A complex fuzzy class is characterized by a pure complex fuzzy grade of membership. Pure complex fuzzy classes are paramount in providing rich semantics for cases where the fuzzy data is periodic with a fuzzy period. Often, however, the available data is contaminated by noise, opposing expert opinions, ambiguity, and false information. This opens the door for using intuitionistic fuzzy sets theory: representing the false information via a degree of non-membership. Several researchers have identified the benefits of integrating the two concepts of complex fuzzy sets and intuitionistic fuzzy sets. Nevertheless, complex fuzzy sets allow for only one component of the degree of membership to be fuzzy. In this paper, we introduce the concept of complex intuitionistic fuzzy classes, which are characterized by pure complex intuitionistic fuzzy grade of membership. We define the basic terms and operations on complex intuitionistic fuzzy classes and provide a motivating example of relevant application.

Switched fuzzy systems can be used to describe the hybrid systems with fuzziness. However, the languages to describe the switching logic and the fuzzy subsystems are, in general, different, and this difference makes the system analysis and implementation hard. In this paper, we use differential Petri net (DPN) as a unified model to represent both the discrete logic and fuzzy dynamic processes. To exam the rationality of the representation, we prove the correctness of the representation for the discrete part and estimate the approximation accuracy of the representation for the dynamic part. Our work provides a way to analyze the switched fuzzy systems by checking the PN and to use the PN model for further system implementation. We demonstrate the benefits of our work via a case study.

The necessity of accurate localization in mobile robotics is obvious—if a robot does not know where it is, it cannot navigate accurately and reach goal locations. Robots learn about their environment via sensors. Small robots require small, efficient, and, if they are to be deployed in large numbers, inexpensive sensors. The sensors used by robots to perceive the world are inherently inaccurate, providing noisy, erroneous data, or even no data at all. Combined with estimation error due to imperfect modeling of the robot, there are many obstacles to successfully localizing in the world. Sensor fusion is used to overcome these difficulties—combining the available sensor data to derive a more accurate pose estimation for the robot. A feeling of “ready-fire-aim'' pervades the discipline—filters are chosen on little to no information, and new filters are simply tested against a few peers and claimed as superior to all others. This is folly—the most appropriate filter is seldom the newest. This article provides an overview and in-depth tutorial of all modern robot localization methods and thoroughly discusses their strengths and weaknesses to assist a robot researcher in the task of choosing the most appropriate filter for their task. © 2012 Wiley Periodicals, Inc.

Web service composition is a distributed model to construct new web service on top of existing primitive or other composite web services. However, current service technologies, including proposed composition languages, do not address the reliability of web service composition. Thus, it is hard to predict the system reliability. In this paper, we propose a method to compute system reliability based on Service Component Architecture (SCA), a standard that provides a language-independent way to define and compose service components in the system. We first present a formal service component signature model with respect to the specification of the SCA assembly model, and then propose a language-independent dynamic behavior model for specifying the interface behavior of the service component by port activities. Then, the failure behaviors of ports are defined through the Nonhomogeneous Poisson Process (NHPP). Based on the semantics of ports, several rules have been generated to compute reliability of port expressions, thus the overall system reliability can be automatically computed. An Online Shop example from IBM web site is given to illustrate our method, together with a testing bed to calculate port reliability.

Software quality is one of the main concerns of software users. Hence, software testing is an utterly important phase in the software development life cycle. Nevertheless, manual evaluation of program compliance with its specification may be prohibitively time consuming. As a remedy, several software testing systems are using an automatic oracle to confirm that the developed software complies with its specification and determine whether a given test case exposes faults. The use of artificial neural networks and info-fuzzy networks as automated oracles has been explored elsewhere. Nevertheless, there is not enough research comparing these two popular approaches to automated evaluation of the test outcome. This paper fills the gap and reports on a set of experiments designed to compare the two methods based on ROC curves, training time, and dispersion analysis.

A key problem in traffic engineering is the optimization of the flow of vehicles through urban intersections by improving the timing policy of traffic signals. Current methods of signal control policy are based on the junction topography and prespecified static traffic volumes. However, the actual daily traffic volumes can be affected by many time-dependent factors making a static policy hardly optimal. In this paper, we induce nonstationary predictive models of traffic flow by applying novel methods of time-series data mining to the traffic sensors data collected from a signalized intersection in Jerusalem over a period of 3 years. Our methodology for modeling dynamic traffic volumes combines clustering and segmentation algorithms. The results of a case study based on real-world traffic data demonstrate that a dynamic signal policy using the data mining approach can produce a decrease of about 33.7% in the total waiting time of drivers during 1 year, in comparison to the existing static traffic policy. The resulting savings for this junction only would be about 13,800 driving hours, which are worth of about $52,000 per annum in terms of Israeli economy. © 2011 Wiley Periodicals, Inc.

Complex fuzzy sets utilize a complex degree of membership, represented in polar coordinates, which is a combination of a degree of membership in a fuzzy set along with a crisp phase value that denotes position within the set. The compound value carries more information than a traditional fuzzy set and enables efficient reasoning. In this paper, we present a new and generalized interpretation of a complex grade of membership, where a complex membership grade defines a complex fuzzy class. The new definition provides rich semantics that is not readily available through traditional fuzzy sets or complex fuzzy sets and is not limited to a compound of crisp cyclical data with fuzzy data. Furthermore, the two components of the complex fuzzy class carry fuzzy information. A complex class is represented either in Cartesian or in polar coordinates where both axes induce fuzzy interpretation. Another novelty of the scheme is that it enables representing an infinite set of fuzzy sets. The paper provides the new definition of complex fuzzy classes along with axiomatic definition of basic operations on complex fuzzy classes. In addition, coordinate transformation as well as an extension from two-dimensional fuzzy classes to n-dimensional fuzzy classes are presented. © 2011 Wiley Periodicals, Inc.

