Fuzzy Logic and Soft Computing

The following sections are included:

• PREFACE

• CONTENTS

Evolutionary algorithms are direct, global optimization algorithms gleaned from the model of organic evolution. The most important representatives, genetic algorithms and evolution strategies, are briefly introduced and compared in this paper, and their major differences are clarified. Furthermore, the paper summarizes the application possibilities of evolutionary algorithms in the design of fuzzy logic controllers. The optimization of fuzzy membership functions turns out to be a promising and successful application domain for evolutionary algorithms, while the automatic learning of fuzzy control rules by means of fuzzy classifier systems is still in an early stage of research.

The problem of generating desirable fuzzy rules is very important in the development of fuzzy systems. The purpose of this paper is to present a generation method of fuzzy control rules by learning from examples using genetic algorithms. We propose a real coded genetic algorithm for learning fuzzy rules, and an iterative process to obtain a set of rules that covers the examples set with a covering value previously defined.

A new method for the automatic design of fuzzy control and decision/diagnosis systems is described. The fuzzy system is extended by a learning ability without changing the traditional fuzzy rule framework. A genetic algorithm is used to find optimal fuzzy rules and membership functions. By choosing the right parameters, the final fuzzy system will use a minimal number of rules and membership functions (fuzzy sets).

An algorithm for inductive learning from erroneous examples is presented. It is assumed that the errors may occur in the attributes' values. However, their location (in which example, and in which attribute) is unknown. Moreover, the errors are assumed incorrigible as it is often the case in practice. A modification of the start-type algorithm is proposed. Importance of the attributes - reflecting, e.g., the attributes' relevance, their proneness to errors, reliability of methods for determining their values, etc. - is elicited from the experts, and weights are determined by Saaty's analytical hierarchy process (AHP). An oncological example illustrating the method proposed is mentioned.

We propose a connectionist approach for temporal reasoning using knowledge such as “If p then q in the future.” The proposed method consists of a network and a logical mechanisms. The knowledge is translated into constraints, and the network mechanism repeatedly searches for solutions consistent with those constraints. Among them, the logical mechanism selects a solution based on preference criterion. Further, we propose an “observe-and-edit” strategy to shorten the time to reach the preferred solution. This division of labor between the network and logical mechanisms contributes to simplifying control in the reasoning process.

In many cases the identification of systems by means of fuzzy rules is given by taking these rules from a predetermined set of those possible. In this case, the correct description of the system is to given by a finite set of rules, each with an associated weight which assesses its correctness or accurancy. Here, we present a method for learning this consistence level or weight using neural networks, for identifying the system, with which to associate, each possible rule with its weight.

A method to acquire knowledge commonly expressed among several cases to make human-computer communication more intelligent is proposed. New design concepts of a natural language communication system in which this method is to be implemented, a new reasoning method, Fuzzy Case-Based Reasoning (F-CBR), used in the system to generate responses, and a problem of human-computer communication by F-CBR causing the necessity of this method are described respectively as well as the proposal.

It is a well known problem with conventional fuzzy reasoning that the fuzziness of inferred results gradually increases according to the progress of the inference. Essentially, the problems are that (1) the combined result obtained with a conventional combination function becomes close to one of the two non-fuzzy values, that is, grade 0 or 1, and never approaches the other non-fuzzy value, and (2) the lack of a reinforcement property. We propose a new combination function which resolves those problems by introducing equilibrium E and dependence factors α and β.

In this paper we propose a novel framework for the design of autonomous fuzzy intelligent systems. The system integrates the following modules into a single autonomous entity:

(1) A fuzzy expert system

(2) Artificial neural network

(3) Genetic algorithm

(4) Case-base reasoning

We describe the integration of these units into one intelligent structure and discuss potential applications to image processing.

