Aggregation of data has many important applications in various fields, especially in economics, planning, optimization, decision-making and ranking of alternatives. There are many approaches to aggregation reflecting the fact that aggregation is application-depend and there are some considerations which are hard to quantify. The simple formula, FA, which is described in the following is easy to apply and lends itself to standardization and generalization. It is convenient to describe the formula in terms of a model in which the data relate to the grades g1, …, gn which a student, call him Robert, received for the collection of courses c1, …, cn . Assume that the data are of the form (c1, g1), (c2, g2), …, (cn, gn). The grades are on a scale from 0 to 100. Upon normalization they become numbers in the unit interval which may be interpreted as grades of membership or degrees. We propose two approaches.

Social networks have gained a lot attention. They are perceived as a vast source of information about their users. Variety of different methods and techniques has been proposed to analyze these networks in order to extract valuable information about the users – things they do and like/dislike. A lot of effort is put into improvement of analytical methods in order to grasp a more accurate and detailed image of users. Such information would have an impact on many aspects of everyday life of people – from politics, via professional life, to shopping and entertainment. The theory of fuzzy sets and systems, introduced in 1965, has the ability to handle imprecise and ambiguous information, and to cope with linguistic terms. The theory has evolved into such areas like possibility theory and computing with words. It is very suitable for processing data in a human-like way, and providing the results in a human-oriented manner. The paper presents a short survey of works that use fuzzy-based technologies for analysis of social networks. We pose an idea that fuzzy-based techniques allow for introduction of humancentric and human-like data analysis processes. We include here detailed descriptions of a few target areas of social network analysis that could benefit from applications of fuzzy sets and systems methods.

The seven papers in this special section provide an overview of the main ideas and research directions for computing with words.

Cognitive informatics (CI) is the transdisciplinary enquiry of cognitive and information sciences that investigates into the internal information processing mechanisms and processes of the brain and natural intelligence, and their engineering applications via an interdisciplinary approach. CI develops a coherent set of fundamental theories and denotational mathematics, which form the foundation for most information and knowledge based science and engineering disciplines such as computer science, cognitive science, neuropsychology, systems science, cybernetics, software engineering, knowledge engineering, and computational intelligence. This paper reviews the central doctrine of CI and its applications. The theoretical framework of CI is described on the architecture of CI and its denotational mathematic means. A set of theories and formal models of CI is presented in order to explore the natural and computational intelligence. A wide range of applications of CI are described in the areas of cognitive computers, cognitive properties of knowledge, simulations of human cognitive behaviors, cognitive complexity of software, autonomous agent systems, and computational intelligence.

The papers in this special section focus on granular computing, defined by Lotfi A. Zadeh as "information granulation involves partitioning a class of objects (points) into granules, with a granule being a clump of objects (points) which are drawn together by indistinguishability, similarity, or functionality."

Can machines think? This question has been an object of discussion and debate for over half-a-century. My interest in the question goes back to the beginning of my academic career. Officially, AI was born in l956. At its birth there was widespread expectation that within a few years it will be possible to build machines that could think like humans. I did not share that belief. To the pioneers, symbolic logic was all that was needed. Anything that involved numerical computations was unwelcome. It took close to thirty years for probability theory to gain grudging acceptance. Clearly, adding probability theory to the armamentarium of AI is a step in the right direction. But is it sufficient? In my view, the answer is: No.

This work discusses the precisiated natural language, which is a subset of a natural language. This subset is equipped with constraint-centered semantics (CSNL) and is translatable into what is called the generalized constraint language. The ways does the concept of a precisiated natural language can serve to enlarge the role of the natural languages in scientific theories were also discussed.

This paper deals with fuzzy logic based approach to question-answering systems. To achieve significant question answering capability it is necessary to develop methods of dealing 1. with the reality 2.with world knowledge-and especially knowledge about underlying probabilities is perception based. In this approach PNL is employed to represent infornation in the world knowledge database (WKD).

Summary form only given. The point of departure in the computational theory of perceptions is the assumption that perceptions are described as propositions in a natural language. Furthermore, computing and reasoning with perceptions is reduced to computing and reasoning with words. In the computational theory of perceptions, representation of meaning is a preliminary to reasoning with perceptions-a process which starts with a collection of perceptions which constitute the initial data set (IDS) and terminates in a proposition or a collection of propositions which play the role of an answer to a query, that is, the terminal data set (TDS). Canonical forms of propositions in IDS constitute the initial constraint set (ICS). The key part of the reasoning process is goal-directed propagation of generalized constraints from ICS to a terminal constraint set (TCS) which plays the role of the canonical form of TDS. The rules governing generalized constraint propagation in the computational theory of perceptions coincide with the roles of inference in fuzzy logic.

Attempts to formulate mathematically precise definitions of basic concepts such as causality, randomness, and probability have a long history. The concept of generalized definability suggests that such definitions may not exist. Furthermore, it suggests that existing definitions of many basic concepts, among them those of stability, statistical independence and Pareto-optimality, may need to be redefined. The point of departure in our approach to definability is the assumption that definability has a hierarchical structure. Furthermore, it is understood that a definition must be unambiguous, precise, operational, general, and coextensive with the concept it defines.

