Showing papers on "Fuzzy mathematics published in 1976"

Resolution of composite fuzzy relation equations

Abstract: This paper provides a methodology for solution of certain basic fuzzy relational equations, with fuzzy sets defined as mappings from sets into complete Brouwerian lattices, covering a large class of types of fuzzy sets.

Applications of Fuzzy Sets to Systems Analysis

Abstract: Ten years ago, Zadeh has brought into vogue the use of a name. Scientists no is an increasing less than poets strike off words that fit a situation. Today there recognition that for understanding vagueness, a fuzzy approach is required. We are just going through ~ transient period. From discussions of general philosophy to practical methods for system analysis. Unfortunately, much of the existing research is scattered. The practitioner interested in these methods face the challenge of sorting through a vast amount of literature to find a core on which to build. One of the objects of this book was to facilitate communication by bringing toge- ther different viewpoints and coloring them from a common viewpoint. Since the romanian version appeared, at the very beginning of 1974, there has been a rapid growth in the literature of fuzzy modelling. A minor revision would have left the book quite out-of-date. The opportunity has been taken to correct, clarify, and update. Inexactness is implicit in human behaviour and erare humanum est. It is a pleasure to acknowledge the help we have received in preparing this version. The opportunity to see an english edition was a powerful stimulus, and we are grateful to Salomon Klaczko for making this possible. Another debt is to all fuzzy authors we have quoted. Their fascinating papers kindled our interest in the subject.

Foundations of fuzzy reasoning

Abstract: This paper gives an overview of the theory of fuzzy sets and fuzzy reasoning as proposed and developed by Lotfi Zadeh. In particular it reviews the philosophical and logical antecedents and foundations for this theory and its applications. The problem of borderline cases in set theory and the two classical approaches of precisifying them out, or admitting them as a third case, are discussed, leading to Zadeh's suggestion of continuous degrees of set membership. The extension of basic set operations to such fuzzy sets, and the relationship to other multivalued logics for set theory, are then outlined. The fuzzification of mathematical structures leads naturally to the concepts of fuzzy logics and inference, and consideration of implication suggests Łukasiewicz infinite-valued logic as a base logic for fuzzy reasoning. The paradoxes of the barber, and of sorites, are then analysed to illustrate fuzzy reasoning in action and lead naturally to Zadeh's theory of linguistic hedges and truth. Finally, the logical, modeltheoretic and psychological derivations of numeric values in fuzzy reasoning are discussed, and the rationale behind interest in fuzzy reasoning is summarized.

A fuzzy set approach to modifiers and vagueness in natural language.

Abstract: SUMMARY Recent developments in semantic theory, such as the work of Labov (1973) and Lakoff (1973), have brought into question the assumption that meanings are precise. It has been proposed that the meanings of all terms are to a lesser or greater degree vague, such that, the boundary of the application of a term is never a point but a region where the term gradually moves from being applicable to nonapplicable. Developments in fuzzy set theory have made it possible to offer a formal treatment of vagueness of natural language concepts. In this article, the proposition that natural language concepts are represented as fuzzy sets of meaning components and that language operators—adverbs, negative markers, and adjectives— can be considered as operators on fuzzy sets was assessed empirically. In a series of experiments, we explored the application of fuzzy set theory to the meaning of phrases such as very small, sort of large, and so on. In Experiment 1, subjects judged the applicability of the set of phrases to a set of squares of varying size. The results indicated that the group interpretation of the phrases can be characterized within the framework of fuzzy set theory. Similar results were obtained in Experiment 2, where each subject's responses were analyzed individually. Although the responses of the subjects, in general, could be interpreted in terms of fuzzy logical operations, one subject responded in a more idiomatic style. Experiments 3 and 4 were attempts to influence the logical-idiomatic distinction in interpretatio n by (a) varying the presentation mode of the phrases and by (b) giving subjects only a single phrase to judge. Overall, the results were consistent with the hypothesis that natural language concepts and operators can be described more completely and more precisely using the framework of fuzzy set theory.