A fuzzy instance-based connectionist learning system is described. Its ability to segment magnetic resonance images of the brain into tissue types is analyzed. Three competing methods are considered for partitioning each real-valued attribute of the given data set into fuzzy subsets of each attribute's range. The system is capable of generating fuzzy rules that classify the regions of an image. We discuss the derivation and interpretation of such rules. The results obtained by applying this learning strategy to a ground truthed image segmented by radiologists are very promising.

There are two models for the representation of linguistic expressions. These are Type I and II fuzzy models. It is suggested that Type II representation of linguistic expressions with fuzzy normal forms do provide a more flexible and richer interpretation of complexities and uncertainties embedded in combination of concepts. However, it adds a computational expense and makes a user responsible for the selection of an appropriate membership function in the “non-specificity” interval defined by fuzzy normal forms. In this context, “implication” and “excluded middle” expressions are reinterpreted with their Type II models. A new “non-specificity” measure is defined for the assessments of semantic uncertainty of Type II fuzzy sets.

The paper discusses the use of techniques of fuzzy neurocomputations in the construction of expert systems. It is revealed how fuzzy sets utilized in this context become advantageous in bridging some fundamental concepts of learning stemming from neural networks and the ideas of explicit knowledge representation residing within symbolic computations. We accentuate an issue of numerical quantification of qualitative relationships being usually available as a part of any domain knowledge encountered in knowledge-based systems. A new class of logically-inclined processing units constructed exclusively with the aid of fuzzy set operators is advantageous in supporting both the explicit form of knowledge representation and learning capabilities. We study generic architectures including those capturing dynamics of the individual features. The issue of handling uncertain and incomplete information is addressed as well.

Uncertainty management techniques offer a formalism well appropriate to deal with data variability. Nonetheless, the multisensor data fusion processing methods developed using uncertainty theories have emphasized the practical difficulty in taking context into account. The question is to exploit the available contextual knowledge so that its effect can be modeled through parameters usable by the decision criterion. The neuro-fuzzy approach proposed here is based on the interpretation of an analytical reference criterion issued from the theory of evidence, so that it can be implemented by a neural network.

After a short introduction of a Linear logic based representation of Petri nets and of Object Petri nets we will propose a characterization of disjunctive sets of sequences using Linear additive connectives and study their relationships with Petri net imprecise markings. An example is considered: it will show pseudo-firings and how imprecision will be decreased on receiving precise information.

In this paper we introduce an automated procedure for extracting information from knowledge bases that contain fuzzy production rules. A backward reasoning algorithm based on the High Level Fuzzy Petri Net model is developed. The algorithm consists of the extraction of a subnet and an evaluation process. We informally verify that the proposed algorithm is similar to another procedure suggested earlier by Yager, with advantages concerning the efficiency in knowledge base information search.

We show that fuzzy controllers are good approximations for control strategies in distributed systems, i.e., plants whose states are described by infinitely many parameters. This result generalizes the previous results on the universal approximation property of fuzzy controllers as approximations of functions (i.e., for plants whose states are described by finitely many parameters).

Fuzzy control applications usually deal with precise inputs. This paper aims at analyzing the behaviour of a fuzzy controller when the input is also fuzzy. For sake of simplicity, only fuzzy proportional controllers are studied. The input is thus represented by a fuzzy number. An accommodation of the general equations of a Mamdani type fuzzy controller is first derived, leading to some easy software implementation. Results are presented for different inference operators and defuzzification methods. Various shapes of the membership function of the input are also studied.

The concept of immediate probabilities is introduced as a modification of the probabilties of the outcomes with information about the payoffs. We investigate the case where this transformation is mediated through dispositional information (optimism/pessimism). We use the Dempster rule of combination to help obtain an expression for these probabilities. We show how immediate probabilities allows us to explain the Allais paradox.

In this paper we will analyze some computational problems related with the use of OWA operators as information aggregators. In particular we will concentrate on ordered hierarchical aggregations of OWA operators as defined in [4].

