Simplification of a fuzzy mechanism with rule combination

 In this paper, a systematic approach to reduce the complexity of a fuzzy controller with the rule combination of a fuzzy rule base is presented. The complexity of a fuzzy controller is defined to be the computation load in this work. The proposed rule combination approach can be applied to the fuzzy mechanisms with product–sum and min–max inferences. With the input membership functions indexed in sequence for each input variable, the n-dimensional fuzzy rule table is represented as vectors so that the combination of the fuzzy rule base is realizable. Then the adjacent fuzzy rules with the same output consequent are combined to have smaller size of fuzzy rule base. The fuzzy mechanism with the combined rule table is shown to have the same output with the original fuzzy mechanism (without rule combination). Thus, in many applications, the rule combination approach presented in this paper can be used to reduce the complexity of the fuzzy mechanism without degrading the performances. Moreover, the Don't Care fuzzy rules are defined and it is indicated that the number of the necessary fuzzy rules might be decreased when the Don't Care fuzzy rules are taken into consideration. Further, the properties of the simplification approach for the fuzzy rule base of the fuzzy mechanism are discussed.     

Coherence checking and propagation of lower probability bounds

 In this paper we use imprecise probabilities, based on a concept of generalized coherence (g-coherence), for the management of uncertain knowledge and vague information. We face the problem of reducing the computational difficulties in g-coherence checking and propagation of lower conditional probability bounds. We examine a procedure, based on linear systems with a reduced number of unknowns, for the checking of g-coherence. We propose an iterative algorithm to determine the reduced linear systems. Based on the same ideas, we give an algorithm for the propagation of lower probability bounds. We also give some theoretical results that allow, by suitably modifying our algorithms, the g-coherence checking and propagation by working with a reduced set of variables and/or with a reduced set of constraints. Finally, we apply our algorithms to some examples. RID="*" ID="*" This paper is a revised and substantially extended version of a previous paper by the same authors, appeared in the Proc. of the 5th Workshop on Uncertainty Processing (WUPES'2000), Jindřichu̇v Hradec, Czech Republic, June 21–24, 1–13, 2000.     

Building ensemble classifiers using belief functions and OWA operators

A pervasive task in many forms of human activity is classification. Recent interest in the classification process has focused on ensemble classifier systems. These types of systems are based on a paradigm of combining the outputs of a number of individual classifiers. In this paper we propose a new approach for obtaining the final output of ensemble classifiers. The method presented here uses the Dempster–Shafer concept of belief functions to represent the confidence in the outputs of the individual classifiers. The combing of the outputs of the individual classifiers is based on an aggregation process which can be seen as a fusion of the Dempster rule of combination with a generalized form of OWA operator. The use of the OWA operator provides an added degree of flexibility in expressing the way the aggregation of the individual classifiers is performed.     

On the exponential representation problem

 The knowledge representation problem is understood as finding a reasonable joint distribution for a collection of marginal distributions. The problem is reformulated as a “moment problem”, and the solution is searched in the exponential form. Further, an equivalent parametrization is introduced with a parameter which can be easily identified even in higher dimensional cases. The problem of evaluating relevant model characteristics is discussed. RID="*" ID="*" Supported by grant GA ČR 102/99/1137.     

Properties of the Realization of Inner Functions

 In this paper we investigate fundamental properties of state-space realizations for inner functions. We derive necessary and sufficient conditions for the inner function to have a realization such that the associated C ₀-semigroup is exponentially stable. Furthermore, we give necessary and sufficient conditions on the inner function such that the C ₀-semigroup is a group. Combining these results, we have that the C ₀-semigroup is an exponentially stable C ₀-group if and only if the inner function is the product of a constant of modulus one and a Blaschke product for which the zeros satisfy the Carleson–Newman condition and the zeros lie in a vertical strip bounded away from the imaginary axis. Date received: January 11, 1999. Date revised: May 16, 2002. RID="*" ID="*"This paper was supported by the Volkswagen Stiftung (RiP program at Oberwolfach) and by the Deutsche Forschungsgemeinschaft.     

