Showing papers on "Fuzzy logic published in 1973"

Outline of a New Approach to the Analysis of Complex Systems and Decision Processes

Abstract: 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 Relative of the ISODATA Process and Its Use in Detecting Compact Well-Separated Clusters

Abstract: Two fuzzy versions of the k-means optimal, least squared error partitioning problem are formulated for finite subsets X of a general inner product space. In both cases, the extremizing solutions are shown to be fixed points of a certain operator T on the class of fuzzy, k-partitions of X, and simple iteration of T provides an algorithm which has the descent property relative to the least squared error criterion function. In the first case, the range of T consists largely of ordinary (i.e. non-fuzzy) partitions of X and the associated iteration scheme is essentially the well known ISODATA process of Ball and Hall. However, in the second case, the range of T consists mainly of fuzzy partitions and the associated algorithm is new; when X consists of k compact well separated (CWS) clusters, Xi , this algorithm generates a limiting partition with membership functions which closely approximate the characteristic functions of the clusters Xi . However, when X is not the union of k CWS clusters, the limi...

A fuzzy relative of the isodata process and its use in detecting compact well-separated clusters

Abstract: Two fuzzy versions of the k-means optimal, least squared error partitioning problem are formulated for finite subsets X of a general inner product space. In both cases, the extremizing solutions are shown to be fixed points of a certain operator T on the class of fuzzy, k-partitions of X, and simple iteration of T provides an algorithm which has the descent property relative to the least squared error criterion function. In the first case, the range of T consists largely of ordinary (i.e. non-fuzzy) partitions of X and the associated iteration scheme is essentially the well known ISODATA process of Ball and Hall. However, in the second case, the range of T consists mainly of fuzzy partitions and the associated algorithm is new; when X consists of k compact well separated (CWS) clusters, Xi , this algorithm generates a limiting partition with membership functions which closely approximate the characteristic functions of the clusters Xi . However, when X is not the union of k CWS clusters, the limi...

Cluster Validity with Fuzzy Sets

Abstract: Given a finite, unlabelled set of real vectors X, one often presumes the existence of (c) subsets (clusters) in X, the members of which somehow bear more similarity to each other than to members of adjoining clusters. In this paper, we use membership function matrices associated with fuzzy c-partitions of X, together with their values in the Euclidean (matrix) norm, to formulate an a posteriori method for evaluating algorithmically suggested clusterings of X. Several numerical examples are offered in support of the proposed technique.

On Fuzzy-Mathematical Programming

Abstract: In problems of system analysis, it is customary to treat imprecision by the use of probability theory. It is becoming increasingly clear, however, that in the case of many real world problems involving large scale systems such as economic systems, social systems, mass service systems, etc., the major source of imprecision should more properly be labeled ‘fuzziness’ rather than ‘randomness.’ By fuzziness, we mean the type of imprecision which is associated with the lack of sharp transition from membership to nonmembership, as in tall men, small numbers, likely events, etc. In this paper our main concern is with the application of the theory of fuzzy sets to decision problems involving fuzzy goals and strategies, etc., as defined by R. E. Bellman and L. A. Zadeh [1]. However, in our approach, the emphasis is on mathematical programming and the use of the concept of a level set to extend some of the classical results to problems involving fuzzy constraints and objective functions.

On Minimization of Fuzzy Functions

Abstract: 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.

Fuzzy multi-stage decision-making, fuzzy state and terminal regulators and their relationship to non-fuzzy quadratic state and terminal regulators†

Abstract: Bellman and Zadeh recently considered fuzzy multi-stage decision processes, the concept of fuzziness emanating from Zadeh's earlier work on fuzzy sets. Here we extend their results to processes inv...

Fuzzy Planner: Reasoning with Inexact Concepts in a Procedural Problem-Solving Language

Abstract: All contemporary deductive problem-solving paradigms deal with a world in which assertions are true (false) and action-rules valid (invalid). This simplified situation is inadequate for realistic applications which include inexact information. This report describes a precise computationally specific method for coupling two different many-valued logics with a procedural problem-solving system (micro-PLANNER). Solutions to deductive problems can be found which meet specific criteria of validity. This particular scheme enables the system to dynamically compute the truth-value of a subgoal during the search process. Thus, the validity of a subgoal may be used to direct the heuristic search procedure. Fuzzy PLANNER is a promising medium for experimenting with different many-valued logics to find the ones most appropriate for different problem domains. However, the notions elaborated here are relevent to any procedural problem-solving language.

