On the generalization of fuzzy rough sets

Rough sets and fuzzy sets have been proved to be powerful mathematical tools to deal with uncertainty, it soon raises a natural question of whether it is possible to connect rough sets and fuzzy sets. The existing generalizations of fuzzy rough sets are all based on special fuzzy relations (fuzzy similarity relations, T-similarity relations), it is advantageous to generalize the fuzzy rough sets by means of arbitrary fuzzy relations and present a general framework for the study of fuzzy rough sets by using both constructive and axiomatic approaches. In this paper, from the viewpoint of constructive approach, we first propose some definitions of upper and lower approximation operators of fuzzy sets by means of arbitrary fuzzy relations and study the relations among them, the connections between special fuzzy relations and upper and lower approximation operators of fuzzy sets are also examined. In axiomatic approach, we characterize different classes of generalized upper and lower approximation operators of fuzzy sets by different sets of axioms. The lattice and topological structures of fuzzy rough sets are also proposed. In order to demonstrate that our proposed generalization of fuzzy rough sets have wider range of applications than the existing fuzzy rough sets, a special lower approximation operator is applied to a fuzzy reasoning system, which coincides with the Mamdani algorithm.     

Necessary conditions on minimal system configuration for general MISO Mamdani fuzzy systems as universal approximators

Recent studies have shown that both Mamdani-type and Takagi-Sugeno-type fuzzy systems are universal approximators in that they can uniformly approximate continuous functions defined on compact domains with arbitrarily high approximation accuracy. In this paper, we investigate necessary conditions for general multiple-input single-output (MISO) Mamdani fuzzy systems as universal approximators with as minimal system configuration as possible. The general MISO fuzzy systems employ almost arbitrary continuous input fuzzy sets, arbitrary singleton output fuzzy sets, arbitrary fuzzy rules, product fuzzy logic AND, and the generalized defuzzifier containing the popular centroid defuzzifier as a special case. Our necessary conditions are developed under the practically sensible assumption that only a finite set of extrema of the multivariate continuous function to be approximated is available. We have first revealed a decomposition property of the general fuzzy systems: A r-input fuzzy system can always be decomposed to the sum of r simpler fuzzy systems where the first system has only one input variable, the second one two input variables, and the last one r input variables. Utilizing this property, we have derived some necessary conditions for the fuzzy systems to be universal approximators with minimal system configuration. The conditions expose the strength as well as limitation of the fuzzy approximation: (1) only a small number of fuzzy rules may be needed to uniformly approximate multivariate continuous functions that have a complicated formulation but a relatively small number of extrema; and (2) the number of fuzzy rules must be large in order to approximate highly oscillatory continuous functions. A numerical example is given to demonstrate our new results.     

The MSFAM: a modified fuzzy ARTMAP system

A fuzzy ARTMAP system is a system for incremental supervised learning of recognition categories and multi-dimensional maps in response to an arbitrary sequence of analog or binary input vectors. Fuzzy ARTMAP systems have been benchmarked against a variety of machine learning, neural networks, and genetic algorithms with considerable success. Owing to many appealing properties, fuzzy ARTMAP systems provide a natural basis for many researchers. Many different approaches have been proposed to modify fuzzy ARTMAP systems. In this paper, we propose a new approach to modifying a fuzzy ARTMAP system. We refer to the new system as the modified and simplified fuzzy ARTMAP (MSFAM) system. The aims of MSFAM systems are not only to reduce the architectural redundancy of the fuzzy ARTMAP system, but also to make extracted rules more comprehensible and concise. Four data sets were used for demonstrating the performance of the proposed MSFAM system.     

Fuzzy relation equations and reduction of fuzzy automata

We show that the state reduction problem for fuzzy automata is related to the problem of finding a solution to a particular system of fuzzy relation equations in the set of all fuzzy equivalences on its set of states. This system may consist of infinitely many equations, and finding its non-trivial solutions may be a very difficult task. For that reason we aim our attention to some instances of this system which consist of finitely many equations and are easier to solve. First, we study right invariant fuzzy equivalences, and their duals, the left invariant ones. We prove that each fuzzy automaton possesses the greatest right (resp. left) invariant fuzzy equivalence, which provides the best reduction by means of fuzzy equivalences of this type, and we give an effective procedure for computing this fuzzy equivalence, which works if the underlying structure of truth values is a locally finite residuated lattice. Moreover, we show that even better reductions can be achieved alternating reductions by means of right and left invariant fuzzy equivalences. We also study strongly right and left invariant fuzzy equivalences, which give worse reductions than right and left invariant ones, but whose computing is much easier. We give an effective procedure for computing the greatest strongly right (resp. left) invariant fuzzy equivalence, which is applicable to fuzzy automata over an arbitrary complete residuated lattice.     