The meaning of multiplicity of n-ary associations in UML

The concept of multiplicity in UML derives from that of cardinality in entity-relationship modeling techniques. The UML documentation defines this concept but at the same time acknowledges some lack of obviousness in the specification of multiplicities for n-ary associations. This paper shows an ambiguity in the definition given by UML documentation and proposes a clarification to this definition, as well as the use of outer and inner multiplicities as a simple extension to the current notation to represent other multiplicity constraints, such as participation constraints, that are equally valuable in understanding n-ary associations. Initial submission: 16 January 2002 / Revised submission: 17 October 2002 Published online: 2 December 2002 RID="*" ID="*"A previous shorter version of this paper was presented under the title “Semantics of the Minimum Multiplicity in Ternary Associations in UML” at The 4th International Conference on the Unified Modeling Language-UML’2001, October 1–5 2001, Toronto, Ontario, Canada, Springer Verlag, LNCS 2185, pp. 329–341.     

An approach to solving classification problems under incomplete information

A new approach to classification problems with incomplete information is described. The decision rule is designed considering the similarity of recognized objects. Designing a decision rule with the help of a general generalized variable is the basis of the approach. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 116–123, November–December 2005.     

基于未知扰动的冲突证据合成方法

自从发现Dempster合成可能导致悖论以来,冲突证据合成一直是Dempster-Shafer理论的重要研究方向之一,迄今尚未有统一的解决方法被广泛接受.提出一种新的冲突证据合成方法,即在Dempster合成之前,基于未知扰动对mass函数进行预处理,并通过预处理来解决标准化问题.与其他相关方法相比,这种新方法不仅和Dempster规则形式上一致,合成过程比较灵活,并且可以通过扩展的Bayes公式得到理论上的解释.     

Conjunctive and disjunctive combination of belief functions induced by nondistinct bodies of evidence

Dempster's rule plays a central role in the theory of belief functions. However, it assumes the combined bodies of evidence to be distinct, an assumption which is not always verified in practice. In this paper, a new operator, the cautious rule of combination, is introduced. This operator is commutative, associative and idempotent. This latter property makes it suitable to combine belief functions induced by reliable, but possibly overlapping bodies of evidence. A dual operator, the bold disjunctive rule, is also introduced. This operator is also commutative, associative and idempotent, and can be used to combine belief functions issues from possibly overlapping and unreliable sources. Finally, the cautious and bold rules are shown to be particular members of infinite families of conjunctive and disjunctive combination rules based on triangular norms and conorms.     

A taxation model

A new approach to taxation modeling is considered that takes into account the relationship between the costs of fixed capital and direct labor. The well-known Balatskii model in which the emphasis is put on fixed capital is substantially generalized. The problem of optimizing the total tax rate with respect to a Laffer-type criterion is solved. Statistical data for Ukraine over 2000–2003 are used. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 1, pp. 160–164, January–February 2007.     

The remainder in the approximation by a generalized Bernstein operator: a representation by a convex combination of second-order divided differences

We investigate the remainder of the approximation formula of a function f∈C[0,1] by means of a generalized Bernstein operator (1.3), depending on two nonnegative integer parameters, introduced by the author in 1984 in the paper [11]. The remainder is expressed in (2.1)–(2.2) by a formula generalizing the author's earlier representation (1.2) of the remainder in Bernstein's classical approximation formula. In (2.13)–(2.14) an expression is given for the remainder involving a linear functional (2.14) which is a convex combination of second-order divided differences. Received: March 1996 / Accepted: April 1996     

On the dempster-shafer framework and new combination rules

We discuss the basic concepts of the Dempster-Shafer approach, basic probability assignments, belief functions, and probability functions. We discuss how to represent various types of knowledge in this framework. We discuss measures of entropy and specificity for belief structures. We discuss the combination and extension of belief structures. We introduce some concerns associated with the Dempster rule of combination inherent in the normalization due to conflict. We introduce two alternative techniques for combining belief structures. The first uses Dempster's rule, while the second is based upon a modification of this rule. We discuss the issue of credibility of a witness.     

A generalization of the Jensen inequality

A new approach for estimating functions and functionals based on the concept of function superconvexity and a generalization of the Jensen inequality is proposed. A new concept is introduced and the associated analytical results are substantiated. Some nontrivial estimates of functions are derived. Translated from Kibemetika i Sistemnyi Analiz, No. 2, pp. 89–99, March–April, 2000.     