Learning of Fuzzy Formal Language

Abstract: A learning model of fuzzy formal language is proposed and discussed. We continue training the learning machine by giving sets of sentences sequentially. As a result of parsing of the given teaching sentences, the learning machine reinforces fuzzy grades of membership of productions in an inherent fuzzy grammar of the machine. The convergence of the proposed model is considered, and it is shown that the grades of membership of desired productions are intensified by choosing an adequate teaching sequence of the sentence set. Furthermore, a concept of ``strongly equivalent,'' in which two grammars are not distinguished by any teaching sequence, is introduced.

A procedure for optimizing development decisions

Abstract: This paper addresses the problem of organizing data and formulating questions to be answered for the purpose of making planning and development decisions. The problem is separated into five distinct parts. Each part is discussed in the context of the planning process and each leads logically to the optimization of development decisions. The five parts are goal definition, establishment of criteria, criteria weighting, alternative rating and alternative ranking. The feature of making development decisions which distinguishes them from other optimization problems is what has been called 'fuzziness'. In any realistic problem formulation, the criteria are not precisely defined; they are fuzzy. The relative importance of each criterion is also fuzzy. Indeed, even when one attempts to rate a particular possible solution, he must deal with fuzzy information. The technique proposed in the paper accounts for this uncertainty in all aspects of the problem and yields probabilistic answers. Thus, when various alternative solutions are proposed for a development problem, the technique yields a probabilistic ranking of the alternatives. Sharper results are obtained if less uncertainty is present in certain parts of the data. However, even in the presence of great uncertainty, realistic problem solutions are obtained. Alternative solutions are rated independent of all others and only after the (fuzzy) ratings are complete are comparative rankings accomplished. Throughout the procedure the realistic uncertainties remain a prominent feature of the procedure.

Constructing Fuzzy Measure and Grading Similarity of Patterns by Fuzzy Integral

Abstract: Lately, studies on fuzzy systems as well as applications of fuzzy set theory have attracted the attention of many researchers. The author has suggested the concept of fuzzy measure and fuzzy integral for representing fuzzy systems. The fuzzy measure without additivity may be regarded as a subjective one by which •gfuzziness•h is measured. At first, this paper explains the fuzzy measure extended onto a family of sets including fuzzy sets and the concrete methods for constructing the fuzzy measure. As an example, the fuzzy measure gƒÉ, -1<ƒÉ<•‡, is defined and its characteristics are clarified. Here, gƒÉ, corresponding to the probability measure when ƒÉ=0, is continuous for ƒÉ. Furthermore, the new concept of the ƒÉcomplement of a fuzzy set is proposed. This has flexibility due to the parameter ƒÉ. It is an extension of the complement defined by L.A. Zadeh. Next, this paper considers the problems appearing when a human grades the similarity of several patterns with no sharp boundaries. The human decision mechanism is represented by a macro model where the fuzzy integral is used. Man's subjective characteristics in grading the similarity are obtained through the fuzzy measure identified so that both, human and model's, outputs agree with each other. A simple expriment on the above problem was performed. The experimental results show that the outputs of the model agree approximately with those obtained by human evaluation.

Comment on an algorithm that generates fuzzy prime implicants by Lee and Chang

Abstract: This correspondence points out the insufficiency of the algorithm to generate the set of all fuzzy prime implicants of a fuzzy formula as presented in Lee and Chang (1971) . Two theorems, which are the basis of a new technique that generates the complete set of fuzzy implicants, and used for the minimization of fuzzy functions, as presented in Kandel (1972) , are discussed.

Fuzzy Sequential Functions

Abstract: Abstrat In this paper, the sequential functions or input-output relations of fuzzy systems which are realizable by maximin machines are studied. Various problems related to these functions are examined, which include characterization and decision Problems, completion problems, extension problems, closure properties and their relations to maximin regular events. Some basic properties of maximin regular events are also presented including the extension of Kleene's theorem to the maximin case.

Comments on "Minimization of Fuzzy Functions"

Abstract: 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.