Fuzzy system and CMAC network with B-spline membership/basis functions are smooth approximators

Due to their universal approximation, fuzzy system with B-spline membership functions and CMAC neural network with B-spline basis functions have been extensively used in control. In many practical applications, they are desired to approximate not only the assigned smooth function as well as its derivatives. In this paper, by designing a fuzzy system and CMAC neural network with B-spline basis functions, we prove that such a fuzzy system and CMAC can universally approximate a smooth function and its derivatives, i.e, for a given accuracy, we can approximate an arbitrary smooth function by such a fuzzy system and CMAC that not only the function is approximate within this accuracy, but its derivatives are approximated as well. The conclusions here provide solid theoretical foundation for their extensive applications. The authors would like to thank the referees for their invaluable suggestions.     

New Robust Stability Conditions and Design of Robust Stabilizing Controllers for Takagi–Sugeno Fuzzy Time-Delay Systems

In this paper, we consider robust stability and stabilization of uncertain Takagi-Sugeno fuzzy time-delay systems where uncertainties come into the state and input matrices. Some stability conditions and robust stability conditions for fuzzy time-delay systems have already been obtained in the literature. However, those conditions are rather conservative and do not guarantee the stability of a wide class of fuzzy systems. This is true in case of designing stabilizing controllers for fuzzy time-delay systems and it thus leads to a conservative fuzzy controller design as well. We first consider rather relaxed robust stability conditions of uncertain fuzzy systems. To this end, we introduce an auxiliary system to the original fuzzy time-delay system to obtain generalized delay-dependent stability conditions. Such an auxiliary system has some arbitrary matrices that generalize not only the system representation but also delay-dependent stability conditions. Conditions we obtain here are delay-dependent conditions that depend on the upper bound of time-delay, and are given in terms of linear matrix inequalities (LMIs). Then, we compare our delay-dependent stability conditions with other conditions in the literature, and show that our conditions guarantee the stability of a wider class of systems than others. Next, we consider the robust stabilization problem with memoryless and delayed state feedback controllers. Based on our generalized robust stability conditions, we obtain delay-dependent sufficient conditions for the closed-loop system to be robustly stable, and give a design method of robustly stabilizing controllers. Finally, we give three examples that illustrate our results.     

基于知识库的开放领域问答系统

问答系统能够理解用户问题,并直接返回答案。现有问答系统大多是面向领域的,仅能回答特定领域的问题。文中提出了基于大规模知识库的开放领域问答系统实现方法。该系统首先采用自定义词典分词和CRF模型相结合的方法识别问句中的主体;其次,采用模糊匹配方法将问句中的主体与知识库中实体建立链接;然后,通过相似度计算以及规则匹配等多种方法识别问句中的谓词并与知识库实体的属性建立关联;最后,进行实体消歧和答案获取。该系统平均F-Measure值为0.695 6,表明所提方法在基于知识库的开放领域问答上具有可行性。     

Sufficient and necessary conditions for Boolean fuzzy systems as universal approximators

A formula is first presented to compute the lower upper bounds on the number of fuzzy sets to achieve pre-specified approximation accuracy for an arbitrary multivariate continuous function. The necessary condition for Boolean fuzzy systems as universal approximators with minimal system configurations is then discussed. Two examples are provided to demonstrate how to design a Boolean fuzzy system in order to approximate a given continuous function with a required approximation accuracy.     

A general technique for deriving analytical structure of fuzzy controllers using arbitrary trapezoidal input fuzzy sets and Zadeh AND operator

Deriving the analytical structure of fuzzy controllers is very important as it creates a solid foundation for better understanding, insightful analysis, and more effective design of fuzzy control systems. We previously developed a technique for deriving the analytical structure of the fuzzy controllers that use Zadeh fuzzy AND operator and the symmetric, identical trapezoidal or triangular input fuzzy sets. Many fuzzy controllers use arbitrary trapezoidal/triangular input fuzzy sets that are asymmetric. At present, there exists no technique capable of deriving the analytical structure of these fuzzy controllers. Extending our original technique, we now present a novel method that can accomplish rigorously the structure derivation for any fuzzy controller, Mamdani type or TS type, that employs the arbitrary trapezoidal input fuzzy sets and Zadeh fuzzy AND operator. The new technique contains our original technique as a special case. Given the importance of PID control, we focus on Mamdani fuzzy PI and PD controllers in this paper and show in detail how to use the new technique for different configurations of the fuzzy PI/PD controllers. The controllers use two arbitrary trapezoidal fuzzy sets for each input variable, four arbitrary singleton output fuzzy sets, four fuzzy rules, Zadeh fuzzy AND operator, and the centroid defuzzifier. This configuration is more general and complicated than the Mamdani fuzzy PI/PD controllers in the current literature. It actually contains them as special cases. We call this configuration the generalized fuzzy PI/PD controller.     

Solving fuzzy linear systems

In this paper, the main aim is to develop a method for solving an arbitrary general fuzzy linear system by using the embedding approach. Considering the existing and uniqueness of fuzzy solution to n × n linear fuzzy system is done. Numerical examples are presented to illustrate the proposed model.