In this paper, a new method is proposed to measure the performance of service composition. Service composition described with Business Process Execution Language (BPEL) is modeled by a group of fuzzy differential equations, where each equation describes the state change of the service composition. Each service state is measured by a time-dependent fuzzy number that indicates the extent to which the state can be reached in execution. This information of measure can help us to conduct performance analysis, such as estimating response time, throughput, efficiency, and maximum load. This method has the following advantages: 1) It can handle fuzziness such as nondeterministic response time; 2) it treats the system as a box and displays a global picture of execution states to the users so that users know exactly where to improve the performance; 3) it can entirely avoid the state explosion problem, which may be hit by the existing performance analysis methods. A case study demonstrates the advantages of our method.

Localization is arguably the most important goal for a robot to solve-without knowledge of its place in the world, a robot cannot do useful work. The current best practice for providing accurate localization given uncertain sensors involves the use of sensor fusion, or combining sensor data in order to derive a better pose estimation for the robot. Small robots add another problem that needs to be solved-limited power, computational ability, and weight/space constrain both the sensors available and the filters that can be used. This paper provides a detailed example in simulation of four filter types: the extended Kalman filter, the Fuzzy EKF, the sigma-point KF (SPKF), and the double fuzzy SPKF, while discussing the strengths and weaknesses of all current state of the art sensor methods. While the field is relatively mature, there has been little to no comparative analysis of different filters performed-what roles do they best serve, how to select the “best” filter, and what tradeoffs must be made for each type. This paper analyzes the current state of the art filters of all categories and determines their applicability to the small robot problem.

The majority of data available in most disciplines is unlabeled and unclassified. The amount of data is often massive, hence scalable processing methods are required. One method of providing structure to unlabeled data is to group it by clustering. Density based methods discover the number of clusters. Additionally, the shape of such clusters can also be irregular. In this paper we examine a version of DBSCAN modified to use fuzzy membership functions (FN-DBSCAN). FN-DBSCAN was implemented using the WEKA data mining framework and a scalable technique (SFN-DBSCAN) is simulated using the framework. Experimental results show that SFN-DBSCAN can be over three times as fast as FN-DBSCAN for small to medium size data. The resulting cluster assignments match at an average rate of 90% when compared with assignments by FN-DBSCAN. SFN-DBSCAN's speed increases proportionally with respect to the number of subsets, but cluster assignment concurrence between FN-DBSCAN and SFN-DBSCAN suffers from degradation as the number of subsets increase.

In this article, we propose the method of the multicriteria alternative selection under uncertainty. The basis of the method is the principle of the Bellman–Zadeh fuzzy measures intersection and nine-point linguistic rating scale of Saaty. The novelty of the method presented here consists of the fact that it does not require labor-intensive procedures, requiring arraying and array processing of paired comparisons matrix. Instead, special correlations are used, which are based on the comparison with the worst alternative and the least important criterion. As an example for the utilization of our method, we use the problem for choosing cars. © 2010 Wiley Periodicals, Inc.

Abstract Terrorist groups use the Web as their infrastructure for various purposes. One example is the forming of new local cells that may later become active and perform acts of terror. The Advanced Terrorist Detection System (ATDS), is aimed at tracking down online access to abnormal content, which may include terrorist‐generated sites, by analyzing the content of information accessed by the Web users. ATDS operates in two modes: the training mode and the detection mode. In the training mode, ATDS determines the typical interests of a prespecified group of users by processing the Web pages accessed by these users over time. In the detection mode, ATDS performs real‐time monitoring of the Web traffic generated by the monitored group, analyzes the content of the accessed Web pages, and issues an alarm if the accessed information is not within the typical interests of that group and similar to the terrorist interests. An experimental version of ATDS was implemented and evaluated in a local network environment. The results suggest that when optimally tuned the system can reach high detection rates of up to 100% in case of continuous access to a series of terrorist Web pages.

Most Web content categorization methods are based on the vector space model of information retrieval. One of the most important advantages of this representation model is that it can be used by both instance-based and model-based classifiers. However, this popular method of document representation does not capture important structural information, such as the order and proximity of word occurrence or the location of a word within the document. It also makes no use of the markup information that can easily be extracted from the Web document HTML tags. A recently developed graph-based Web document representation model can preserve Web document structural information. It was shown to outperform the traditional vector representation using the k-Nearest Neighbor (k-NN) classification algorithm. The problem, however, is that the eager (model-based) classifiers cannot work with this representation directly. In this article, three new hybrid approaches to Web document classification are presented, built upon both graph and vector space representations, thus preserving the benefits and overcoming the limitations of each. The hybrid methods presented here are compared to vector-based models using the C4.5 decision tree and the probabilistic Naïve Bayes classifiers on several benchmark Web document collections. The results demonstrate that the hybrid methods presented in this article outperform, in most cases, existing approaches in terms of classification accuracy, and in addition, achieve a significant reduction in the classification time. © 2008 Wiley Periodicals, Inc.

This paper presents a novel framework for predicting stock trends and making financial trading decisions based on a combination of data and text mining techniques. The prediction models of the proposed system are based on the textual content of time-stamped Web documents in addition to traditional numerical time series data, which is also available from the Web. The financial trading system based on the model predictions (ADMIRAL) is using three different trading strategies. In this paper, the ADMIRAL system is simulated and evaluated on real-world series of news stories and stocks data using the C4.5 decision tree induction algorithm. The main performance measures are the predictive accuracy of the induced models and, more importantly, the profitability of each trading strategy using these predictions

Software-reliability models (SRMs) are used for the assessment and improvement of reliability in software systems. These models are normally based on stochastic processes, with the nonhomogeneous Poisson process being one of the most prominent model forms. An underlying assumption of these models is that software failures occur randomly in time. This assumption has never been quantitatively tested. Our contribution in this paper is to conduct an experimental investigation that contrasts random processes with nonlinear deterministic processes as a model for software failures. We study two sets of real-world software-reliability data using the techniques of chaotic time-series analysis. We have found that both appear to arise from a deterministic process, rather than a stochastic process, and that both show some evidence of chaotic dynamics. In addition, we have conducted a series of k-steps-ahead forecasting experiments in the datasets, pitting a number of well-known stochastic SRMs against radial basis function networks (RBFNs), which are deterministic in nature. The out-of-sample prediction results from the RBFNs showed an improvement of roughly 25% over the best of the stochastic models, for both of our datasets. Finally, we propose a causal model to explain these results, which hypothesizes that faults in a program are distributed over a fractal subset of the program's input space

