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In this paper, we propose the notion of precise sets in texture spaces. Precise sets are defined by using textural sections and presections under a direlation. We obtain some properties of definability; it is proved that the family of precise sets under reflexive and transitive direlation is an Alexandroff ditopology. It is observed that sections and presections, which are approximation operators in the textural meaning, are Galois connections. Finally, effective results are given for definability by using textural precise sets. [ABSTRACT FROM AUTHOR]

Looking back to Prof. Zadeh's paradigm of Computing withWords (CWW) [28, 29, 30], one can notice that the initial attempt of such an endeavour was to set up a basic vocabulary of linguistic words, and fix their semantics based on fuzzy sets. Then a grammar was proposed to generate compound linguistic expressions based on the primitive ones, and simultaneously based on the semantic interpretations of those basic linguistic expressions a general scheme for the semantics of the rest of linguistic expressions were proposed. Sentences involving linguistic quantifiers and vague predicates constitute a fragment of natural language. In this paper, we choose this fragment of the natural language, and explore the semantics from the perspective of rough sets [13, 14, 16, 17, 18, 21]. We fix a set of basic crisp quantifiers, mainly of proportional kind. A set of vague quantifiers are proposed to lie in a close vicinity of those crisp quantifiers in the sense that a particular vague quantifier can be visualized as a blurred, may be called rough, image of a set of crisp quantifiers. Semantics of the rest of the vague quantifiers can be obtained based on the subjective perception of the interrelations among the (vague) quantifiers. [ABSTRACT FROM AUTHOR]

This paper presents a new class of Petri nets called generalised fuzzy Petri nets. The new class extends the existing fuzzy Petri nets by introducing three operators in the form of triangular norms, which are supposed to function as substitute for the min, max and * (algebraic product) operators. To demonstrate the power and the usefulness of this model, an application of the generalised fuzzy Petri nets in the domain of train traffic control is provided. The new model is more flexible than the classical one as in the former class the user has the chance to define the input/output operators. The proposed approach can be used for knowledge representation and reasoning in decision support systems. [ABSTRACT FROM AUTHOR]

The article analyses prevalent definitions of rough sets from the foundational and mathematical perspectives. In particular, the issue of language dependency in the definitions, and implications of the definitions on the issue of vagueness are discussed in detail. [ABSTRACT FROM AUTHOR]

Information system of net structures based on their calculus (a distributive lattice) is introduced and, in this context, basic notions of rough set theory are re-formulated and exemplified. [ABSTRACT FROM AUTHOR]

The aim of this paper is to present the methods and algorithms of information systems decomposition. In the paper, decomposition with respect to reducts and the so-called global decomposition are considered. Moreover, coverings of information systems by components are discussed. An essential difference between two kinds of decomposition can be observed. In general, global decomposition can deliver more components of a given information system. This fact can be treated as some kind of additional knowledge about the system. The proposed approach is based on rough set theory. To demonstrate the usefulness of this approach, we present an illustrative example coming from the economy domain. The discussed decomposition methods can be applied e.g. for design and analysis of concurrent systems specified by information systems, for automatic feature extraction, as well as for control design of systems represented by experimental data tables. [ABSTRACT FROM AUTHOR]

This study aims to create an algorithm for assessing the degree to which songs belong to genres defined a priori. Such an algorithm is not aimed at providing unambiguous classification-labelling of songs, but at producing a multidimensional description encompassing all of the defined genres. The algorithm utilized data derived from the most relevant examples belonging to a particular genre of music. For this condition to be met, data must be appropriately selected. It is based on the fuzzy logic principles, which will be addressed further. The paper describes all steps of experiments along with examples of analyses and results obtained. [ABSTRACT FROM AUTHOR]

In this article we present three inclusion functions which characterise the nearness relation between finite sets of objects defined in line with J. F. Peters, A. Skowron, and J. Stepaniuk [26]. By means of these functions we extend the notion of nearness to the graded case where one can measure the degree to which one set is near to another one. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

With the gaining popularity of rough clustering, soft computing research community is studying relationships between rough and fuzzy clustering as well as their relative advantages. Both rough and fuzzy clustering are less restrictive than conventional clustering. Fuzzy clustering memberships are more descriptive than rough clustering. In some cases, descriptive fuzzy clustering may be advantageous, while in other cases it may lead to information overload. Many applications demand use of combined approach to exploit inherent strengths of each technique. Our objective is to examine correlation between these two techniques. This paper provides an experimental description of how rough clustering results can be correlated with fuzzy clustering results. We illustrate procedural steps to map fuzzy membership clustering to rough clustering. However, such a conversion is not always necessary, especially if one only needs lower and upper approximations. Experiments also show that descriptive fuzzy clustering may not always (particularly for high dimensional objects) produce results that are as accurate as direct application of rough clustering. We present analysis of the results from both the techniques. [ABSTRACT FROM AUTHOR]

