from data and domain knowledge. The contribution of this article is a proposed solution approach to evaluating perception that provides a basis for optimizing various tasks related to discovery of compound granules representing rough integrals, process models, their interaction, or approximation of trajectories of discovered models of processes.

9

Distance Measures Induced by Finite Approximation Spaces and Approximation Operators

Wolski M.

Fundamenta Informaticae

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2008

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Vol. 85, nr 1-4

497-512

EN

In this paper we study metric properties of finite approximation spaces and approximation operators from Rough Set Theory. In the first part of the article we examine finite approximation spaces and finite approximation topological spaces regarded as particular instances of two basic types of information structures from the framework of Information Quanta: information quantum relational systems and property systems, respectively. In the second part of the paper is the Marczewski-Steinhaus metric discussed as a certain distance of sets defined with respect to the approximation operators. We propose two types of á la Marczewski-Steinhaus distance functions: the first type is based on the lower approximation operator; the second one is based on the upper approximation operator. These types can be defined with respect to both finite approximation spaces (information quantum relational systems) and finite approximation topological spaces (property systems), giving us four distance measure functions. In order to define a distance of sets which preserves their lower and upper approximations, one can take the sum of two respective functions.

10

Relational Data and Rough Sets

Stepaniuk J.

Fundamenta Informaticae

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2007

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Vol. 79, nr 3-4

525-539

EN

In this paper, we show that approximation spaces are basic structures for knowledge discovery from multi-relational data. The utility of approximation spaces as fundamental objects constructed for concept approximation is emphasized. Examples of basic concepts are given throughout this paper to illustrate how approximation spaces can be beneficially used in many settings.

11

Nearness of Objects: Extension of Approximation Space Model

Peters J.F., Skowron A.

Fundamenta Informaticae

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2007

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Vol. 79, nr 3-4

497-512

EN

The problem considered in this paper is the extension of an approximation space to include a nearness relation. Approximation spaces were introduced by Zdzisaw Pawlak during the early 1980s as frameworks for classifying objects by means of attributes. Pawlak introduced approximations as a means of approximating one set of objects with another set of objects using an indiscernibility relation that is based on a comparison between the feature values of objects. Until now, the focus has been on the overlap between sets. It is possible to introduce a nearness relation that can be used to determine the "nearness" of sets of objects that are possibly disjoint and, yet, qualitatively near to each other. Several members of a family of nearness relations are introduced in this article. The contribution of this article is the introduction of a nearness relation that makes it possible to extend Pawlak's model for an approximation space and to consider the extension of generalized approximations spaces.

12

Approximation Spaces Based on Relations of Similarity and Dissimilarity of Objects

Gomolińska A.

Fundamenta Informaticae

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2007

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Vol. 79, nr 3-4

319-333

EN

In this article, we aim at extension of similarity-based approximation spaces to the case, where both similarity and dissimilarity of objects are taken into account. Apart from the well-known notions of lower rough approximation, upper rough approximation, and variable-precision positive regions of concepts, adapted to our case, the notions of exterior, possibly negative region, and ignorance region of concepts are introduced and investigated.

13

Approximation Spaces and Nearness Type Structures

Wolski M.

Fundamenta Informaticae

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2007

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Vol. 79, nr 3-4

567-577

EN

The present paper investigates approximation spaces in the context of mathematical structures which axiomatise the notion of nearness. Starting with the framework of information quanta which distinguishes two levels of information structures, namely property systems (the first level) and information quantum relational systems (the second level), we shall introduce the notion of Pawlak property system. These systems correspond bijectively to finite approximation spaces, i.e. their respective information quantum relational systems. Then we characterise Pawlak property systems in terms of symmetric topological spaces. In the second part of the paper, these systems are defined by means of topological structures based on the concept of nearness. We prove that the category of Pawlak property systems is isomorphic to the category of finite topological nearness spaces and provide its additional topological characterisation.

14

Approaches to Conflict Dynamics Based on Rough Sets

Ramanna S., Peters J.F., Skowron A.

Fundamenta Informaticae

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2007

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Vol. 75, nr 1-4

453-468

EN

Conflict analysis and conflict resolution play an important role in negotiation during contract-management situations in many organizations. The issue here is how to model a combination of complex situations among agents where there are disagreements leading to a conflict situation, and there is a need for an acceptable set of agreements. Conflict situations also result due to different sets of view points about issues under negotiation. The solution to this problem stems from pioneering work on this subject by Zdzisaw Pawlak, which provides a basis for a complex conflict model encapsulating a decision system with complex decisions. Several approaches to the analysis of conflicts situations are presented in this paper, namely, conflict graphs, approximation spaces and risk patterns. An illustrative example of a requirements scope negotiation for an automated lighting system is presented. The contribution of this paper is a rough set-based requirements scope determination model and assessment mechanisms using a complex conflict model.