In this paper some results on group decision making under linguistic preferences and fuzzy linguistic quantifiers are presented. Assuming a set of individual linguistic preferences, representing the preferences of the particular individuals, we develop a solution method for the choice process. We define a linguistic ordered weighted averaging operator, and use it for deriving a collective linguistic preference where the weights are defined using a fuzzy linguistic quantifier. Finally, we use the concept of nondominated alternatives for obtaning a set of maximal nondominated alternatives from the collective linguistic preference, that is, the solution to the decision process.

This paper presents a general strategy for the management of decision making under uncertainty in industry, and its application to the specific problem of route generation in an electric engineering company. The general strategy is based on the fusion of several complementary approaches, using both linguistic and numerical information. The overall goal is to produce a system which is highly interactive, allows the use of as much information as possible, and leads the decision making process without forcing it.

It is shown that, under some rather general conditions, any comparison meaningful mean on a strict simple order is an order statistic.

We investigate the aggregation phase of multicriteria decision making procedures. Characterizations of some classes of nonconventional aggregation operators are established. The first class consists of the ordered weighted averaging operators (OWA) introduced by Yager. The second class corresponds to the weighted maximum defined by Dubois and Prade. The dual class (weighted minimum) and some ordered versions are also characterized. Results are obtained via solutions of functional equations.

In this paper on the utility and preference based decision process is presented in unified way. We first deal with the utility based decision making comparing with the preference model. We define an universal preference function with useful properties and we show that the utility based decision making gives the same result as the preference based using the universal preference function. The preference based decision making procedures (the outranking approaches) can be get also with the help of the universal preference function i.e. we proved that there exist a unary function applying to the universal preference function getting the preference relations of the different outranking approaches.

The aim of this paper is to show how multivalued logics underlying fuzzy sets theories allow some concepts and properties used in decision aid procedures to be extended. In particular, some procedures based on the use of a conventional covering relation are presented and generalized to the fuzzy case. The main interest of these procedures is their ability to build transitive preference structures from intransitive pairwise comparisons of objects. This typical feature should be useful not only to rank fuzzy numbers or to discriminate between the alternatives of a decision problem, but also to classify multidimensional objects.

This paper focuses on the acquisition of abstract information, i.e. information which are not analytically related to conventional physical quantities as for example the comfort. In these complex cases, we propose to use fuzzy sensors which compute and report linguistic assessment of numerical acquired values. Two methods are proposed to realize the aggregation from basic measurements. The first one performs the combination of the relevant features by means of a rule based description of the relations between them. With the second one, the aggregation is realised through an interpolation mechanism that make a fuzzy partition of the numeric multi-dimensional space of the basic features.

As a generalization of the additive clustering model [6], we discuss the following three additive fuzzy clustering models: a simple additive fuzzy clustering model, an overlapping fuzzy clustering model and a fuzzy clustering model for ordinal scaled similarity [4]. The essential merits of fuzzy clustering models are 1) the amount of computations for the identification of the models are much fewer than a hard clustering model and 2) fewer number of clusters are needed to get a suitable fitness.

These fuzzy clustering models are extended to the model for asymmetric similarity. In this model, the concept of the similarity among clusters is introduced. The crucial assumption of this model is that the asymmetry of the similarity between the pair of objects is caused by the asymmetric similarity among clusters. The validity of this model is shown by some examples.

In this paper the quotient operation in fuzzy relational database is analyzed. With the basis on the classical case a fuzzy quotient definition is given in a general fuzzy database context. Furthermore this concept in generalized by introducing linguistic quantifiers different of ∀ and developing the quotient operations by using them.

In this paper, we consider the evaluation of queries addressed to usual relations and involving imprecise predicates. The queries considered here are modelled as alpha-cuts of complex fuzzy predicates and we study to what extent query processing can be at least partly performed by a regular database management system. In this respect, we show that any tuple must comply with some Boolean conditions in order to belong to the alpha-cut of a compound predicate. The transformation process looks like a rewriting mechanism based on rules. As a matter of fact, some selected tuples do not really belong to the desired alpha-cut and measures, reported here, were made in order to estimate the difference depending on the connectors and the predicates appearing in the query.