Publication of the Special Issue of the Proceedings of the IEEE - an issue dedicated to "Industrial Innovations Using Soft Computing," is an important milestone in the evolution of soft computing (SC). It is a symbol of recognition that a decade after its debut, SC is emerging as a coherent body of concepts and techniques that has wide-ranging applications to the conception, design, and utilization of information/intelligent systems. Progress in science and technology has led to brilliant successes. We have sent men to the moon, built the web, and are on the brink of mapping the structure of genes. But what we have not learned is how to mimic the remarkable human capability to perform a wide variety of physical and mental tasks without any measurements and any computations. In performing such tasks, e.g., driving in city traffic or playing golf, we employ perceptions-rather than measurements-of time, distance, speed, shape, color, likelihood, and other attributes of physical and mental objects. It is this human capability that existing scientific theories do not possess. The computational theory of perceptions is aimed at adding this capability to measurement-based theories. I believe that, in time, the capability to operate on perception- based information will be added to all members of the SC coalition.

A fuzzy algorithm is an ordered set of fuzzy instructions that upon execution yield an approximate solution to a given problem. Two unrelated aspects of fuzzy algorithms are considered in this paper. The first is concerned with the problem of maximization of a reward function. It is argued that the conventional notion of a maximizing value for a function is not sufficiently informative and that a more useful notion is that of a maximizing set. Essentially, a maximizing set serves to provide information not only concerning the point or points at which a function is maximized, but also about the extent to which the values of the reward function approximate to its supremum at other points in its range. The second is concerned with the formalization of the notion of a fuzzy algorithm. In this connection, the notion of a fuzzy Markoff algorithm is introduced and illustrated by an example. It is shown that the generation of strings by a fuzzy algorithm bears a resemblance to a birth-and-death process and that the execution of the algorithm terminates when no more "live" strings are left.

As its name suggests, computing with words (CW) is a methodology in which words are used in place of numbers for computing and reasoning. The point of this note is that fuzzy logic plays a pivotal role in CW and vice-versa. Thus, as an approximation, fuzzy logic may be equated to CW. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers, and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW. In CW, a word is viewed as a label of a granule; that is, a fuzzy set of points drawn together by similarity, with the fuzzy set playing the role of a fuzzy constraint on a variable. The premises are assumed to be expressed as propositions in a natural language. In coming years, computing with words is likely to evolve into a basic methodology in its own right with wide-ranging ramifications on both basic and applied levels.

There has been a rapid growth in the number and variety of applications of fuzzy logic. The successes of fuzzy logic have also generated a skeptical reaction. Most of the criticisms directed at fuzzy logic are rooted in a misunderstanding of what it is and/or a lack of familiarity with it. In many cases, what is not recognized is that the term fuzzy logic (FL) is actually used in two different senses. In a narrow sense, fuzzy logic (FLn) is a logical system which is an extension of multivalued logic and is intended to serve as a logic of approximate reasoning. But in a wider sense, fuzzy logic (FLw) is more or less synonymous with the theory of fuzzy sets (FST). Today the term fuzzy logic is used predominantly in its wider sense. It is in this sense that any field X can be fuzzified-and hence generalized by replacing the concept of a crisp set in X by a fuzzy set. What is gained through fuzzification is greater generality, higher expressive power, an enhanced ability to model real-world phenomena and a methodology for exploiting the tolerance for imprecision. Most of the applications of fuzzy logic relate to control in the context of industrial systems and consumer products. What is discernible, however, is (a) the trend toward the use of fuzzy logic in task-oriented-rather than set-point-oriented-control; and (b) the incorporation of fuzzy logic and neural network techniques in the conception and design of complex systems in which control and expert system techniques are used in combination.>

The author presents a summary of the basic concepts and techniques underlying the application of fuzzy logic to knowledge representation. He then describes a number of examples relating to its use as a computational system for dealing with uncertainty and imprecision in the context of knowledge, meaning, and inference. It is noted that one of the basic aims of fuzzy logic is to provide a computational framework for knowledge representation and inference in an environment of uncertainty and imprecision. In such environments, fuzzy logic is effective when the solutions need not be precise and/or it is acceptable for a conclusion to have a dispositional rather than categorical validity. The importance of fuzzy logic derives from the fact that there are many real-world applications which fit these conditions, especially in the realm of knowledge-based systems for decision-making and control.>

The concept of a generalized assignment statement provides a basis for a unified approach to knowledge representation and inference. The basic idea underlying this approach is that a proposition, p, in a natural language may be viewed as an elastic constraint on a variable, X. More specifically, the meaning of p may be represented as p → X isr C in which → should be read as “translates into” and the right-hand member is a representation of the meaning of p in the form of a generalized assignment statement. In this statement, X plays the role of a constrained variable; C is the constraint on X; and r in the copula isr is a variable whose values define the role of C in relation to X. The prinipal values of r are d, standing for disjunctive; c, standing for conjunctive; p, standing for probabilistic; g, standing for granular; and h, standing for hybrid. Since in most cases the value of r is d, it is convenient to abbreviate isd to is. With this understanding, X is C should be interpreted as X isd C.