15

Reinforcement Learning with Approximation Spaces

Peters J.F., Henry Ch.

Fundamenta Informaticae

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2006

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Vol. 71, nr 2,3

323-349

EN

This paper introduces a rough set approach to reinforcement learning by swarms of cooperating agents. The problem considered in this paper is how to guide reinforcement learning based on knowledge of acceptable behavior patterns. This is made possible by considering behavior patterns of swarms in the context of approximation spaces. Rough set theory introduced by Zdzisaw Pawlak in the early 1980s provides a ground for deriving pattern-based rewards within approximation spaces. Both conventional and approximation space-based forms of reinforcement comparison and the actor-critic method as well as two forms of the off-policy Monte Carlo learning control method are investigated in this article. The study of swarm behavior by collections of biologically-inspired bots is carried out in the context of an artificial ecosystem testbed. This ecosystem has an ethological basis that makes it possible to observe and explain the behavior of biological organisms that carries over into the study of reinforcement learning by interacting robotic devices. The results of ecosystem experiments with six forms of reinforcement learning are given. The contribution of this article is the presentation of several viable alternatives to conventional reinforcement learning methods defined in the context of approximation spaces.

16

Possible Rough Ingredients of Concepts in Approximation Spaces

Gomolińska A.

Fundamenta Informaticae

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2006

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Vol. 72, nr 1-3

139-154

EN

We discuss the problem of rough ingredients and parts of concepts of an indiscernibility-based approximation space. The notion of a (rough) ingredient is extended to the notion of a possible (rough) ingredient, and analogously in the case of parts. The term "possible" means that a concept is perceived as a candidate for a future substitute of some ingredient. Our approach is in line with rough mereology except for allowing the empty concept for the sake of simplicity.

17

Calculi of Approximation Spaces

Skowron A., Stepaniuk J., Peters J., Swinarski R.

Fundamenta Informaticae

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2006

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Vol. 72, nr 1-3

363-378

EN

This paper considers the problem of how to establish calculi of approximation spaces. Approximation spaces considered in the context of rough sets were introduced by Zdzisaw Pawlak more than two decades ago. In general, a calculus of approximation spaces is a system for combining, describing, measuring, reasoning about, and performing operations on approximation spaces. An approach to achieving a calculus of approximation spaces that provides a basis for approximating reasoning in distributed systems of cooperating agents is considered in this paper. Examples of basic concepts are given throughout this paper to illustrate how approximation spaces can be beneficially used in many settings, in particular for complex concept approximation. The contribution of this paper is the presentation of a framework for calculi of approximation spaces useful for approximate reasoning by cooperating agents.

18

Satisfiability and Meaning of Formulas and Sets of Formulas in Approximation Spaces

Gomolińska A.

Fundamenta Informaticae

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2005

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Vol. 67, nr 1-3

77--92

EN

In this paper, we study general notions of satisfiability and meaning of formulas and sets of formulas in approximation spaces. Rather than proposing one particular form of rough satisfiability and meaning, we present a number of alternative approaches. Approximate satisfiability and meaning are important, among others, for modelling of complex systems like systems of adaptive social agents. Finally, we also touch upon derivative concepts of meaning and applicability of rules.

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Rough Sets and Vague Concepts

Skowron A.

Fundamenta Informaticae

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2005

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Vol. 64, nr 1-4

417-431

EN

The approximation space definition has evolved in rough set theory over the last 15 years. The aim was to build a unified framework for concept approximations. We present an overview of this evolution together with some operations on approximation spaces that are used in searching for relevant approximation spaces. Among such operations are inductive extensions and granulations of approximation spaces. We emphasize important consequences of the paper for research on approximation of vague concepts and reasoning about them in the framework of adaptive learning. This requires developing new approach to vague concepts going beyond the traditional rough or fuzzy approaches.

20

Modelling Complex Patterns by Information Systems

Stepaniuk J., Bazan J.G., Skowron A.

Fundamenta Informaticae

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2005

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Vol. 67, nr 1-3

203--217

EN

We outline an approach to hierarchical modelling of complex patterns that is based on operations of sums with constraints on information systems. We show that such operations can be treated as a universal tool in hierarchical modelling of complex patterns.