In the present work we will attempt an extension of an object oriented data model:

• to describe fuzzy objects with a new attribute type that we termed fuzzy complex attribute. A complex attribute can be either calculated (i.e., its value is calculated from values of other attributes) or aggregate (i.e., it is composed by a set of attributes). All manipulated values can be fuzzy.

• to extend the retrieval process with the new attribute type in order to allow more flexibility in queries.

The approach we propose is illustrated by a real example in human resources management.

In future CIS systems, the knowledge representation of the information issued from the outside world, will require to have at our disposal some DBMS including the notions of time, space, but also imprecision, uncertainty and incompleteness of information. In this article, we present a data model adapted to the representation and the manipulation of imperfect data in the context of CIS. We also present some concepts in order to describe conceptual schemata featuring the notion of fuzziness on its constitutive elements. The proposed concepts are an extension of existing standards with which a compatibility has been studied (ODMG 93, SQL2)

The aim of this paper is to introduce the idea of fuzzy betweenness relation on a set X. This is done by generalizing the definition of betweenness relation proposed by Menger. It is proved that a separating T-indistinguishability operator on X (with T a strict archimedean t-norm) generates a fuzzy betweenness relation on X and reciprocally, every fuzzy betweenness relation on X defines a separating T-indistinguishability operator on X. Moreover, it is proved that the crisp part of a fuzzy betweenness relation is a classical (metric) betweenness relation.

The main idea of this paper is the introduction of a duality between the elements of a set X and its fuzzy subsets. This duality induces a useful interpretation of the elements as fuzzy subsets and viceversa. As a natural consequence of this interpretation, the set of the fuzzy sets over [0, 1]^X can be viewed as a classical extension of X. In the case that X is endowed with a T-indistinguishability operator E and considering only the set H of the generators of E, the operator E can be naturally defined over the mentioned extension. On the other hand, a dual interpretation of the representation theorem, leads to a similar theorem but with a set of points as generators. Finally, it is also shown that T-similarities over [0,1]^H are a suitable tool in order to built a sound theoretical background for the study of the inference by similarity.

The similarity-based model of possibilistic and fuzzy reasoning mainly relies on the use of the socalled pairs of implication-consistency measures, that are shown to be possibility envelopes. This notion is the possibilistic counterpart of lower and upper probabilities and it is used in this paper to introduce within the similarity framework several conditioning methods. A preliminary and comparative study of them is also presented in a general setting, as well as the application of the different methods to a typical reasoning scenario with fuzzy variables.

In this paper, we like to present a general, complete and easily applicable nonclassical cardinality theory which makes use of the infinite-valued Łukasiewicz logic. The theory pertains to vaguely defined objects, understood as objects that are separated from a universe by means of arbitrary sharp or vague properties, including subdefinite sets (incompletely known sets, in other words). We will discuss questions of equipotency of vaguely defined objects, inequalities between and arithmetical operations on the resulting generalized cardinal numbers which express the powers (cardinalities) of vaguely defined objects. In particular, the presented theory offers a common generalization of different existing approaches to the problem of cardinality of fuzzy sets, twofold fuzzy sets, partial sets, etc.

We give a characterization of Inv(R), the class of invariant fuzzy sets of a fuzzy relation R on X. It is shown that Inv(R) can be described in terms of a lattice-theoretic hull of no more than N extreme elements, where N = |X| is the cardinality of X. These ideas are used to investigate the Lyapunov stability of a class of fuzzy control systems.

We introduce a notion of partial order for fuzzy sets in connection with their greater or smaller fuzziness. This allows us to give a simple basis to the theory of fuzziness measures.

Sugeno's integral permits to build very large classes of fuzziness measures.

