Modularization of fuzzy relational equations

 In this study, we are concerned with a modularization of fuzzy relational equations that is converting a highly dimensional (multivariable) relational equation into a series of single input fuzzy relational equations. The problem originates from a need of handling (estimating) highly dimensional relational structures and is inherently associated with the curse of dimensionality present in relational fuzzy models. We propose a two-layer architecture and discuss a detailed optimization scheme leading to the determination of the fuzzy relations occurring there. Illustrative numerical studies are also included.     

A structure of fuzzy systems for support of decision making

 The aim of this paper is to introduce the notions of operation and mapping between general fuzzy decision systems (GFDS) over some decision space (𝕍, ℂ), where 𝕍 is a set of variants and ℂ is a set of criteria. The operations between two decision systems make possible to combine several decision system of different experts and the mappings between two decision systems enable to study structural properties of such systems. Relations between utility function h:𝕍→[0,1] and operations, and further, mappings between GFDS are investigated, too.     

Analog and digital image enhancement using fuzzy expert systems

 Image enhancement is a field that is being used in various areas and disciplines. Advances in computers, microcontrollers and DSP boards have opened new horizons to digital image processing, and have opened many avenues to the design and implementation of new innovative techniques. This paper compares image enhancement via the modification of the probability density function of the gray levels with the new techniques that involves the use of knowledge-base (fuzzy expert) systems that are capable of mimicking the behavior of a human expert. A fuzzy expert system based software for image enhancement, called SmartPhotoLab has been introduced for the above purpose. Present address: A. El-Osery Dept. of Electrical Engineering, New Mexico Tech, Workman Center Rm. 247 801 Leroy place, Socorro, NM 87801 e-mail: elosery@ee.nmt.edu. This work was supported in parts by NASA grants no. NAG2–1196 and 2-1480.     

Constructions of fuzzy functions based on fuzzy equalities

 The strong (perfect) fuzzy function have been applied to approximate reasoning and vague algebra in the literature of fuzzy sets. The construction of strong (perfect) fuzzy functions possesses an important role for their applications. In the presented paper, some of the results on the construction of strong (perfect) fuzzy functions are improved, and several new and desirable results in this direction are obtained. Furthermore, it is also shown that how these results can be used to point out the connections between fuzzy functions in the classical sense and the strong (perfect) fuzzy functions.     

Adaptive fuzzy spiking neural P systems for fuzzy inference and learning

Fuzzy spiking neural P systems (in short, FSN P systems) are a novel class of distributed parallel computing models, which can model fuzzy production rules and apply their dynamic firing mechanism to achieve fuzzy reasoning. However, these systems lack adaptive/learning ability. Addressing this problem, a class of FSN P systems are proposed by introducing some new features, called adaptive fuzzy spiking neural P systems (in short, AFSN P systems). AFSN P systems not only can model weighted fuzzy production rules in fuzzy knowledge base but also can perform dynamically fuzzy reasoning. It is important to note that AFSN P systems have learning ability like neural networks. Based on neuron's firing mechanisms, a fuzzy reasoning algorithm and a learning algorithm are developed. Moreover, an example is included to illustrate the learning ability of AFSN P systems.     

Interpolation and extrapolation of fuzzy quantities – the multiple-dimensional case

 This paper deals with the problem of rule interpolation and rule extrapolation for fuzzy and possibilistic systems. Such systems are used for representing and processing vague linguistic If-Then-rules, and they have been increasingly applied in the field of control engineering, pattern recognition and expert systems. The methodology of rule interpolation is required for deducing plausible conclusions from sparse (incomplete) rule bases. The interpolation/extrapolation method which was proposed for one-dimensional input space in [4] is extended in this paper to the general n-dimensional case by using the concept of aggregation operators. A characterization of the class of aggregation operators with which the extended method preserves all the nice features of the one- dimensional method is given.     