Many of the computational intelligence techniques currently used do not scale well in data type or computational performance, so selecting the right dimensionality reduction technique for the data is essential. By employing a dimensionality reduction technique called representative dissimilarity to create an embedded space, large spaces of complex patterns can be simplified to a fixed-dimensional Euclidean space of points. The only current suggestions as to how the representatives should be selected are principal component analysis, projection pursuit, and factor analysis. Several alternative representative strategies are proposed and empirically evaluated on a set of term vectors constructed from HTML documents. The results indicate that using a representative dissimilarity representation with at least 50 representatives can achieve a significant increase in classification speed, with a minimal sacrifice in accuracy, and when the representatives are selected randomly, the time required to create the embedded space is significantly reduced, also with a small penalty in accuracy. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 1093–1109, 2006.

We present the design and implementation of a web mining system that creates a hierarchical clustering of web documents retrieved by commercial web search engines. The cluster hierarchy is produced by a novel method called the Cluster Hierarchy Construction Algorithm (CHCA) and it can be used to explore the topics of interest related to the search query and their relationships. We discuss important design issues for our system, including stemming and dimensionality reduction, as well as some implementation details. We show examples of system results, compare them with results from similar systems, and analyze the responses to a survey of the system's users. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 607–625, 2005.

For original paper see ibid., vol.7, no.6, p.634-42 (1999). We have no problem with the comments presented by Mathew and Kaimal (see ibid., p.415, 2005) regarding the investigation discussed in our original paper. The fact is that the general result in the investigated cases, that for a prisoner in both situations (noncooperative and selfish) the optional strategy is still to "confess" irrespective of the strategic choice made by the opponent party, really is in agreement with our original discussions. We thank the authors of the comments for their interest in our work, which has been recently expanded into "software testing via fuzzy strategic games" as well as "fighting terrorism in cyberspace using fuzzy strategic games." It is interesting to note that, especially in these two applications, the comments made by Mathew and Kaimal may make a significant impact.

Most existing methods of document clustering are based on the classical vector-space model, which represents each document by a fixed-size vector of key terms or key phrases. In large and diverse document collections such as the World Wide Web, this approach suffers from a tremendous computational overload, since the constant size of the term vector equals to the total number of key terms in all documents. We propose a new fuzzy-based approach to clustering documents that are represented by vectors of variable size. Each entry in a vector consists of two fields. The first field is the name of a key phrase in the document and the second denotes an importance weight associated with this key phrase within the particular document. We will describe the proposed approach in detail and show how it is implemented in a real world application from the area of web monitoring.

In this paper we describe a classification method that allows the use of graph-based representations of data instead of traditional vector-based representations. We compare the vector approach combined with the k-Nearest Neighbor (k-NN) algorithm to the graph-matching approach when classifying three different web document collections, using the leave-one-out approach for measuring classification accuracy. We also compare the performance of different graph distance measures as well as various document representations that utilize graphs. The results show the graph-based approach can outperform traditional vector-based methods in terms of accuracy, dimensionality and execution time.

In today's software industry, the design of test cases is mostly based on human expertise, while test automation tools are limited to execution of pre-planned tests only. Evaluation of test outcomes is also associated with a considerable effort by human testers who often have imperfect knowledge of the requirements specification. Not surprisingly, this manual approach to software testing results in heavy losses to the world's economy. In this paper, we demonstrate the potential use of data mining algorithms for automated modeling of tested systems. The data mining models can be utilized for recovering system requirements, designing a minimal set of regression tests, and evaluating the correctness of software outputs. To study the feasibility of the proposed approach, we have applied a state-of-the-art data mining algorithm called Info-Fuzzy Network (IFN) to execution data of a complex mathematical package. The IFN method has shown a clear capability to identify faults in the tested program.

We investigate the use of fuzzy clustering for the analysis of software metrics databases. Software metrics are collected at various points during software development, in order to monitor and control the quality of a software product. We use fuzzy clustering to examine three collections of software metrics. This is one of the very few attempts to use unsupervised learning in the software metrics domain, even though unsupervised learning seems more appropriate for this application domain. Some characteristics of this application domain that have significant implications for machine learning are highlighted and discussed. Our results illustrate how unsupervised learning can be used in software quality control.

In this paper we describe work relating to classification of Web documents using a graph-based model instead of the traditional vector-based model for document representation. We compare the classification accuracy of the vector model approach using the k-nearest neighbor (k-NN) algorithm to a novel approach which allows the use of graphs for document representation in the k-NN algorithm. The proposed method is evaluated on three different Web document collections using the leave-one-out approach for measuring classification accuracy. The results show that the graph-based k-NN approach can outperform traditional vector-based k-NN methods in terms of both accuracy and execution time.

In today’s industry, the design of software tests is mostly based on the testers’ expertise, while test automation tools are limited to execution of pre-planned tests only. Evaluation of test outputs is also associated with a considerable effort by human testers who often have imperfect knowledge of the requirements specification. Not surprisingly, this manual approach to software testing results in heavy losses to the world’s economy. The costs of the so-called "catastrophic " software failures (such as Mars Polar Lander shutdown in 1999) are even hard to measure. In this paper, we demonstrate the potential use of data mining algorithms for automated induction of functional requirements from execution data. The induced data mining models of tested software can be utilized for recovering missing and incomplete specifications, designing a minimal set of regression tests, and evaluating the correctness of software outputs when testing new, potentially flawed releases of the system. To study the feasibility of the proposed approach, we have applied a novel data mining algorithm called Info-Fuzzy Network (IFN) to execution data of a general-purpose code for solving partial differential equations. After being trained on a relatively small number of randomly generated input-output examples, the model constructed by the IFN algorithm has shown a clear capability to discriminate between correct and faulty versions of the program.