This paper is devoted to the interpretation of quantified statements of the type “Q X are A”, where Q is a fuzzy quantifier, X is a set of elements and A a fuzzy predicate. The overall objective is to point out the connections which are existing between quantified statements and fuzzy integrals. In this paper, we show that Prade's interpretation of “Q X are A”, when Q is a monotonous quantifier, is equivalent to a Sugeno fuzzy integral. One of the advantages of this view resides in the possibility to derive improved algorithms in order to compute quantified expressions.

Difference posets of fuzzy subsets are studied. It is shown that the difference Θ of fuzzy subsets is generated by a normed generator g. Consequently the classes of full fuzzy g-difference posets and of semisimple MV-algebras coincide up to an isomorphism. Finally, states, observables and their expected values on fuzzy difference posets are studied.

In this paper the infomation associated to an event à (crisp or fuzzy) of a space (Ω, Ѕ) measures the degree of surprise which arises when this event occurs. J.Kampé dc Fèriet and B.Forte gave in 1967 ([5]) an axiomatic definition of this concept, in which the information measure J is associated directly to the crisp event A of a measurable space (Ω, Ѕ). We will propose here a possible way to measure the information in the fuzzy context. We proceed by supposing that an axiomatic information measure is defined on the crisp events (crisp subsets): the information of a fuzzy subset will be constructed by means of the informations of their α-cuts.

In a recent conference on fuzzy sets [4], Lofti Zadeh stressed the modest level or penetration of Fuzzy Set Theory in Artificial Intelligence. The possible causes of this are briefly investigated herein. Also possible extensions of some critical points of this theory are suggested, so that we can try to make it more flexible and suitable for different specific purposes, as for instance for inductive classification.

In this work we comment on the necessity of having appropriate mechanisms for the representation and manipulation of uncertainty and vagueness in medicine. We present fuzzy theory as an adequate formal tool for this purpose, considering different examples in the domain of cardiology. The discussion of these examples follows an order of increasing complexity with respect to the knowledge representation and manipulation level in each case.

In this work we deal with medical diagnostic problems including visual inspection as a fundamental procedure with which to evaluate clinical signs and to formulate diagnostic conclusions. In this case discrepancies among physicians may arise in formulating diagnosis.

Our approach addresses this problem by combining the use of fuzzy sets as the representational framework with a machine-aided knowledge acquisition strategy which includes, neural networks.

In the paper, specifically, we illustrate the use of this approach in building a system that functions as an active support in the diagnosis of hormonal disorders.

For this application the solutions adopted have been integrated in a unified framework and implemented in an hybrid system.

This paper describes a decision making process in a fuzzy context. Decisions are considered as a vote result. Votes are done by specific production rules : decision rules. We build an expert system shell called MENTA, with a particular decisional engine for this vote process. This paper presents the engine, its rules and a medical application.

This paper addresses the problem of image content retrieval from imprecise queries in the context of a digital angiographies Database Management System. We present a flexible retrieval system that evaluates to what extent an image in the database corresponds to a query containing imprecise radiological criteria. The three main modules of the system are described: the image content representation module provides high level symbolic representations of the renal arteries depicted in the image, the query representation module is based on the fuzzy set theory and the retrieval process is described as a matching process between a query including imprecise criteria and symbolic image descriptions.

This paper describes a method that exploits fuzzy-set concepts to model successfully ambiguities and uncertainties inherent in assigning an image pixel to an object of interest for the purpose of image segmentation. The method relies on the choice of a set of seed points (one for each object of interest) and exhibits the particular novelty of being entirely independent of input parameters. Some results on medical images are reported, which illustrate the method's validity.

Human glycaemia regulation is a complex phenomenon in which factors of different nature occur, for instance: glucose level and its variation, physical activity, morphology, time and composition of the meals, etc. In case of pancreas deficiency, diabetics have to control their glucose level thanks to injections of insulin, using empirical rules to determine the infusion rate. We propose a predictive fuzzy model of glycaemic variations, as a part of an automatic control system for diabetics. An associated learning procedure makes it possible to deal with interpersonal variations. This work about the treatment of diabetes is achieved in cooperation with the University Hospital Center of Rennes.