Interrogating the structure of fuzzy cognitive maps

 Causal algebra in fuzzy cognitive maps (FCMs) plays a critical role in the analysis and design of FCMs. Improving causal algebra in FCMs to model complicated situations has been one of the major research topics in this area. In this paper we propose a dynamic causal algebra in FCMs which can improve FCMs' inference and representation capability. The dynamic causal algebra shows that the indirect, strongest, weakest and total effects a vertex influences another in the FCM not only depend on the weights along all directed paths between the two vertices but also the states of the vertices on the directed paths. Therefore, these effects are nonlinear dynamic processes determined by initial conditions and propagated in the FCM to reach a static or cyclic pattern. We test our theory with a simple example.     

Analytic fuzzy tableaux

 In this paper Beth–Smullyan's tableaux method is extended to the fuzzy propositional logic. The fuzzy tableaux method is based on the concepts of t-truth and extended graded formula. As in classical logic, it is a refutation procedure. A closed fuzzy tableau beginning with the extended graded formula [r, A] asserting that this is not t-true, is a tableau proof of the graded formula (A, r). The theorems of soundness, completeness, and decidability are proved.     

Non-commutative fuzzy Galois connections

 Fuzzy Galois connections were introduced by Bělohlávek in [4]. The structure considered there for the set of truth values is a complete residuated lattice, which places the discussion in a “commutative fuzzy world”. What we are doing in this paper is dropping down the commutativity, getting the corresponding notion of Galois connection and generalizing some results obtained by Bělohlávek in [4] and [7]. The lack of the commutative law in the structure of truth values makes it appropriate for dealing with a sentences conjunction where the order between the terms of the conjunction counts, gaining thus a temporal dimension for the statements. In this “non-commutative world”, we have not one, but two implications ([15]). As a consequence, a Galois connection will not be a pair, but a quadruple of functions, which is in fact two pairs of functions, each function being in a symmetric situation to his pair. Stating that these two pairs are compatible in some sense, we get the notion of strong L-Galois connection, a more operative and prolific notion, repairing the “damage” done by non-commutativity. Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday.     

Computer networks design using hybrid fuzzy expert systems

 Designing and configuring large computer networks to support a variety of applications and computational environments is difficult, as it not only requires highly specialized technical skills and knowledge, but also a deep understanding of a dynamic commercial market. Hybrid fuzzy expert systems integrate fuzzy expert systems and neural networks methods replacing classical hard decision methods and providing better performance than traditional techniques. In this paper, we present an integrated fuzzy expert system, machine learning, and neural networks approach to large structured computer networks design and evaluation. After presenting an overview of the system and the major research choices, we describe in detail the system's modules and present examples of its potential use.     

An entropy construction inspired by fuzzy sets

 Fuzzy dynamics is considered as a model of a general algebraic scheme based on two binary operations fulfilling a very weak distributive law. The main result is the existence of the limit defining the entropy of the general dynamical system. Present address: Katedra matematiky Fakulty prírodnych vied UMB, Tajovského 40, SK-97 401 Banská Bystrica, Slovakia E-mail: riecan@fpv.umb.sk Dedicated to Prof. Ján Jakubík on the occasion of his 80th birthday Supported by grant VEGA 1/9056/02.     

A design for a class of nonlinear systems with delay time using fuzzy logic

 The goal of this paper is to design a controller for a class of nonlinear systems with delay time using fuzzy logic. The control scheme considered in this paper integrates a fuzzy component and a sliding control component. In the former, the fuzzy system can be considered as a universal approximator to approximate the unknown functions in plant. In the latter, a variable structure control with a sector guarantees the global stability of the closed-loop system when a variable, involving tracking error, travels outside of the sector. The adaptive laws to adjust the parameters in the system are developed based on the Lyapunov synthesis approach. It is shown that the proposed adaptive controller guarantees tracking error, between the outputs of the considered system and desired␣values, to be asymptotical in decay.     