The paper studies complete and incomplete information systems, i.e. basic structures of rough set theory (RST), from the standpoint of monadic Boolean algebras (MBAs). In the first part we recall a simple characterisation of RST as a theory of MBAs. In the second part the paper discusses approximation operators obtained from incomplete information systems. In this case we lose a direct link to MBAs and thus our aim is to obtain an MBA which will additionally be interesting from the perspective of RST. In the third part we consider a simple application of approximation operators based on this MBA. [ABSTRACT FROM AUTHOR]

The paper aims to establish topological links between perception of objects (as it is defined in the framework of near sets) and classification of these objects (as it is defined in the framework of rough sets). In the near set approach, the discovery of near sets (i.e. sets containing objects with similar descriptions) starts with the selection of probe functions which provide a basis for describing and discerning objects. On the other hand, in the rough set approach, the classification of objects is based on object attributes which are collected into information systems (or data tables). As is well-known, an information system can be represented as a topological space (U, τE). If we pass froman approximation space (U,E) to the quotient space U/E, where points represent indiscernible objects of U, then U/E will be endowed with the discrete topology induced (via the canonical projection) by τE. The main objective of this paper is to show how probe functions can provide new topologies on the quotient set U/E and, in consequence, new (perceptual) topologies on U. [ABSTRACT FROM AUTHOR]

This paper establishes an explicit embedding functor from the category of the proximity spaces into the category of nearness spaces. This is done by defining methods to constructively inducing a proximity structure with a given nearness structure and vis versa. Those induction methods provide useful tools to understand the inter-relations and to translate the result between the two types of spaces. [ABSTRACT FROM AUTHOR]

In this article we discuss judgment of satisfiability of formulas of a knowledge representation language as an object classification task. Our viewpoint is that of the rough set theory (RST), and the descriptor language for Pawlak's information systems of a basic kind is taken as the study case. We show how certain analogy-based methods can be employed to judge satisfiability of formulas of that language. [ABSTRACT FROM AUTHOR]

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 Orłowska 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 Zdzisław 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. [ABSTRACT FROM AUTHOR]

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 wellknown 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. [ABSTRACT FROM AUTHOR]

The problem considered is how to model perception and identify behavioral patterns of objects changing over time in complex dynamical systems. An approach to solving this problem has been found in the context of rough set theory and methods. Rough set theory introduced by Zdzis?aw Pawlak during the early 1980s provides the foundation for the construction of classifiers, relative to what are known as temporal pattern tables. Temporal patterns can be treated as features that make it possible to approximate complex concepts. This article introduces some rough set tools for perception modeling that are developed for a system for modeling networks of classifiers. Such networks make it possible to identify behavioral patterns of objects changing over time. They are constructed using an ontology of concepts delivered by experts that engage in approximate reasoning about concepts embedded in such an ontology. We also present a method that we call a method for on-line elimination of non-relevant parts (ENP). This method was developed for on-line elimination of complex object parts that are irrelevant for identifying a given behavioral pattern. The article includes results of experiments that have been performed on data from a vehicular traffic simulator and on medical data obtained from Neonatal Intensive Care Unit in the Department of Pediatrics, Collegium Medicum, Jagiellonian University. The contribution of this article is the introduction of a network of classifiers that make it possible to identify the behavioral patterns of objects that change over time. [ABSTRACT FROM AUTHOR]

The problem considered in this paper is how to approximate sets of objects that are qualitatively but not necessarily spatially near each other. The term qualitatively near is used here to mean closeness of descriptions or distinctive characteristics of objects. The solution to this problem is inspired by the work of Zdzisław Pawlak during the early 1980s on the classification of objects by means of their attributes. This article introduces a special theory of the nearness of objects that are either static (do not change) or dynamic (change over time). The basic approach is to consider a link relation, which is defined relative to measurements associated with features shared by objects independent of their spatial relations. One of the outcomes of this work is the introduction of new forms of approximations of objects and sets of objects. The nearness of objects can be approximated using rough set methods. The proposed approach to approximation of objects is a straightforward extension of the rough set approach to approximating objects, where approximation can be considered in the context of information granules (neighborhoods). In addition, the usual rough set approach to concept approximation has been enriched by an increase in the number of granules (neighborhoods) associated with the classification of a concept as near to its approximation. A byproduct of the proposed approximation method is what we call a near set. It should also be observed that what is presented in this paper is considered a special (not a general) theory about nearness of objects. The contribution of this article is an approach to nearness as a vague concept which can be approximated from the state of objects and domain knowledge. [ABSTRACT FROM AUTHOR]

