Between Mind and Computer: Fuzzy Science and Engineering

The following sections are included:

• FOREWORD

• PREFACE

• CONTENTS

Possibilistic ATMS are truth maintenance systems oriented towards hypothetical reasoning where both assumptions and justifications can bear an uncertainty weight. Uncertainty is represented in the framework of possibility theory. In possibilistic logic uncertain clauses are handled as such and then in possibilistic ATMS the management of uncertainty is not separated from the other classical capabilities of the ATMS. The main interest of a possibilistic ATMS is to take advantage of the uncertainty pervading the available knowledge so as to rank-order environments in which a given statement is true. The approach is illustrated on a fault diagnosis problem. Basic algorithms associated with possibilistic ATMS are given.

An automatic start-up and shut-down control system is designed for a drum-type boiler of a fossil power plant using fuzzy logic. The control system consists of seven continuous parts and one decision maker for a sequential action. In each part of the fuzzy logic controller, the knowledge for control is acquired from the expertized operator and recorded data. The designed control system is tested via a model of a drum-type boiler of the controlled plant, the Seoul Fossil Power Plant Boiler Unit #4.

In this paper we show that there appears to be an abundance of universal fuzzy controllers. A universal fuzzy controller is a type of fuzzy controller that can control any controllable process.

Two essential components of systems modeling are i) representation and ii) inference. We review recent developments in fuzzy systems modeling from a perspective of: a) knowledge representation with fuzzy sets including measurement and acquisition of membership functions for a system parameter identification, as well as combination of knowledge with fuzzy sets for the formation of rules in a system structure identification, and b) approximate reasoning with fuzzy logic including properties of reasoning, combination of rules and/or their consequences, and three heuristics that have been proposed during the course of development. This review is restricted to point-valued fuzzy sets and logics.

This article describes a realization of fuzzy logic by a neural network. Each node in the network represents a premise or a conclusion. Let x be a member of the universal set, and let A be a node in the network. The value of activation of node A is taken to be the value of the membership function at point x, m_A (x). A logical operation is defined by a set of weights which are independent of x.

Given any value of x, a preproceesor will determine the values of the membership function for all the premises that correspond to the input nodes. These are treated as input to the network.

A propagation algorithm is used to emulate the inference process. When the network stabilizes, the value of activation at an output node represents the value of the membership function that indicates the degree that the given conclusion is true.

Weight assignment for the standard logical operations will be discussed. It will also be shown that the scheme allows us to define more general logical operations.

When Zadeh originally defined the fussy extension of crisp set theoretical (and logical) operation, he introduced a very special pair, max and min, very likely became of their simplicity. Later, the examination of various t-norms and conorms showed that there is an infinite number of possible extensions, among which max and min represent only the extremal case. The strictly monotonous norms are often better models of real life phenomena, and natural human thinking, than the original definitions. I-fuzzy algebra is an attempt to axiomatise a rather broad class of norms of which, the most typical representatives are the well known algebraic connectives. In this system, some astonishing behaviour of the norms can be observed: as an example we mention the existence of inverse set theoretical operations! The stress of this Chapter is on the discussion of mathematical properties, but a few hints to applications are also given. The section on fuzzy flip-flops able to store fuzzy information has also a strong practical motivation.

Since the classical paper of Bellman and Zadeh on fuzzy decision making, many research papers have been published. It is clearly impossible to do a detailed survey in this limited space. Thus, in this survey, the methodologies and principles for fuzzy decision making are emphasized. Topics covered are fuzzy linear programming (FLP), fuzzy multiple criteria decision making (FMCDM), and fuzzy multiple person decision making (FMPDM). The problem of duality and sensitivity analysis are also discussed.

Topology on lattices which combines topology structure and order structure well has two rather mature branches: locale theory and fuzzy topology. In this paper we survey the development of fuzzy topology over the last twenty years just from the view point of the above combination of the structures. For a completely distributive lattice L and an ordinary set X, topology on lattice L^x is just fuzzy topology. It differs from locale theory in that it possesses naturally pointlike elements and stratification (i.e. level structure). Emphasizing the above two features we introduce the basic concepts, concrete constructions and the main results of fuzzy topology. In particular we explore the contribution of the pointed approach in fuzzy topology. For the two branches of topology on lattices the connection between locale theory and fuzzy topology is also touched upon. We investigate briefly the category foundations of L-fuzzy topology and discuss the categorical topology approach in fuzzy topology.

Fuzzy logic programming is mathematical foundations of fuzzy programming languages based on fuzzy logic, such as fuzzy prolog. We apply the following definitions: (1) The fuzzy definite clause is a logic formula for representing an uncertain assertion. (2) The fuzzy logical consequence of fuzzy definite clauses is intended to be the correct conclusion. (3) The SLD-resolution is used to compute whether a given fuzzy definite clause is a fuzzy logical consequence under a set of fuzzy definite clauses. (4) The degree of resolution, is used to produce the fuzzy logical consequence with the highest confidence. The main results concern the soundness and completeness of the SLD-resolution, i.e., every answer computed in the SLD-resolution is correct and every correct answer is an instance of a computed answer.

The following sections are included:

• Introductory Remarks

• The Paradigm of Knowledge Representation and Processing With Fuzzy Sets

• Input Interface

• Processing Phase
  • Fuzzy neural structures
    • Aggregative and matching functions of logical neurons
    • Logic processors as basic, processing units
    • Learning algorithms
    • Generalization properties of logical processors
  • Genetic algorithms in optimization of fuzzy models

• Output Interface: Linguistic to Numerical Transformation

• Concluding Remarks

• Acknowledgment

• References

New non standard operations of subtraction (⊝) and division (Ø), acting on fuzzy numbers, are introduced to solve fuzzy equations with the effect of reducing fuzziness in successive computations. Their properties are presented and it is shown how these operations are combined or related with the usual operations of addition (+) and multiplication (×) in a hybrid fuzzy arithmetic.

In engineering practice, the external excitation and internal parameters of a dynamic system are often fuzzy or fuzzy random owing to their ambiguous definition and unpredictability. In this chapter, the mathematical simulation of fuzzy and fuzzy random excitations and the analysis methods of fuzzy and fuzzy random response and reliability are discussed respectively for the dynamic systems described by differential equations. Firstly, fuzzy and fuzzy random processes are defined and employed to simulate the fuzzy and fuzzy random excitations. Secondly, the solution techniques of differential equations with a fuzzy or fuzzy random nonhomogeneous part are presented and used to analyse the response of a dynamic system subjected to fuzzy or fuzzy random excitation. Finally, several methods are put forward for the reliability analysis of fuzzy and fuzzy random systems.

Truth valued flow inference (TVFI) theory treats fuzzy inferences as the processes of the truth values flowing among propositions, and focus intention to the analysis of structures of inference channels. In this paper we give a survey to a mathematical theory for TVFI, in which the basic concepts of channel lattice and background graph, the essential reason of reasoning, the formula of how to determine the confidence degree of channels, and the idea of how to combine knowledge from different sources such as experts' experience, statistical data and causality laws, …, those contents have been introduced.

In this work we describe an implementation of the procedure proposed by Yager to make inferences in inheritance systems that have both default and absolute knowledge and allow for multiple inheritance with exceptions. The procedure is based on Theory of Approximate Reasoning and uses possibility qualification for representing typical or default values.