A fuzzy model of document retrieval systems

Abstract: This paper is concerned with the organization and retrieval of records in document retrieval systems which admit of imprecision in the form of fuzziness in document characterization and retrieval rules. A mathematical model for such systems, based on the theory of fuzzy sets, is introduced. A document retrieval system, as defined in this paper, is a quadruple (X, D, Q, γ), where X is a collection of the document descriptions (also referred to as index records, or records); D is the descriptor set; Q is a query set; γ: QxX → [0, 1], (called the matching function) assigns to each pair (q, x) where q ϵ Q and x ϵ X, a number γ(q, x) in the interval [0, 1], called the matching index for the query q and the document description x. In our system model, each document description x is defined as a fuzzy set in the descriptor set D. As a fuzzy subset of D, each x is characterized by a membership function μx: D → [0, 1], where μx(d), representing the grade of membership of d in x, is referred to as the index weight of the descriptor d for the document representation x. The retrieval response of the system is defined in terms of the matching function γ. More specifically, given a query q, the index record retrieval response, f(q), is defined to be a fuzzy set in X whose membership function is given by μ ƒ(q) (x) = γ(q, x) . To deal with the organization problems of data in our conceptual model, the conventional concept of a list is extended to a fuzzy list. Specifically, L(d), the fuzzy list corresponding to a descriptor d, is defined as a fuzzy set in the document description set X whose membership function is given by μ l (d) (x) = μ x , (d) . In this way, the notion of an inverted file structure can be extended to the fuzzy data in our retrieval model.

Analysis of fuzzy control algorithms using the relation matrix

Abstract: This paper uses the relation matrix to examine the structure of fuzzy control algorithms. After introducing some basic notations and definitions, it presents a theorem which allows an arbitrary relation matrix to be translated into a set of fuzzy control rules. Using this result it is possible to show that the implementation of the algorithm does not affect its structure and that the most meaningful way of altering its performance is to change the rules themselves.

Probability measures on fuzzy events in phase space

Abstract: The notion of fuzzy sample point is introduced, and generalized probability measures on fuzzy events are defined. This leads to the concept of spectral measure on fuzzy events. It is shown that such measures can be associated with quantum‐mechanical states when the fuzzy elementary events are represented by Gaussian distributions on phase space.

Outline of an approach for the analysis of fuzzy systems

Abstract: The behaviour of the humanistic systems is too complex or too ill-defined to be analysed using definitive mathematics. In this paper we propose a method for the analysis of such systems using the concept of fuzzy sets. The methods for the manipulation of fuzzy variables are considered. These methods enable us to represent quantitatively the sum (or product) of fuzzy variables like large and small, and the convolution of fuzzy sequences. Since, while working with fuzzy variables, the data may become enormous, a technique for the data reduction is proposed. Using these techniques for the manipulation of fuzzy variables we are in a position to analyse fuzzy systems.

Fuzzy interpretive structural modeling

Abstract: It is natural to inquire if fuzzy set theoretic methods could enhance the structuring method. It is shown in this paper that it is possible to develop fuzzy structuring methods. These are based on a fuzzy reach ability matrix and fuzzy graphs. Panel members respond to a relational question “is siRsj ?”along a scale from 0% to 100% representing a set of polar concepts, such as “dense-sparse.” The characteristic logic equation has its fuzzy version in the matrix form. However, in the element form, the problem is more complicated. A reachability matrix of an inter connection matrix is developed to aid the solution process. Measures of incertitude (fuzzy entropy) guide the panel members in choosing appropriate entries. Incertitudes of the set S of issues and of the fuzzy graph ┌R are defined.

N-variable fuzzy maps with application to disjunctive decomposition of fuzzy switching functions

Abstract: A graphical scheme (map) for representation and manipulation of fuzzy switching functions of N-variables is described. Properties of the map and relations between represented implicants are discussed. Emphasis is placed on illustrations of the use of the map for graphical minimization and decomposition of fuzzy switching functions.

Convolution of Fuzzy Variables

Abstract: The behaviour of the humanistic systems is too complex or too ill-defined to be analysed using definitive mathematics. It has been shown that using fuzzy sets, one may analyse such systems. The concept of fuzzy sets has been applied in many fields for taking care of ill-defined variables. It is well known that in the analysis of the systems with memory, the convolution plays a very important role. In this paper, we present a method for the convolution of the fuzzy variables. Different cases of fuzziness for the variables taking part in the convolution are considered. It is observed that the result of the convolution may lead to a fuzzy set having too many elements. A method is presented for representing a fuzzy set having large number of elements by a set having small number of elements. Considering a simple example, these techniques are illustrated.

Fuzzy maps and their application in the simplification of fuzzy switching functions

Abstract: In Boolean logic, a Karnaugh map may be regarded either as a pictorial form of a trugh table, or as an extension of the Venn diagram.However, when fuzzy logic is concerned another minimization method is required, and therefore an extension of a Karnaugh map is investigated.In this paper a new minimization algorithm is developed in order to remove the existing disadvantage of simplifying fuzzy forms. The algorithm is based on a new representation of fuzzy forms that assures a very suitable automatic generation of the fuzzy prime implicants and of the essential fuzzy prime implicants.