In the software testing process, selecting the test cases and verifying their results requires a lot of subjective decisions and human intervention. For a program having a large number of inputs, the number of corresponding combinatorial black-box test cases is huge. A method needs to be established in order to limit the number of test cases and to choose the most important ones. In this research effort we present a novel methodology for identifying important test cases automatically. These test cases involve input attributes which contribute to the value of an output and hence are significant. The reduction in the number of test cases is attributed to identifying input-output relationships. A ranked list of features and equivalence classes for input attributes of a given code are the main outcomes of this methodology. Reducing the number of test cases results directly in the saving of software testing resources.

In this paper, compensatory granular reasoning methods are proposed based on fuzzy logic, neural networks, genetic algorithms, compensatory computing and granular computing. The compensatory operation can be reduced to the relevant general compensatory AND operations which replace the traditional AND operations in the compensatory granular reasoning. Different compensatory AND operations such as fuzzy compensatory AND, neural compensatory AND, genetic compensatory AND, linear compensatory AND, and exponential compensatory AND are defined. The compensatory granular reasoning methods can be used to make more reliable decisions. For example, the compensatory granular reasoning methods can make more fault-tolerant fuzzy moves in a fuzzy game if a player carefully chooses a reasonable compensatory operation. In the future, different compensatory granular reasoning methods will be used in different decision-making applications.

The basic operations of fuzzy sets, such as negation, intersection, and union, usually are computed by applying the one-complement, minimum, and maximum operators to the membership functions of fuzzy sets. However, different decision agents may have different perceptions for these fuzzy operations. In this article, the concept of parameterized fuzzy operators will be introduced. A parameter α will be used to represent the degree of softness. The variance of α captures the differences of decision agents' subjective attitudes and characteristics, which result in their differing perceptions. The defined parameterized fuzzy operators also should satisfy the axiomatic requirements for the traditional fuzzy operators. A learning algorithm will be proposed to obtain the parameter α given a set of training data for each agent. In this article, the proposed parameterized fuzzy operators will be used in individual decision-making problems. An example is given to show the concept and application of the parameterized fuzzy operators. © 2003 Wiley Periodicals, Inc.

A novel framework for logical reasoning, termed complex fuzzy logic, is presented in this paper. Complex fuzzy logic is a generalization of traditional fuzzy logic, based on complex fuzzy sets. In complex fuzzy logic, inference rules are constructed and "fired" in a manner that closely parallels traditional fuzzy logic. The novelty of complex fuzzy logic is that the sets used in the reasoning process are complex fuzzy sets, characterized by complex-valued membership functions. The range of these membership functions is extended from the traditional fuzzy range of [0,1] to the unit circle in the complex plane, thus providing a method for describing membership in a set in terms of a complex number. Several mathematical properties of complex fuzzy sets, which serve as a basis for the derivation of complex fuzzy logic, are reviewed in this paper. These properties include basic set theoretic operations on complex fuzzy sets - namely complex fuzzy union and intersection, complex fuzzy relations and their composition, and a novel form of set aggregation - vector aggregation. Complex fuzzy logic is designed to maintain the advantages of traditional fuzzy logic, while benefiting from the properties of complex numbers and complex fuzzy sets. The introduction of complex-valued grades of membership to the realm of fuzzy logic generates a framework with unique mathematical properties, and considerable potential for further research and application.

Software testing forms an integral part of the software development life cycle. Since the objective of testing is to ensure the conformity of an application to its specification, a test “oracle” is needed to determine whether a given test case exposes a fault or not. Using an automated oracle to support the activities of human testers can reduce the actual cost of the testing process and the related maintenance costs. In this paper, we present a new concept of using an artificial neural network as an automated oracle for a tested software system. A neural network is trained by the backpropagation algorithm on a set of test cases applied to the original version of the system. The network training is based on the “black-box” approach, since only inputs and outputs of the system are presented to the algorithm. The trained network can be used as an artificial oracle for evaluating the correctness of the output produced by new and possibly faulty versions of the software. We present experimental results of using a two-layer neural network to detect faults within mutated code of a small credit approval application. The results appear to be promising for a wide range of injected faults. ? 2002 John Wiley & Sons, Inc.

The objective of this paper is to investigate the innovative concept of complex fuzzy sets. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0, 1], but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. The inherent difficulty in acquiring intuition for the concept of complex-valued membership presents a significant obstacle to the realization of its full potential. Consequently, a major part of this work is dedicated to a discussion of the intuitive interpretation of complex-valued grades of membership. Examples of possible applications, which demonstrate the new concept, include a complex fuzzy representation of solar activity (via measurements of the sunspot number), and a signal processing application. A comprehensive study of the mathematical properties of the complex fuzzy set is presented. Basic set theoretic operations on complex fuzzy sets, such as complex fuzzy complement, union, and intersection, are discussed at length. Two novel operations, namely set rotation and set reflection, are introduced. Complex fuzzy relations are also considered. Index Terms-Complex fuzzy intersection, complex fuzzy relations, complex fuzzy sets, complex fuzzy union, complex-valued grades of membership, fuzzy complex numbers.

We describe a computing-with-words algorithm for changing the granularity of an existing rulebase. A fuzzy rulebase is regranulated by manually selecting the granularity of each input dimension, and then automatically calculating the rulebase consequents. We demonstrate this algorithm by studying the common "fuzzy-PD" rulebase at various granularities. We find evidence that the "3/spl times/3" fuzzy-PD rulebase may be too coarse a granularity for this problem.

The innovative concept of complex fuzzy sets is introduced. The novelty of the complex fuzzy set lies in the range of values its membership function may attain. In contrast to a traditional fuzzy membership function, this range is not limited to [0,1] but extended to the unit circle in the complex plane. Thus, the complex fuzzy set provides a mathematical framework for describing membership in a set in terms of a complex number. A study of this original concept is presented, including examples of possible applications, which demonstrate the new theory.

This paper describes a new method that utilizes technology from soft computing to understand words after any OCR software system generated the proper database. First, the problem will be defined very precisely, and then, the algorithm for solving the problem of recognizing words will be presented. Finally, examples are provided to show how the algorithm is applied to recognize different types of words. © 2001 John Wiley & Sons, Inc.