Neural network modeling has found many applications in the last decade, especially in medicine. It offers an efficient method for the development of decision tools directly from database information. This methodology has replaced in many cases the knowledge-based approach that gained popularity in the mid-1970's. Neural networks in most instances are not capable of symbolic processing. In this paper, a neural network structure is combined with a symbolic layer to produce a hybrid system for medical decision making which is illustrated in diagnosis and treatment of carcinoma of the lung.

The paper proposes a multi-level fuzzy evaluation function[1,2] which imposes a total order on crop alternatives. The model can be used to consider various input conditions and allows for flexible treatment of such issues as soil degradation, erosion and pollution. These issues and others are graded subjectively in terms of a membership function and partitioned into a matrix which is crossed with an aggregated weighted index for crop yield.

The resulting multi-level comprehensive fuzzy evaluation function considers environmental issues, management skills and economic benefits with biophysical suitability. It allows us to make recommendations for land usage.

This research on the longitudinal control of a motor vehicle was carried out as part of the Autonomous Intelligent Cruise Control (AICC) project, which is itself part of the PROMETHEUS European Community program. It has two objectives : a speed regulation : the driver sets a cruising speed, the vehicle reaches that speed and keeps it; a distance control : when the AICC vehicle detecs a target vehicle, it decides whether it is necessary to modify the actuators instructions so as to place itself at a safety distance behind the followed vehicle.

These two operating modes are permanently active and must satisfy subjective criteria of comfort while ensuring a high robustness in control. These constraints are one reason why we had a hybrid approach in the conception of the regulator. Indeed, a classic type of control (by internal model) ensures a deceleration/acceleration regulation of the vehicle; the deceleration/acceleration instruction being generated by a controller based on Fuzzy Logic methods. This original hybrid control guarantees the robustness criterion even if there is a variation of the structural parameters such as mass, and if there are model errors as well. It also enables the driver to personalize the running of his vehicle by selecting a driving style (“confortable”, “sportslike”,…).With the controller thus conceived we obtain quite satisfactory results within the desired limits of use.

A study was made to design an automatic system to find the required adjustments of a car seat according to the anthropometric features of the users. The main problems of car seating comfort were presented, and a fuzzy knowledge-based system was proposed as a solution. The construction of the knowledge base, the performance and the conditions of use of the system were described. A comparison with a program obtained through a conventional approach facilitated the validation of the system and showed the advantages of a fuzzy knowledge-based system for the Bertrand Faure industrial corporation.

This paper presents two applications of a multiple attribute fuzzy decision support system (MAF-DSS) addressing different contexts. One determines the extent to which a person is a hero in ancient mythology, and the second is a military vehicle pattern recognition problem. The aim is to use the applications to demonstrate the generality, flexibility and main properties of the fuzzy multiple attribute decision support tool.

The recognition of 3-D objects from sequences of their 2-D views is modeled by a family of self-organizing neural architectures, called VIEWNET, that use View Information Encoded With NET works. VIEWNET incorporates a preprocessor that generates a compressed but 2-D invariant representation of an image, a supervised incremental learning system (Fuzzy ARTMAP) that classifies the preprocessed representations into 2-D view categories whose outputs are combined into 3-D invariant object categories, and a working memory that makes a 3-D object prediction by accumulating evidence over time from 3-D object category nodes as multiple 2-D views are experienced. VIEWNET was benchmarked on an MIT Lincoln Laboratory database of 128x128 2-D views of aircraft, including small frontal views, with and without additive noise. A recognition rate of up to 90% is achieved with one 2-D view and of up to 98.5% correct with three 2-D views. The properties of 2-D view and 3-D object category nodes are compared with those of cells in monkey inferotemporal cortex.