The representation of fuzzy relational production rules

We look at the representation within the framework of the approximate reasoning of relational type rules. A relational production rule consists of a rule in which one of the antecedent requirements involves the satisfaction of a relationship between two variables. An example of this type of rule is if lower and upper bounds are close then the uncertainty is low.     

How to make constrained fuzzy arithmetic efficient

 In the standard fuzzy arithmetic, the vagueness of fuzzy quantities always increases. G. J. Klir [2, 3] suggests an alternative – the constrained fuzzy arithmetic – which reduces this effect. On the other hand, it significantly increases the complexity of computations in comparison to the classical calculus of fuzzy quantities. So far, little attention was paid to the problems of implementation of the constrained fuzzy arithmetic, especially to its computational efficiency. We point out the related problems and outline the ways of their solution. We suggest to decompose the whole expression, classify all its subexpressions with respect to their individual computational complexity and precompute the corresponding subresults according to this classification.     

Genetic algorithms-based fuzzy regression analysis

 This paper describes the concept of fuzzy regression analysis based on genetic algorithms. It is shown that the performance of fuzzy regression models may be improved and fuzzy modeling technique can be simplified by incorporating genetic algorithms into regression analysis procedure. The effectiveness of the proposed approach is illustrated through simulation of fuzzy linear regression model obtained by other authors and comparison of the results. The paper further demonstrates the applications of the approach to the manufacturing and business problems.     

Solving fuzzy equations using evolutionary algorithms and neural nets

 In this paper we use evolutionary algorithms and neural nets to solve fuzzy equations. In Part I we: (1) first introduce our three solution methods for solving the fuzzy linear equation AˉXˉ + Bˉ= Cˉ; for Xˉ and (2) then survey the results for the fuzzy quadratic equations, fuzzy differential equations, fuzzy difference equations, fuzzy partial differential equations, systems of fuzzy linear equations, and fuzzy integral equations; and (3) apply an evolutionary algorithm to construct one of the solution types for the fuzzy eigenvalue problem. In Part II we: (1) first discuss how to design and train a neural net to solve AˉXˉ + Bˉ= Cˉ for Xˉ and (2) then survey the results for systems of fuzzy linear equations and the fuzzy quadratic.     

Fuzzy arithmetic-based interpolative reasoning for nonlinear dynamic fuzzy systems

FAIR (fuzzy arithmetic-based interpolative reasoning)—a fuzzy reasoning scheme based on fuzzy arithmetic, is presented here. Linguistic rules of the Mamdani type, with fuzzy numbers as consequents, are used in an inference mechanism similar to that of a Takagi–Sugeno model. The inference result is a weighted sum of fuzzy numbers, calculated by means of the extension principle. Both fuzzy and crisp inputs and outputs can be used, and the chaining of rule bases is supported without increasing the spread of the output fuzzy sets in each step. This provides a setting for modeling dynamic fuzzy systems using fuzzy recursion. The matching in the rule antecedents is done by means of a compatibility measure that can be selected to suit the application at hand. Different compatibility measures can be used for different antecedent variables, and reasoning with sparse rule bases is supported. The application of FAIR to the modeling of a nonlinear dynamic system based on a combination of knowledge-driven and data-driven approaches is presented as an example.     

基于递阶分解聚类的非线性系统递推模糊辨识

提出一种基于递阶分解聚类的递推模糊辨识方法．采用半模糊化方法对论域内的样本进行归类,根据各子集“线性化”程度评判模糊聚类的有效性,通过对性能最差的子集进行分解并辨识新增子模型的参数,逐步完成整个样本空间的模糊划分和模型辨识过程．在线辨识时采用递推最小二乘算法对模糊规则进行修正,同时可根据建模精度的要求删除性能最差的规则,并确立新模糊规则．仿真研究表明了该方法的有效性．     

About axiomatic systems of product fuzzy logic

 This short paper has two goals. The first is to show a new axiomatic system of product fuzzy logic with only one non-BL axiom which has only two variables. The second goal is to prove that there cannot be any axiomatic system of the product fuzzy logic with single non-BL axiom with only one variable.