Adding the dimension of time to databases produces time series databases (TSDB) and introduces new aspects and difficulties to data mining and knowledge discovery. In this correspondence, we introduce a general methodology for knowledge discovery in TSDB. The process of knowledge discovery in TSDR includes cleaning and filtering of time series data, identifying the most important predicting attributes, and extracting a set of association rules that can be used to predict the time series behavior in the future. Our method is based on signal processing techniques and the information-theoretic fuzzy approach to knowledge discovery. The computational theory of perception (CTP) is used to reduce the set of extracted rules by fuzzification and aggregation. We demonstrate our approach on two types of time series: stock-market data and weather data.

A general framework for dealing with numerical measurements in an approximate, uncertain, or fuzzy environment is presented. A fuzzy measurement is defined. It possesses several unique properties, which arise from its physical nature and distinguish it from concepts such as the fuzzy number. These properties, which include the fuzzy correlation term and the fuzzy Equality relation, follow directly from physical considerations. The introduction of the fuzzy correlation term provides a mathematical tool for representing any correlation relations, which may exist between different fuzzy measurements. The main function of the fuzzy correlation term is to eliminate, or filter out, measurement values that are unlikely, given other fuzzy measurements. Thus, using the fuzzy correlation term, the range of possible measurement values is limited by physical realities. The information represented by the fuzzy correlation term is shown to be of great value in providing a wider picture of reality than it is possible to obtain by simply considering individual fuzzy measurements. Arithmetic operations on fuzzy measurements and functions of fuzzy measurements are also discussed, leading to the derivation of the fuzzy Riemann integral and its applications.

We introduce the linguistic gradient operator, a novel concept in fuzzy systems theory. A multiple-input, single-output fuzzy rule-base of the Mamdani type is modeled in a linguistic state space. The consequents of the rules are treated as a function, whose domain is the linguistic state space representing the rule antecedents. The linguistic gradient operator operates on the consequent function, and produces an n-dimensional vector pointing in the direction of the greatest local increase in the rule consequents. Thus, the linguistic gradient shows how a given rule relates to its neighbors in linguistic space. An example demonstrates that the linguistic gradient vector is a linguistic analogue of the classical gradient vector.

Uncertainty of complex-valued physical quantities z=x+y can be described by complex fuzzy sets. Such sets can be described by membership functions /spl mu/(x, y) which map the universe of discourse (complex plane) into the interval [0, 1]. The problem with this description is that it is difficult to directly translate into words from natural language. To make this translation easier, several authors have proposed to use, instead of a single membership function for describing the complex number, several membership functions which describe different real-valued characteristics of this number, such as its real part, its imaginary part, its absolute value, etc. The quality of this new description strongly depends on the choice of these real-valued functions, so it is important to choose them optimally. We formulate the problem of optimal choice of these functions and show that, for all reasonable optimality criteria, the level sets of optimal functions are straight lines and circles. This theoretical result is in good accordance with our numerical experiments, according to which such functions indeed lead to a good description of complex fuzzy sets.

A new approach for analyzing the similarity of dynamical systems is presented, with applications to speech analysis. This approach is based on a temporal fuzzy set representation of the trajectories of the dynamical system. The similarity between segments of the speech signal is determined via similarity measures of the corresponding temporal fuzzy sets. We present an application of the method to vowel recognition in the samples (amplitude–time) space. © 2000 John Wiley & Sons, Inc.

In this paper, we present a neural-networks-based knowledge discovery and data mining (KDDM) methodology based on granular computing, neural computing, fuzzy computing, linguistic computing, and pattern recognition. The major issues include 1) how to make neural networks process both numerical and linguistic data in a data base, 2) how to convert fuzzy linguistic data into related numerical features, 3) how to use neural networks to do numerical-linguistic data fusion, 4) how to use neural networks to discover granular knowledge from numerical-linguistic data bases, and 5) how to use discovered granular knowledge to predict missing data. In order to answer the above concerns, a granular neural network (GNN) is designed to deal with numerical-linguistic data fusion and granular knowledge discovery in numerical-linguistic databases. From a data granulation point of view, the GNN can process granular data in a database. From a data fusion point of view, the GNN makes decisions based on different kinds of granular data. From a KDDM point of view, the GNN is able to learn internal granular relations between numerical-linguistic inputs and outputs, and predict new relations in a database. The GNN is also capable of greatly compressing low-level granular data to high-level granular knowledge with some compression error and a data compression rate. To do KDDM in huge data bases, parallel GNN and distributed GNN will be investigated in the future.

Using the embedding method, the fuzzy integral is represented as a parametric Riemann integral. An algorithm which approximates this integral uniformly, is incorporated to design a soft computing tool for solving a fuzzy Fredholm integral equation of

By merging the theory of moves (TOM) and the fuzzy sets theory, we have developed the theory of fuzzy moves (TFM) to make better fuzzy moves for fuzzy games. Since the data granularity of conventionally used fuzzy sets is too low to contain more heuristic information and mined knowledge, we take primary fuzzy sets with higher data granularity as fundamental elements for fuzzy reasoning so as to make more reasonable moves. The simulation results indicate that (1) TFM with normal fuzzy reasoning can make better and more reasonable moves than TOM with precise reasoning since different global strategies are taken into account by TFM and (2) the novel fuzzy reasoning methodology is more reasonable and more useful to make fuzzy moves than the conventional one. ©1999 John Wiley & Sons, Inc.

The "curse of dimensionality" is one of the key problems facing fuzzy systems theory today. Fuzzy systems are based on a set of IF-THEN rules and the structure of these fuzzy rules causes an exponential growth in the number of rules when more inputs are added, resulting in unwieldy rulebases. Many strategies have been proposed to alleviate the curse of dimensionality. Combs and Andrews (ibid., vol.6, p.1-11, 1998) attempt to reduce the growth of the fuzzy rulebase by proposing a disjunctive form for fuzzy rules. In the current paper, we comment on this approach and show that it is mathematically invalid, as illustrated by an example. Combs replies that the authors are addressing 2 different issues; whether the union and intersection rules of configuration be equivalent in fuzzy logic, and whether the latter can overcome the curse of dimensionality.