Consistent specification of interface suites in UML

The paper motivates and describes a model oriented approach for consistent specification of interface suites in UML. An interface suite is a coherent collection of interfaces defining interactions that transcend component boundaries. The specification of interface suites contains diagrammatic views and documentation, but it is extended with templates for structured specifications deriving from the ISpec approach. To guarantee that the specification views, documentation and templates are consistent, a specification model has been constructed. The model contains both structural and behavioural information, represented in the form of sequences of carefully designed tuples. The model provides the underlying structure for the tool supporting the design process. The tool directs the designer to specify all elements of the model in a consistent way. The specification is collected both by customized specification templates and by diagrams. The documentation and the diagram elements – both derived from the template information – are automatically generated. This prevents errors and provides specification consistency. Initial submission: 15 February 2002 / Revised submission: 20 September 2002 Published online: 2 December 2002 RID="*" ID="*"Supported by PROGRESS grant EES.5141 and ITEA DESS grant IT990211.     

RM approach for ranking of L–R type generalized fuzzy numbers

Ranking of fuzzy numbers play an important role in decision making, optimization, forecasting etc. Fuzzy numbers must be ranked before an action is taken by a decision maker. In this paper, with the help of several counter examples it is proved that ranking method proposed by Chen and Chen (Expert Syst Appl 36:6833–6842, 2009) is incorrect. The main aim of this paper is to propose a new approach for the ranking of L–R type generalized fuzzy numbers. The proposed ranking approach is based on rank and mode so it is named as RM approach. The main advantage of the proposed approach is that it provides the correct ordering of generalized and normal fuzzy numbers and it is very simple and easy to apply in the real life problems. It is shown that proposed ranking function satisfies all the reasonable properties of fuzzy quantities proposed by Wang and Kerre (Fuzzy Sets Syst 118:375–385, 2001).     

Geometry of relative plausibility and relative belief of singletons

The study of the interplay between belief and probability can be posed in a geometric framework, in which belief and plausibility functions are represented as points of simplices in a Cartesian space. Probability approximations of belief functions form two homogeneous groups, which we call “affine” and “epistemic” families. In this paper we focus on relative plausibility, belief, and uncertainty of probabilities of singletons, the “epistemic” family. They form a coherent collection of probability transformations in terms of their behavior with respect to Dempster’s rule of combination. We investigate here their geometry in both the space of all pseudo belief functions and the probability simplex, and compare it with that of the affine family. We provide sufficient conditions under which probabilities of both families coincide.     

Developing the concept of one-way functions for cryptographic security systems using achievements in chaotic dynamics

The paper shows that an “informal” interpretation of one-way functions in modern cryptography is inadequate and defines such functions in terms of information theory. This combination of complexity and information theories opens new opportunities for constructing one-way functions, whose one-way transformation is based on the ambiguity of their inverse mappings. It is shown that random mappings are promising candidates for constructing such functions. __________ Translated from Kibernetika i Sistemnyi Analiz, No. 6, pp. 136–146, November–December 2006.     

Generalized Mittag-Leffler matrix functions in game problems for evolutionary equations of fractional order

A general method for solving game problems of approach for dynamic Volterra-evolution systems is presented. This method is based on the method of resolving functions [5] and the techniques of the theory of multivalued mappings. Properties of resolving functions are studied in more detail. Cases are separated where resolving functions can be derived in an analytical form. The scheme proposed covers a wide range of functional-differential systems, in particular, integral, integro-differential, and differential-difference systems of equations that describe the dynamics of a conflict controlled process. Game problems for systems with fractional Riemann-Liouville derivatives and regularized Dzhrbashyan-Nersesyan derivatives are studied in more detail. We will call them fractal games. An important role in the presentation of solutions of such systems is played by the generalized Mittag-Leffler matrix functions, which are introduced here. The use of asymptotic representations of these functions within the framework of the scheme of the method allows us to establish sufficient conditions of resolvability of game problems. A formal definition of parallel approach is given and illustrated by game problems for systems with fractional derivatives. Translated from Kibernetika i Sistemnyi Analiz, No. 3, pp. 3–32, May–June, 2000.