This paper introduces an approach to behavioral pattern identification as a part of a study of temporal patterns in complex dynamical systems. Rough set theory introduced by Zdzisław Pawlak during the early 1980s provides the foundation for the construction of classifiers relative to what are known as temporal pattern tables. It is quite remarkable that temporal patterns can be treated as features that make it possible to approximate complex concepts. This article introduces what are known as behavior graphs. Temporal concepts approximated by approximate reasoning schemes become nodes in behavioral graphs. In addition, we discuss some rough set tools for perception modeling that are developed for a system for modeling networks of classifiers. Such networks make it possible to recognize behavioral patterns of objects changing over time. They are constructed using an ontology of concepts delivered by experts that engage in approximate reasoning about concepts embedded in such an ontology. We also present a method that we call a method for on-line elimination of non-relevant parts (ENP). This method was developed for on-line elimination of complex object parts that are irrelevant for identifying a given behavioral pattern. The article includes results of experiments that have been performed on data from a vehicular traffic simulator useful in the identification of behavioral patterns by drivers. [ABSTRACT FROM AUTHOR]

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 Zdzisław 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. [ABSTRACT FROM AUTHOR]

Attribute reduction is an important issue in rough set theory and has already been studied from the algebra viewpoint and information viewpoint of rough set theory respectively. However, the concepts of attribute reduction based on these two different viewpoints are not equivalent to each other. In this paper, we make a comparative study on the quantitative relationship between some basic concepts of rough set theory like attribute reduction, attribute significance and core defined from these two viewpoints. The results show that the relationship between these conceptions from the two viewpoints is rather an inclusion than an equivalence due to the fact that the rough set theory discussed from the information point of view restricts attributes and decision tables more specifically than it does when considered from the algebra point of view. The identity of the two viewpoints will hold in consistent information decision tables only. That is, the algebra viewpoint and information viewpoint are equivalent for a consistent decision table, while different for an inconsistent decision table. The results are significant for the design and development of methods for information reduction. [ABSTRACT FROM AUTHOR]

The indiscernibility relation is a fundamental concept of the rough set theory. The original definition of the indiscernibility relation does not capture the situation where some of the attribute values are missing. This paper tries to enhance former works by proposing an individual treatment of missing values at the attribute or value level. The main assumption of the theses presented in this paper considers that not all missing values are semantically equal. We propose two different approaches to create an individual indiscernibility relation for a particular information system. The first relation assumes variable, but fixed semantics of missing attribute values in different columns. The second relation assumes different semantics of missing attribute values, although this variability is limited with expressive power of formulas utilizing descriptors. We provide also a comparison of flexible indiscernibility relations and missing value imputation methods. Finally we present a simple algorithm for inducing sub-optimal relations from data. [ABSTRACT FROM AUTHOR]

The Khepera robot belongs to the family of miniature mobile robots of the K-Team firm. It is used in a number of places for scientific and educational purposes. Considering its advantages (such as small size, precision of movement, ease of control), it is applied to testing different approaches in the domain of artificial intelligence. This paper describes the methodology of a control system design for the Khepera robot based on a rough set approach. The proposed approach entails a study of robot behaviour insofar as its movements are influenced by measurements fromits sensors and the choice of actions that make it possible for the robot to achieve its system goals. The constructed controller concerns the realization of some tasks such as avoiding the obstacles, reaching a target, following an obstacle, finding the way out of a labyrinth. The proposed controller has been tested on both a robot simulator and on a real robot. Our experimental results show that the proposed rough set methodology can be applied to the design of a controller for the Khepera robot. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We outline some results of our current research on developing a methodology for solving problems of spatio-temporal reasoning. We consider classifiers for complex concepts in spatio-temporal reasoning that are constructed hierarchically. We emphasise the fact that the construction of such hierarchical classifiers should be supported by domain knowledge. Approximate reasoning networks (AR networks) are proposed for approximation of reasoning schemes expressed in natural language. Such reasoning schemes are extracted from knowledge bases representing domain knowledge. This approach makes it possible to induce classifiers for complex concepts by constructing them along schemes of reasoning extracted from domain knowledge. [ABSTRACT FROM AUTHOR]