A game is a decision-making situation with many players, each having objectives that conflict with each other. The players involved in the game usually make their decisions under conditions of risk or uncertainty. In the paper, a fuzzy approach is proposed to solve the strategic game problem in which the pure strategy set for each player is already defined. Based on the concepts of fuzzy set theory, the approach uses a multicriteria decision-making method to obtain the optimal strategy in the game, a method which shows more advantages than the classical game methods. Moreover, with this approach, some useful conclusions are reached concerning the famous "prisoner's dilemma" problem in game theory.

The paper focuses on the use of fuzzy logic in network management, ATM congestion control, and ATM switching. The ATM network management ensures traffic agreement at the source level in order to prevent congestion situations from occurring. The paper reviews work in the area of traffic policing and introduces a scheme for improving the performance of a fuzzy policer. The ATM fuzzy switch introduced, enhances the capabilities of the ATM switch, one of the most critical elements in the ATM network.

First, we would like to thank V. Kratschmer for his validation of our results in the paper regarding the belief measure by using a topological approach. Though assertions (1) and (3) are presented in a weakened fashion, our results still remain valid, as he claims. It is true that assertion (2) has been proved by us only for Borel sets B, which have at most countable components. We were not able to prove the same result for Borel sets with uncountable components (such as the irrational numbers, for example) using our line of reasoning. We therefore applaud the proof presented by V. Kratschmer for the more general Borel sets using an interesting use of some topological properties induced by the Hansdorff metric defined on the space of closed intervals of the real numbers. This certainly makes our original approach to fuzzy data analysis combining fuzzy sets theory and Dempster-Shafer even more useful.

The expressions of "excluded middle" and "crisp contradiction" are reexamined starting with their original linguistic expressions which are first restated in propositional and then predicate forms. It is shown that, in order to generalize the truth tables and hence the normal forms, the membership assignments in predicate expressions must be separated from their truth qualification. In two-valued logic, there is no need to separate them from each other due to reductionist Aristotalean dichotomy. Whereas, in infinite (fuzzy) valued set and logic, the separation of membership assignments from their truth qualification forms the bases of a new reconstruction of the truth tables. The results obtained from these extended truth tables are reducible to their Boolean equivalents under the axioms of Boolean theory. Whereas, in fuzzy set and logic theory, we obtain a richer and more complex interpretations of the "fuzzy middle" and "fuzzy contradiction."

The analysis of chaotic components in movements during hands tremor and more classic analysis of the movements is considered in relation to its application to diagnosis and rehabilitation techniques. Fuzzy methods are used to characterize the movements and to assess their normality. Visual and auditory feedback is provided to help controlling the movements in a rehabilitation system that incorporates elements of virtual reality.

A fuzzy neural network with knowledge discovery FNNKD is designed to perform adaptive compensatory fuzzy reasoning based on more useful and more heuristic primary fuzzy sets. In order to overcome the weakness of the conventional crisp neural network and the fuzzy operation oriented neural network, we have developed a general fuzzy reasoning oriented fuzzy neural network called a crisp-fuzzy neural network CFNN that is capable of extracting high-level knowledge such as fuzzy IF-THEN rules from either crisp data or fuzzy data. A CFNN can effectively compress a 5 5 fuzzy IF-THEN rule base of a cart-pole balancing system to a 3 3 one, then to a 2 2 one, and finally to a 1 1 one, and can expand on invalid sparse 3 3 fuzzy IF-THEN rule base of a cart-pole balancing system to a valid 5 5 one. In addition, a CFNN can control a more complex cart-pole balancing system with random fuzzy noise inputs and outputs i.e., nonconventional using crisp inputs and outputs without any noise . The simulations have indicated that a CFNN is an efficient neurofuzzy system with abilities to discover new fuzzy knowledge from either numerical data or fuzzy data, compress and expand fuzzy knowledge, and do fuzzy reasoning.

Based on more useful and more heuristic primary fuzzy sets contained by conventionally used fuzzy sets, a novel normal fuzzy reasoning methodology is proposed in order to overcome the weakness of many conventional fuzzy systems. Analysis indicated that the compositional rule of inference Zadeh [IEEE Transactions on Systems, Man, and Cybernetics, 3, 28–44 (1973)] and related defuzzification schemes are not effective since they may generate unreasonable results such as the constant-speed problem and the unstable cart-pole problem. The new normal fuzzy system is a generalized framework of many conventional fuzzy systems such as Takagi–Sugeno's fuzzy system Sugeno and Yasukawa [IEEE Transactions on Fuzzy Systems, 1, 7–30 (1993)] and Takagi and Sugeno [IEEE Transactions on Systems, Man, and Cybernetics, SMC-15, 116–132 (1985)], Wang's fuzzy system Wang [Adaptive Fuzzy Systems and Control Design and Stability Analysis, Prentice-Hall, Englewood Cliffs, 1994], Lin's fuzzy system Lin [Neural Fuzzy Control Systems with Structure and Parameter Learning, World Scientific, Singapore, 1994] and the fuzzy system using Yager's level set method Figueirdo et al. [IEEE Transactions on Fuzzy Systems, 1, 156–159 (1993)]. According to the normal fuzzy reasoning methodology, Takagi–Sugeno's fuzzy system can be constructed directly from high-level fuzzy rules without using the complex parameter estimation algorithm Lin [Neural Fuzzy Control Systems with Structure and Parameter Learning, World Scientific, Singapore, 1994] and Sugeno and Yasukawa [IEEE Transactions on Fuzzy Systems, 1, 7–30 (1993)]. Simulations of the constant-speed problems, the fuzzy multiplications and the cart-pole system strongly indicated that the new normal fuzzy system is more effective and more robust than many conventional fuzzy systems. Therefore, the normal fuzzy reasoning methodology can be used to efficiently construct robust fuzzy systems. © 1998 John Wiley & Sons, Inc.

We developed three linguistic statements to describe user information desires in a battlefield information environment. These rules are based on end-user interest in each track report generated from radars across the battlefield. Along with these rules of user interest, a linguistic statement describing communications systems capabilities at each node was created. These linguistic statements were converted to fuzzy variables and these variables were used as network control devices in a simulation model. The model results show that effective communications control can be exercised by these simple rules.

Turksen's (1995, 1986) fuzzy normal forms and Kandel et al.'s (1995) and Schwede and Kandel's (1977) fuzzy maps are unified to establish universal truth tables and normal forms. After an introduction to the topic, we start out with a discussion on linguistic assertions. Next, we develop a new universal truth table construction in order to generate normal forms for both the 16 main concepts of two variables, as well as the subconcepts that can be generated from them. It is shown how fuzzy propositional and fuzzy predicate expressions can be generated with the universal truth table construction. Furthermore, it is shown that the laws of conservation hold in a general way on the "fuzzified laws of exclude middle and contradiction" generated by the class of t norms and their dual t conorms.

In this paper, a new adaptive fuzzy reasoning method using compensatory fuzzy operators is proposed to make a fuzzy logic system more adaptive and more effective. Such a compensatory fuzzy logic system is proved to be a universal approximator. The compensatory neural fuzzy networks built by both control-oriented fuzzy neurons and decision-oriented fuzzy neurons cannot only adaptively adjust fuzzy membership functions but also dynamically optimize the adaptive fuzzy reasoning by using a compensatory learning algorithm. The simulation results of a cart-pole balancing system and nonlinear system modeling have shown that: 1) the compensatory neurofuzzy system can effectively learn commonly used fuzzy IF-THEN rules from either well-defined initial data or ill-defined data; 2) the convergence speed of the compensatory learning algorithm is faster than that of the conventional backpropagation algorithm; and 3) the efficiency of the compensatory learning algorithm can be improved by choosing an appropriate compensatory degree.

A new approach for the defuzzification process, based on the Most Typical Value (MTV) concept, is being introduced and investigated. It assumes that the single decision, based on the output fuzzy subset, should represent a typical value of this set, rather than a numerical "central tendency" of its elements. The MTV defuzzification method is compared with previous defuzzification procedures and its applicability is illustrated.

As a mathematical representation of the linguistic variable, fuzzy number theory with its applications was extensively developed. We investigate two popular representations of fuzzy numbers, symmetric-triangular and bell-shaped, and compare their arithmetic, dimensions and metrics. It is shown that, as far as linear structures are concerned, both representations are equivalent.

A new approach for obtaining a typical value of a fuzzy set as a fuzzy integral is developed by replacing the classic Lebesgue measure with the "typicality" measure. A new method for representing and calculating the one-dimensional fuzzy integral with respect to arbitrary measure derived by monotonic increasing function is proposed and its applicability is illustrated with examples.

In this article we address the issues brought up by Elkan in his article, “The paradoxical success of fuzzy logic,” [IEEE Expert, 3–8 (1994)]. Elkan's work has caused concern since it purportedly reveals a Fuzzy Logic weakness regarding its theoretical foundations. A further investigation of Elkan's theorem (“Theorem 1”) revealed that its conclusion is not correct. After indicating the points where we disagree with Elkan, we reformulate Theorem 1, calling this new version “Theorem 2.” Theorems 1 and 2 have the same hypotheses but different conclusions. According to Theorem 2 there is a region of points that do hold the equivalence in the hypotheses of Theorem 1. In other words, one does not need to change the definition of logical equivalence in Theorem 1 in order to prove that Fuzzy Logic does not collapse to a two-valued logic. In a further analysis of Theorem 2 we show that Elkan's work does not affect the power of Fuzzy Logic to model vagueness. © 1996 John Wiley & Sons, Inc.

In this article we investigate knowledge acquisition (KA) and its relationships to random sets. Based on random set theory, we develop some estimation theorems and procedures for set-valued statistics such as nonparametric estimators. Under random interval assumption, we establish some special possibility distributions that can be easily implemented in KA tools. The knowledge studied here are rules describing relationships between various concepts, as used in diagnosis (pattern recognition) expert systems. © 1996 John Wiley & Sons, Inc.

In this article, we investigate knowledge acquisition and pattern recognition by using the principles of random sets. Based on random set theory, we develop some estimation theorems and procedures for set-valued statistics, such as nonparametric estimators and set-valuedization techniques. Under random interval assumption, we establish some special possibility distributions that can be easily implemented in knowledge acquisition tools. The knowledge studied here are rules describing relationships between various concepts, as used in diagnosis (pattern recognition) expert systems. Several examples are given to illustrate the estimation theorems and procedures for the acquisition of concepts and relationships. We use our acquisition techniques on a modeling prediction example in two different ways: One is by acquiring the concepts and relationships simultaneously; another is by acquiring rules for predefined concepts. On two classification problems, we use our methods to acquire classification rules. The results are compared with several machine learning methods. © 1996 John Wiley & Sons, Inc.

In this work we define a conjugate fuzzy Cauchy problem and discuss the major differences between an ordinary and a conjugate fuzzy Cauchy problems. A necessary condition for the existence of a fuzzy solution to the conjugate fuzzy Cauchy problem is proposed.

The classical definitions of ‘expected value’ and ‘standard deviation’ may sometimes lead to quantities which fail to represent a ‘typical’ feature of a given data set, whenever this set consists of more than one cluster. The use of the fuzzy expected value (FEV) and the clustering fuzzy expected value (CFEV) also yield central tendency and in general cannot represent a typical value of the given data. In this work a new quantity—a Most Typical Value (MTV) is defined and investigated. A given fuzzy set in Rn is first clustered and replaced by a finite set of clusters. This set is then represented by a single vector—the most typical value of the set.

The generalization of Dempster-Shafer (D-S) theory to fuzzy sets provides means for evidential reasoning on basis of fuzzy information. Fuzzy implication operators are used to turn partial fuzzy knowledge about a physical system into an inference engine. Presently there are more than 72 fuzzy implication operators known and investigated, evoking demand for strategies of finding the best performing fuzzy implication operator for the physical system under investigation. In this paper the authors propose such a concept based on generalized D-S theory. A convention to update prior knowledge regarding the applicability of fuzzy implication operators, with respect to fuzzy evidences pertaining to the physical system under consideration, is presented.

Dempster's upper and lower probabilities occur when dealing with multivalued mappings and qualify as Shafer's belief and plausibility measures. In this paper we generalize the notion of upper and lower probabilities while maintaining their properties as belief and plausibility measures, and subsequently apply this concept on Kwakernaak's fuzzy random variables. It is shown how fuzzy random variables induce belief and plausibility functions and it is pointed out what relations exist between fuzzy probabilities and degrees of belief and plausibility. Finally we focus on empirical fuzzy random variables and investigate their asymptotic behavior in terms of belief and plausibility constraints and gain a framework for nonparametric statistical reasoning on the basis of uncertain evidences.>

In this paper, a generalization of parameter hypotheses tests used in statistical inference is presented. This leads to fuzzy hypotheses defined as imprecise predictions about the unknown parameter of a family of distribution functions in question, rather than exactly and sharply given assertions about its location in the parameter space. The notion of fuzzy partitions is discussed in the first part of this paper and details regarding their structure in relation to possibility theoretical interpretations are investigated subsequently. In the second part the authors introduce the notions of fuzzy tests and disclose the impacts of an ill defined counter hypothesis (i.e., hypotheses, partly supporting the same subset of the parameter space) on the reliability of such a fuzzy test. An example demonstrates the consistency of the notion of fuzzy tests with the classical crisp case in statistical inference.>

The successful application of fuzzy reasoning models to fuzzy control systems depends on a number of parameters, such as fuzzy membership functions, that are usually decided upon subjectively. It is shown in this paper that the performance of fuzzy control systems may be improved if the fuzzy reasoning model is supplemented by a genetic-based learning mechanism. The genetic algorithm enables us to generate an optimal set of parameters for the fuzzy reasoning model based either on their initial subjective selection or on a random selection. It is shown that if knowledge of the domain is available, it is exploited by the genetic algorithm leading to an even better performance of the fuzzy controller.>

A hierarchical data structure for analyzing visual motion is presented. Although the literature on perception is abundant with studies on visual motion, none of the studies investigated the importance of a hierarchical model in the analysis of visual motion. The model was implemented on a supercomputer (Cyber 205). The algorithms of hierarchical correlation were performed on binary images. The results are compared with those obtained using similar serial algorithms. The impact of such a hierarchy on component directional selectivity and on pattern directional selectivity is studied.>

The representation of knowledge (concepts, facts, and rules) is discussed. In the representation schema, the main tools used are factors spaces and fuzzy sets. Factors are used to capture the characteristics of a concept (intension), and fuzzy sets are used to describe the membership degree of objects to a concept (extension). Various forms of facts and rules are presented in the factor space structure. Fuzzy reasoning is discussed. The nature of fuzzy reasoning is analyzed by dissecting a fuzzy rule in two different ways. By using level cuts of rules and classic implications, some implication operators are obtained. Some reasoning principles are provided in factor spaces.>

A reusable expert system called Fess was designed to be easy to validate and has been used in several different domains. The validation and testing of the system from the building stage onward is discussed.>

No abstract available.

A fuzzy-valued switching function F over n variables is a mapping from Vn to V where V represents the closed unit interval [0,1] such that F is representable by a logic formula defined under the operations of fuzzy algebra.

An example is presented to show that minimal form may not always be unique as asserted by Neff and Kandel. A corect approach to minimization for fuzzy functions is given, and further work needed in the area is indicated. Some relevant information is pointed out regarding the treatment and simplification of fuzzy-valued switching functions. Examples are discussed in order to illustrate the correct approach for the minimization of these functions.

A graphical scheme (map) for representation and manipulation of fuzzy switching functions is described. Properties of the map and relationships between represented implicants are discussed. Emphasis is placed on illustrations of the use of the map in manipulation and minimization of fuzzy switching functions.

An algorithm for obtaining simple disjunctive decompositions of fuzzy functions is described. The insufficiency of Boolean methods for simplification of fuzzy functions is discussed. Special properties of the decomposed fuzzy structures are introduced and their relationships to two-valued logic are investigated. This new algorithm is the first one in a containing research in the area of multivalued and fuzzy decompositions of complex systems.

A concise matrix notation is introduced, leading to a very simple statement of the resolution principle of mechanical theorem proving in the propositional calculus. The refinements of general resolution can also be stated easily using this notation. In addition, the notation has lead to the development of three new techniques of theorem proving which are described and proved complete.

In this correspondence, we are concerned with an algorithm used to find the set of all inexact prime implicants of an inexact switching function as defined with the help of fuzzy algebra. The principal advantage claimed over existing methods is that it is suitable for an efficient computer implementation. The steps are easy to apply and well adapted to being programmed without any heuristic methods.

This correspondence discusses a restrictive structure of the customary hierarchy of formal languages attained by imposing restrictions on a set of productions and on their Iuse by means of codewords. A coded fuzzy language (CFL) is defined in order gap between formal languages and natural languages. Somte properties of CFLs and their relationships to the restrictive device are studied. A cyclic language of order n is defined to investigate properties of formal and fuzzy languages with regard to classes of recognizers.

This correspondence points out the insufficiency of the method for minimization of fuzzy functions as presented in a recent short note.1 Two lemmas, which are the basis for a new minimization technique [2], are represented.

The study of fuzziness in combinational switching systems by means of a suitable fuzzy algebra is discussed. The insufficiency of the methods for simplification of fuzzy functions as presented by Lee and Chang [9] and by Siy and Chen [10] is discussed. A new technique for minimization of fuzzy functions is developed. Special properties of fuzzy functions are discussed and their relationships to two-valued logic are investigated.

This correspondence is concerned with the study of static hazards in combinational systems by means of ( n, k, i) trees and digraphs [1],[2].