An April 1984 workshop on research directions was sponsored by the Decision and Management Science Program of the National Science Foundation at the University of Texas at Dallas. One session concerned the relationship of research to phenomena in decisionmaking and operations. Nine presentations from workers in very diverse areas comprised the session. Together, they provide a very useful compendium of commonalities and differences in this important area of contemporary thought that offers much insight into future research directions. What follows was initially prepared by the nine presenters at the session, edited by the editor of this Transactions, distributed to all session presenters for their thorough review and commentary, and then revised for publication by the editor. It is our hope that this provides the research community at large with the results of a very useful dialog.

A fuzzy syllogism in fuzzy logic is defined as an inference schema in which the major premise, the minor premise, and the conclusion are propositions containing fuzzy quantifiers. A basic fuzzy syllogism in fuzzy logic is the intersection/product syllogism. Several other basic syllogisms are developed that may be used as rules of combination of evidence in expert systems. Among these is the consequent conjunction syllogism. Furthermore, it is shown that syllogistic reasoning in fuzzy logic provides a basis for reasoning with dispositions, that is, with propositions that are preponderantly but not necessarily always true. It is also shown that the concept of dispositionality is closely related to the notion of usuality and serves as a gateway to what might be called a theory of usuality, a theory that may eventually provide a computational framework for commonsense reasoning.

Informally, a disposition is a proposition which is preponderantly, but no necessarily always, true. For example, birds can fly is a disposition, as are the propositions Swedes are blond and Spaniards are dark.An idea which underlies the theory described in this paper is that a disposition may be viewed as a proposition with implicit fuzzy quantifiers which are approximations to all and always, e.g., almost all, almost always, most, frequently, etc. For example, birds can fly may be interpreted as the result of supressing the fuzzy quantifier most in the proposition most birds can fly. Similarly, young men like young women may be read as most young men like mostly young women. The process of transforming a disposition into a proposition is referred to as explicitation or restoration.Explicitation sets the stage for representing the meaning of a proposition through the use of test-score semantics (Zadeh, 1978, 1982). In this approach to semantics, the meaning of a proposition, p, is represented as a procedure which tests, scores and aggregates the elastic constraints which are induced by p.The paper closes with a description of an approach to reasoning with dispositions which is based on the concept of a fuzzy syllogism. Syllogistic reasoning with dispositions has an important bearing on commonsense reasoning as well as on the management of uncertainty in expert systems. As a simple application of the techniques described in this paper, we formulate a definition of typicality -- a concept which plays an important role in human cognition and is of relevance to default reasoning.

The approach described in this paper represents a substantive departure from the conventional quantitative techniques of system analysis. It has three main distinguishing features: 1) use of so-called ``linguistic'' variables in place of or in addition to numerical variables; 2) characterization of simple relations between variables by fuzzy conditional statements; and 3) characterization of complex relations by fuzzy algorithms. A linguistic variable is defined as a variable whose values are sentences in a natural or artificial language. Thus, if tall, not tall, very tall, very very tall, etc. are values of height, then height is a linguistic variable. Fuzzy conditional statements are expressions of the form IF A THEN B, where A and B have fuzzy meaning, e.g., IF x is small THEN y is large, where small and large are viewed as labels of fuzzy sets. A fuzzy algorithm is an ordered sequence of instructions which may contain fuzzy assignment and conditional statements, e.g., x = very small, IF x is small THEN Y is large. The execution of such instructions is governed by the compositional rule of inference and the rule of the preponderant alternative. By relying on the use of linguistic variables and fuzzy algorithms, the approach provides an approximate and yet effective means of describing the behavior of systems which are too complex or too ill-defined to admit of precise mathematical analysis.

A fuzzy mapping from X to Y is a fuzzy set on X × Y. The concept is extended to fuzzy mappings of fuzzy sets on X to Y, fuzzy function and its inverse, fuzzy parametric functions, fuzzy observation, and control. Set theoretical relations are obtained for fuzzy mappings, fuzzy functions, and fuzzy parametric functions. It is shown that under certain conditions a precise control goal can be attained with fuzzy observation and control as long as the observations become sufficiently precise when the goal is approached.

In many of the theories of prediction and detection developed during the past decade, one encounters linear integral equations which can be subsumed under the general form\int_a^b R(t, \tau) x(\tau) d\tau = f(t), a \leqq t \leqq b. This equation includes as special cases the Wiener-Hopf equation and the modified Wiener-Hopf equation\int_0^T R(\mid t - \tau \mid ) x(\tau) d\tau = f(t), 0 \leqq t \leqq T. The type of kernel considered in this note occurs when the noise can be regarded as the result of operating on white noise with a succession of not necessarily time-invariant linear differential and inverse-differential operators. For this type of noise, which is essentially a generalization of the stationary noise with a rational spectral density function, it is shown that the solution of the integral equation can be expressed in terms of solution of a certain linear differential equation with variable coefficients.