. A rapidly growing number of user and student modeling systems have employed numerical techniques for uncertainty management. The three major paradigms are those of Bayesian networks, the Dempster-Shafer theory of evidence, and fuzzy logic. In this overview, each of the first three main sections focuses on one of these paradigms. It first introduces the basic concepts by showing how they can be applied to a relatively simple user modeling problem. It then surveys systems that have applied techniques from the paradigm to user or student modeling, characterizing each system within a common framework. The final main section discusses several aspects of the usability of these techniques for user and student modeling, such as their knowledge engineering requirements, their need for computational resources, and the communicability of their results. Key words: numerical uncertainty management, Bayesian networks, Dempster-Shafer theory, fuzzy logic, user modeling, student modeling 1. Introdu...

: We present some interesting properties related to the area compensation procedure to compare fuzzy numbers. It has been proved that this method produces more than a fuzzy interval order: it induces a ranking of fuzzy numbers. Some further results are given about the transitivity property and about computational aspects. Extensions to non-normal fuzzy numbers and fuzzy quantities are also proposed. Keywords: Fuzzy numbers, ordering 1 Introduction The problem of choosing among alternatives frequently appears in various fields as approximate reasoning, mathematical programming with fuzzy information, muticriteria decision making,... Many authors have investigated the use of fuzzy sets in ranking alternatives. A canonical way to extend the natural ordering of real numbers to fuzzy numbers was suggested by Baas and Kwakernaak [2] as early as 1977. Since that time, many researchers have developed methods to compare and to rank fuzzy numbers. Some of these methods have been reviewed by Bo...

We generalize the notion of an approximation space introduced in [3]. In generalized approximation spaces we define the lower and upper set approximations. We illustrate the introduced notions with different types of relation approximation. 1 Introduction Investigations on relation approximation are well motivated both from theoretical and practical points of view. The equality approximation is fundamental for a generalization of the rough set approach [3] to the case of an indiscernibility relation being based on an approximation of the equality relations in the value sets of attributes rather than on the exact equality relations in these sets. Applications of rough set methods in process control require some good tools for function approximation. Finally, let us also mention some applications of relation approximation to discrete optimization problems [7] where approximations of input-output relations of programs are investigated. The relation approximation based on the rough set ap...

. The present state of rough set theory and its applications is presented by articles in this collection as well as by research papers listed in APPENDIX 1 to which we refer the reader. We would like to discuss here some directions for further research as well as to point to some recent results not mentioned earlier which seem to us to be of importance for development of rough set theory and its applications. In our discussion we will be guided by the following three main topics: (i) Rough Sets in Knowledge Representation; (ii) Rough Sets in Approximate Reasoning about Knowledge; (iii) New Applications and New Hardware/Software. 1 A view on extended rough set theory New applications demand new ideas and their implementations. We would like to point to important new extensions of the nowadays rough set theory. 1.1 Concept Approximation The central theme in our discussion on the lines of the first two topics is that of Concept Approximation. It does involve a description process of ...

A formal analysis of probabilities, possibilities, and fuzzy sets is presented in this paper. A number of theorems proved show that probabilities carry more information per bit than both possibilities and fuzzy sets. The cost of this higher capacity is increased computational complexity and reduced computational efficiency. The resulting tradeoff of high complexity and information capacity versus computational efficiency is discussed under the spectrum of experimental systems and applications. 1 Introduction Probabilities, possibilities, and fuzzy sets are all measures used to formalize and quantify uncertainty. There is an on-going debate regarding the appropriateness of each measure in formalizing uncertainty. Arguments in favour of probabilities can be found in [17, 2] while arguments more in favour of possibilities and fuzzy sets are presented in [16, 13]. A brief presentation and qualitative comparison of the above measures as well as MYCIN's certainty factors ([26]) and Dempster...

We generalize the notion of an approximation space introduced in [8]. In tolerance approximation spaces we define the lower and upper set approximations. We investigate some attribute reduction problems for tolerance approximation spaces determined by tolerance information systems. The tolerance relation defined by the so called uncertainty function or the positive region of a given partition of objects have been chosen as invariants in the attribute reduction process. We obtain the solutions of the reduction problems by applying boolean reasoning [1]. The solutions are represented by tolerance reducts and relative tolerance reducts. 1 Introduction We discuss a generalization of the approximation space definition introduced in [8]. Our investigations are motivated by the results of [3], [7], [18] and [20] concerning sets with the boundary regions less crisp than in the case presented in [8] as well as by papers [6], [13], [17] on relation approximation. Investigations on relati...

: We discuss problems related to information granule calculus (computing with words) [26, 27, 28, 29, 30, 31, 10, 17, 18, 19]. Our approach to information granule construction and general scheme of approximate reasoning on granules is based on rough mereology [14, 15, 16]. The schemes are constructed by agents. Our approach may be applied in the problems of approximate synthesis of complex objects (solutions) in distributed systems of intelligent agents. 1 Introduction There are three main tasks in information granule construction. First one should define information granules. Next it is necessary to show how to measure the closeness (similarity) of granules and finally how to compose information granules into more complex information granules. In the first part of the paper we introduce a notion of information granule. We will start from rough set approach [9]. A basic assumption of rough set theory is that any object x from the object universe U is perceived through an available i...

Usually the discretization problem is defined as the problem of searching for cuts (boundaries). Any set of cuts defines a partition of real values into intervals in such a way that values from one interval are not discernible. In this situation we say that the boundaries between intervals are "sharp". In the paper we consider the discretization problem defined by "soft cuts" and the reasoning scheme based on rough sets and fuzzy sets. We also propose some modifications of existing rule induction methods by using soft cuts. These methods are efficient on relatively large data tables. This paper concentrates on applications of our method for control problems. We demonstrate applications of our approach to the following two control problems: inverted pendulum balancing [5], and aircraft steering (autopilot)[9]. Keywords--- Rough Sets, Discretization, Control problems I. Introduction Some strategies for discretization of real value attributes often have to be used when we want to apply l...

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High successful applications of soft computing and fuzzy logic can be found in engineering disciplines. Current research is done for computational subject, e. g. computing with words, or computing with colors, which is one of the new ideas for the processing of color information. The main fact of this approach is the adequate representation of color and its linguistic description. This approach can be applied to different engineering fields. This work continues the idea of linguistic color processing, that we presented in [8] for the first time. Keywords: