Fuzzy Sets, Fuzzy Logic, and Fuzzy Systems

The following sections are included:

• CONTENTS

• PREFACE

• NOTE TO THE READER

• ACKNOWLEDGMENTS

• INTRODUCTION

• BIBLIOGRAPHY OF PAPERS BY LOTFI A. ZADEH

A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint.

The notion of fuzziness as defined in this paper relates to situations in which the source of imprecision is not a random variable or a stochastic process, but rather a class or classes which do not possess sharply defined boundaries, e.g., the “class of bald men,” or the “class of numbers which are much greater than 10,” or the “class of adaptive systems,” etc.

A basic concept which makes it possible to treat fuzziness in a quantitative manner is that of a fuzzy set, that is, a class in which there may be grades of membership intermediate between full membership and non-membership. Thus, a fuzzy set is characterized by a membership function which assigns to each object its grade of membership (a number lying between 0 and 1) in the fuzzy set.

After a review of some of the relevant properties of fuzzy sets, the notions of a fuzzy system and a fuzzy class of systems are introduced and briefly analyzed. The paper closes with a section dealing with optimization under fuzzy constraints in which an approach to problems of this type is briefly sketched.

The following sections are included:

• INTRODUCTION

• ABSTRACTION AND GENERALIZATION

• PATTERN CLASSIFICATION

• REFERENCES

In information theory as well as in many other fields of science it is customary to treat uncertainty and imprecision through the concepts and methods of probability theory. The almost exclusive reliance on probability theory for this purpose obscures the fact that there are many situations in which the source of imprecision is not a random variable but a class or classes which do not possess sharply defined boundaries. For example, the “class” of real number which are much greater than 10 is clearly not a precisely defined set of objects. The same is true of the “class” of good strategies in a game, the “class” of handwritten characters representing the letter A, the “class” of intelligent men, the “class” of systems which are approximately equivalent to a specified system, etc. In fact, on closer examination it appears that most of the classes of objects encountered in the real world are of this fuzzy, not sharply defined, type. In such classes, an object need not necessarily either belong or not belong to a class; there may be intermediate grades of membership. Thus, to describe the degree of belonging to such a class requires the use of a multivalued logic with a possibly continuous infinity of truth values…

The following sections are included:

• INTRODUCTION

• EXECUTION OF FUZZY INSTRUCTIONS
  • Probabilistic Execution
  • Nondeterministic Execution with Threshold

• FUZZY ALGORITHMS AND FUZZY TURING MACHINES

• RATIONALE FOR FUZZY ALGORITHMS

• REFERENCES

A fuzzy language is defined to be a fuzzy subset of the set of strings over a finite alphabet. The notions of union, intersection, concatenation, Kleene closure, and grammar for such languages are defined as extensions of the corresponding notions in the theory of formal languages. An explicit expression for the membership function of the language L(G) generated by a fuzzy grammar G is given, and it is shown that any context-sensitive fuzzy grammar is recursive. For fuzzy context-free grammars, procedures for constructing the Chomsky and Greibach normal forms are outlined and illustrated by examples.

The following sections are included:

• INTRODUCTION

• ELEMENTARY PROPERTIES OF FUZZY SETS

• SYSTEM, AGGREGATE, AND STATE

• STATE EQUATIONS FOR FUZZY SYSTEMS

• FUZZY SYSTEMS AND FUZZY ALGORITHMS

• THE CONCEPT OF AGGREGATE

• REFERENCES

The point of departure in this paper is the definition of a language, L, as a fuzzy relation from a set of terms, T = {x}, to a universe of discourse, U = {y}. As a fuzzy relation, L is characterized by its membership function μ_L:T × U → [0,1], which associates with each ordered pair (x,y) its grade of membership, μ_L(x,y), in L.

Given a particular x in T, the membership function μ_L(x,y) defines a fuzzy set, M(x), in U whose membership function is given by μ_M(x)(y) = μ_L(x,y). The fuzzy set M(x) is defined to be the meaning of the term x, with x playing the role of a name for M(x).

If a term x in T is a concatenation of other terms in T, that IS, x = x₁ ⋯ x_n, x_i ∈ T, i = 1,…,n, then the meaning of x can be expressed in terms of the meanings of x₁,…,x_n through the use of a lambda-expression or by solving a system of equations in the membership functions of the x_i which are deduced from the syntax tree of x. The use of this approach is illustrated by examples.

The past decade has witnessed a rapidly growing mathematization of control theory, with the result that the relation between the theory and its application has become more tenuous than ever.

Looking backward, in the years following World War II when the theory of control systems was still in its infancy, much of the control literature centered on the steady-state or transient analysis of time-invariant linear systems. The level of mathematical sophistication of the theory was relatively low, and only a few papers contained theorems and their proofs…

A fuzzy algorithm is an ordered set of fuzzy instructions which 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 Markoff 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.

A fuzzy language as defined in this paper is a quadruple L = (U,T,E,N) in which U is a non-fuzzy universe of discourse; T (called term set) is a fuzzy set of terms which serve as names of fuzzy subsets of U; E (called an embedding set for T) is a collection of symbols and their combinations from which the terms are drawn, i.e., T is a fuzzy subset of E; and N is a fuzzy relation from E (or the support of T) to U called a naming relation.

As a fuzzy subset of E, T is characterized by a membership function μ_T : E → [0,1], with μ_T(x) representing the grade of membership of a term x in T. Similarly, the naming relation N is characterized by a bivariate membership function μ_N : E × U → [0,1] in which μ_N(x,y) represents the strength of the relation between a term x and an object y in U.

The syntax and semantics of L are viewed as collections of rules for the computation of μ_T and μ_N, respectively. The meaning of a term x is denned to be a fuzzy subset, M(x), of U, whose membership function is given by μ_M(x)(y) = μ_N(x,y).

Various concepts relating to fuzzy languages are introduced and their relevance to natural languages and human intelligence is pointed out. In particular, it is suggested that the theory of fuzzy languages may have the potential of providing better models for natural languages than is possible within the framework of the classical theory of formal languages.

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.

To someone like myself, steeped in the quantitative analyses of inanimate systems, the principal ideas in Skinner's Beyond Freedom and Dignity are difficult to translate into assertions that are capable of proof or refutation. Nevertheless, I find them highly interesting and thought-provoking.

It is a truism that human behavior is vastly more complex than the behavior of man-conceived systems. This is reflected in the fact that such basic concepts as control, reinforcement, feedback, goal, constraint, decision, strategy, adaptation, and environment, which are central to the discussion of human behavior, are much better understood and more clearly defined in system theory—which deals with abstract systems from an axiomatic point of view—than in psychology or philosophy. Unfortunately, high precision is rarely compatible with high complexity. Thus, the precision and determinism of system theory have the effect of severely restricting its capability to deal with the complexities of human behavior…

In recent years, systems analysis has become a widely used technique for dealing with problems relating to the design, management and operation of large-scale systems.

A thesis advanced in this paper is that the conventional techniques of systems analysis are of limited applicability to societal systems, because such systems are, in general, much too complex and much too ill-defined to be amenable to quantitative analyses. It is suggested that the applicability of systems analysis may be enhanced through the use of the so-called linguistic approach, in which words rather than numbers serve as values of variables. The basic elements of this approach are outlined and illustrated by examples.

A fuzzy restriction may be visualized as an elastic constraint on the values that may be assigned to a variable. In terms of such restrictions, the meaning of a proposition of the form “x is P,” where x is the name of an object and P is a fuzzy set, may be expressed as a relational assignment equation of the form R(A(x)) = P, where A(x) is an implied attribute of x, R is a fuzzy restriction on x, and P is the unary fuzzy relation which is assigned to R. For example, “Stella is young,” where young is a fuzzy subset of the real line, translates into R(Age(Stella)) = young.

The calculus of fuzzy restrictions is concerned, in the main, with (a) translation of propositions of various types into relational assignment equations, and (b) the study of transformations of fuzzy restrictions which are induced by linguistic modifiers, truth-functional modifiers, compositions, projections and other operations. An important application of the calculus of fuzzy restrictions relates to what might be called approximate reasoning, that is, a type of reasoning which is neither very exact nor very inexact. The main ideas behind this application are outlined and illustrated by examples.

The term fuzzy logic is used in this paper to describe an imprecise logical system, FL, in which the truth-values are fuzzy subsets of the unit interval with linguistic labels such as true, false, not true, very true, quite true, not very true and not very false, etc. The truth-value set, , of FL is assumed to be generated by a context-free grammar, with a semantic rule providing a means of computing the meaning of each linguistic truth-value in as a fuzzy subset of [0, 1].

Since is not closed under the operations of negation, conjunction, disjunction and implication, the result of an operation on truth-values in requires, in general, a linguistic approximation by a truth-value in . As a consequence, the truth tables and the rules of inference in fuzzy logic are (i) inexact and (ii) dependent on the meaning associated with the primary truth-value true as well as the modifiers very, quite, more or less, etc.

Approximate reasoning is viewed as a process of approximate solution of a system of relational assignment equations. This process is formulated as a compositional rule of inference which subsumes modus ponens as a special case. A characteristic feature of approximate reasoning is the fuzziness and nonuniqueness of consequents of fuzzy premisses. Simple examples of approximate reasoning are: (a) Most men are vain; Socrates is a man; therefore, it is very likely that Socrates is vain, (b) x is small; x and y are approximately equal; therefore y is more or less small, where italicized words are labels of fuzzy sets.

In a sharp departure from the conventional approaches to decision analysis, the linguistic approach abandons the use of numbers and relies instead on a systematic use of words to characterize the values of variables, the values of probabilities, the relations between variables, and the truth-values of assertions about them.

The linguistic approach is intended to be used in situations in which the system under analysis is too complex or too ill-defined to be amenable to quantitative characterization. It may be used, in particular, to define an objective function in linguistic terms as a function of the linguistic values of decision variables.

In cases in which the objective function is vector-valued, the linguistic approach provides a language for an approximate linguistic characterization of the trade-offs between its components. Such characterizations result in a fuzzy set of Pareto-optimal solutions, with the grade of membership of a solution representing the complement of the degree to which it is dominated by other solutions.

Fuzzy logic differs from conventional logical systems in that it aims at providing a model for approximate rather than precise reasoning.

The fuzzy logic, FL, which is described in this paper has the following principal features, (a) The truth-values of FL are fuzzy subsets of the unit interval carrying labels such as true, very true, not very true, false, more or less true, etc.; (b) The truth-values of FL are structured in the sense that they may be generated by a grammar and interpreted by a semantic rule; (c) FL is a local logic in that, in FL, the truth-values as well as the connectives such as and, or, if … then have a variable rather than fixed meaning; and (d) The rules of inference in FL are approximate rather than exact.

The central concept in FL is that of a fuzzy restriction, by which is meant a fuzzy relation which acts as an elastic constraint on the values that may be assigned to a variable. Thus, a fuzzy proposition such as ‘Nina is young’ translates into a relational assignment quation of the form R(Age(Nina)) = young in which Age(Nina) is a variable, R(Age(Nina)) is a fuzzy restriction on the values of Age (Nina), and young is a fuzzy unary relation which is assigned as a value to R(Age(Nina)).

In general, a composite fuzzy proposition translates into a system of relational assignment equations. In this paper, translation rules are developed for propositions of four basic types: Type I, of the general form ‘X is mF,’ where X is the name of an object or a variable, m is a linguistic modifier, e.g., not, very, more or less, quite, etc., and F is a fuzz subset of a universe of discourse. Type II, of the general form ‘X is F * Y is G’ or ‘X is in relation R to Y,’ where * is a binary connective, e.g., and, or, if. then, etc., and R is a fuzzy relation, e.g., much greater. Type III, of the general form ‘QX are F,’ where Q is a fuzzy quantifier, e.g., some, most, many, several, etc., and F is a fuzzy subset of a universe of discourse. And, Type IV, of the general form ‘X is F is τ,’ where τ is a linguistic truth-value such as true, very true, more or less true, etc. These rules may be used in combination to translate composite propositions whose constituents are instances of some of the four types in question, e.g., ‘“Most tall men are stronger than most short men” is more or less true,’ where the italicized words denote labels of fuzzy sets.

The translation rules for fuzzy propositions of Types I, II, III and IV induce corresponding truth valuation rules which serve to express the fuzzy truth-value of a fuzzy proposition in terms of the truth-values of its constituents. In conjunction with linguistic approximation, these rules provide a basis for approximate inference from fuzzy premises, several forms of which are described and illustrated by examples.

The following sections are included:

• INTRODUCTION

• FUZZY ORDERINGS

• TRANSLATION RULES FOR LINGUISTIC PROPOSITIONS

• NOTES

• REFERENCES

The following sections are included:

• Introduction

• Pattern Classification in a Fuzzy-Set-Theoretic Framework
  • Opaque vs. Transparent Algorithms
  • Pattern Classification

• The Linguistic Approach to Pattern Classification

• Translation Rules and the Interpolation of a Relational Tableau
  • The Mapping Rule

• Cluster Analysis
  • Fuzzy Affinity Property
  • Fuzzy Level-Sets

• Concluding Remarks

• References

The following sections are included:

• NOTATION, TERMINOLOGY, AND BASIC OPERATIONS
  • Containment
  • Level-Sets of a Fuzzy Set
  • Operations on Fuzzy Sets
  • Fuzzy Relations
  • Projections and Cylindrical Fuzzy Sets
  • The Extension Principle
  • Fuzzy Sets with Fuzzy Membership Functions

• THE CONCEPT OF A FUZZY RESTRICTION AND TRANSLATION RULES FOR FUZZY PROPOSITIONS
  • Translation Rules of Type I
  • Translation Rules of Type II
  • Translation Rules for Likelihood- and Possibility-Qualified Propositions

• REFERENCES

• BIBLIOGRAPHY

The following sections are included:

• Introduction

• Information Granularity and Possibility Distributions

• Analysis of Granular Evidence

• Concluding Remarks

• References and Related Papers

The following sections are included:

• Introduction

• Possibility distributions, truth-qualification principle and Liar's paradox

• Notes

• References

The following sections are included:

• BASIC CONCEPTS

• FUZZY SYSTEMS THEORY AND FUZZY LOGIC

• CONCLUDING REMARKS

• NOTES

• REFERENCES

A thesis advanced in this paper is that much of the uncertainty which is associated with soft data is nonstatistical in nature. Based on this premise, an approach to the representation and manipulation of soft data—in which the recently developed theory of possibility plays a central role—is described and illustrated with examples.

The following sections are included:

• Introduction

• The Concept of Possibility Distribution

• Test-Score Semantics: Nature of Tests
  • Notational preliminaries
  • Projection
  • Particularization
  • Particularization/projection (transduction)
  • Cardinality of fuzzy sets

• Test-Score Semantics: Meaning Representation
  • Focused translation
  • Translation rules
  • Vector test scores and presuppositions

• Examples of Translation
  • Margaret is slim and very attractive
  • Ellen resides in a small city near Oslo
  • Gary earns much more than his youngest brother
  • Several large balls
  • Patricia has many acquaintances and a few close friends most of whom are highly intelligent
  • During the past few months three large tankers carried a total of 500,000 tons of oil to Naples
  • Our last example involves a command rather than a proposition

• Concluding Remark

• References

In a recent paper, Osherson and Smith (1981) present an insightful critique of some of the contending approaches to prototype theory and arrive at the conclusion that the theory of fuzzy sets does not provide a satisfactory solution to the problems that stand in the way of constructing an adequate theory of prototypes.

In what follows, the issues raised by Osherson and Smith are commented upon and an alternative definition of the concept of a prototype is proposed. In contrast to some of the conventional definitions, the proposed definition does not associate with a given set A a unique prototypical element of A. Rather, the prototypes of A constitute a fuzzy set, PT(A), whose elements, in general, are not elements of A. Viewed in this perspective, the concept of a prototype is a fuzzy concept, and the theory of fuzzy sets provides an appropriate framework for its formulation and applications…

In its traditional interpretation, Frege's principle of compositionality is not sufficiently flexible to have a wide applicability to natural languages. In a fuzzy-set-theoretic setting which is outlined in this paper, Frege's principle is modified and broadened by allowing the meaning of a proposition, p, to be composed not from the meaning of the constituents of p, but, more generally, from the meaning of a collection of fuzzy relations which form a so-called explanatory database that is associated with p. More specifically, through the application of test-score: semantics, the meaning of p is represented as a procedure which tests, scores and aggregates the elastic constraints which are implicit in p. The employment of fuzzy sets in this semantics allows p to contain fuzzy predicates such as tall, kind, much richer, etc.; fuzzy quantifiers such as most, several, few, usually etc.; modifiers such as very, more or less, quite somewhat, etc.; and other types of semantic entities which cannot be dealt with within the framework of classical logic.

The approach described in the paper suggests a way of representing the meaning of dispositions, e.g., Overeating causes obesity, Icy roads are slippery, Young men like young women, etc. Specifically, by viewing a disposition, d, as a proposition with implicit fuzzy quantifiers, the problem of representing the meaning of d may be decomposed into (a) restoring the suppressed fuzzy quantifiers and/or fuzzifying the nonfuzzy quantifiers in the body of d; and (b) representing the meaning of the resulting dispositional proposition through the use of test-score semantics.

To place in evidence the logical structure of p and, at the same time, provide a high-level description of the composition process, p may be expressed in the canonical form “X is F” where X = (X1,.,Xn) is an explicit n-ary variable which is constrained by p, and ⁻F is a fuzzy n-ary relation which may be interpreted as an elastic constraint on X. This canonical form and the meaning-composition process for propositions and dispositions are illustrated by several examples among which is the proposition p≜ Over the past few years Naomi earned far more than most of her close friends.

It is suggested that communication between humans – as well as between humans and machines – may be made more precise by the employment of a meaning representation language PRUF which is based on the concept of a possibility distribution. A brief exposition of PRUF is presented and its application to precisiation of meaning is illustrated by a number of examples.

The conventional approaches to decision analysis are based on the assumption that the probabilities which enter into the assessment of the consequences of a decision are known numbers. In most realistic settings, this assumption is of questionable validity since the data from which the probabilities must be estimated are usually incomplete, imprecise or not totally reliable.

In the approach outlined in this paper, the probabilities are assumed to be fuzzy rather than real numbers. It is shown how such probabilities may be estimated from fuzzy data and a basic relation between joint, conditional and marginal fuzzy probabilities is established. Manipulation of fuzzy probabilities requires, in general, the use of fuzzy arithmetic, and many of the properties of fuzzy probabilities are simple generalizations of the corresponding properties of real-valued probabilities.

The basic idea underlying the approach outlined in this paper is that commonsense knowledge may be regarded as a collection of dispositions, that is, propositions which are preponderantly, but not necessarily always, true. Technically, a disposition may be interpreted as a proposition with implicit fuzzy quantifiers, e.g., most, almost all, usually, often, etc. For example, a disposition such as Swedes are blond may be interpreted as most Swedes are blond. For purposes of inference from commonsense knowledge, the conversion of a disposition into a proposition with explicit fuzzy quantifiers sets the stage for an application of syllogistic reasoning in which the premises are allowed to be of the form Q A's are B's, where A and B are fuzzy predicates and Q is a fuzzy quantifier. In general, the conclusion yielded by such reasoning is a proposition which may be converted into a disposition through the suppression of fuzzy quantifiers.

During the past several years, the question of how to deal with uncertainty in the context of expert systems has attracted a great deal of attention because much of the information which is resident in the knowledge base of a typical expert system is imprecise, incomplete or not totally reliable.

The existing approaches to the management of uncertainty in expert systems are based for the most part on probability theory or its variants. However, it may be argued, as it is done in this paper, that probability theory is not sufficiently expressive as a language of uncertainty to represent the meaning of the imprecise facts and rules that form the knowledge base of a typical expert system. In an alternative approach which is outlined in this paper, fuzzy logic forms the basis for both meaning representation and inference. In particular, syllogistic reasoning is used to formulate a collection of rules for combination of evidence, with fuzzy quantifiers replacing probabilities and certainty factors as indicators of the degree of uncertainty.

During the past two years, the Dempster-Shafer theory of evidence has attracted considerable attention within the AI community as a promising method of dealing with uncertainty in expert systems. As presented in the literature, the theory is hard to master. In a simple approach that is outlined in this paper, the Dempster-Shafer theory is viewed in the context of relational databases as the application of familiar retrieval techniques to second-order relations, that is, relations in which the data entries are relations in first normal form. The relational viewpoint clarifies some of the controversial issues in the Dempster-Shafer theory and facilitates its use in AI-oriented applications.

The following sections are included:

• Introduction

• Meaning Representation

• Inference

• REFERENCES AND RELATED PUBLICATIONS

The concept of usuality relates to propositions which are usually true or, more precisely, to events which have a high probability of occurrence. For example, usually Cait is very cheerful, usually a TV set weighs about fifty pounds, etc. Such propositions are said to be usuality-qualified. A usuality-qualified proposition may be expressed in the form usually (X is F), in which X is a variable taking values in a universe of discourse U and F is a fuzzy subset of U which may be interpreted as a usual value of X. In general, a usual value of variable, X, is not unique, and any fuzzy subset of U qualifies to a degree to be a usual value of X. A usuality-qualified proposition in which usually is implicit rather than explicit is said to be a disposition. Simple examples of dispositions are snow is white, a cup of coffee costs about fifty cents and Swedes are taller than Italians.

In this paper, we outline a theory of usuality in which the point of departure is a method of representing the meaning of usuality-qualified propositions. Based on this method, a system of inference for usuality-qualified propositions may be developed. As examples, a dispositional version of the Aristotelian Barbara syllogism as well as a dispositional version of the modus ponens are described. Such dispositional rules of inference are of direct relevance to commonsense reasoning and, in particular, to commonsense decision analysis. A potentially important application area for the theory of usuality is the management of uncertainty in expert systems.

A disposition may be interpreted as a proposition which is preponderantly, but not necessarily always, true. In this sense, birds can fly is a disposition, as are the propositions Swedes are blond, snow is white, and slimness is attractive. An idea which underlies the theory described in this article 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, usually, etc. For example, birds can fly may be interpreted as the result of suppressing 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 with explicit fuzzy quantifiers is referred to as explicitation or restoration. Explicitation sets the stage for representing the meaning of a disposition through the use of test-score semantics [see L.A. Zadeh: International J. Man–Machine Studies, 10 395–460 (1978); Empirical Semantics, 281–349 (1982)]. In this approach to semantics, a proposition, p, is viewed as a collection of interrelated elastic constraints, and the meaning of p is represented as a procedure which tests, scores and aggregates the constraints which are induced by p. The article 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 article, we formulate a definition of typicality and establish a connection between the typical and usual values of a variable.

The following sections are included:

• Introduction

• Fuzzy sets and linguistic variables

• Representation of complex concepts

• Fuzzy logic control

• Reasoning with commonsense knowledge
  • Usuality

The following sections are included:

• Introduction

• Dispositionality and Usuality

• References and Related Publications

The conventional approaches to knowledge representation, e.g., semantic networks, frames, predicate calculus, and Prolog, are based on bivalent logic. A serious shortcoming of such approaches is their inability to come to grips with the issue of uncertainty and imprecision. As a consequence, the conventional approaches do not provide an adequate model for modes of reasoning which are approximate rather than exact. Most modes of human reasoning and all of common sense reasoning fall into this category.

Fuzzy logic, which may be viewed as an extension of classical logical systems, provides an effective conceptual framework for dealing with the problem of knowledge representation in an environment of uncertainty and imprecision. Meaning representation in fuzzy logic is based on test-score semantics. In this semantics, a proposition is interpreted as a system of elastic constraints, and reasoning is viewed as elastic constraint propagation. Our paper presents a summary of the basic concepts and techniques underlying the application of fuzzy logic to knowledge representation and 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.

In retrospect, the year 1990 may well be viewed as the beginning of a new trend in the design of household appliances, consumer electronics, cameras, and other types of widely used consumer products. The trend in question relates to a marked increase in what might be called the Machine Intelligence Quotient (MIQ) of such products compared to what it was before 1990. Today, we have microwave ovens and washing machines that can figure out on their own what settings to use to perform their tasks optimally; cameras that come close to professional photographers in picture-taking ability; and many other products that manifest an impressive capability to reason, make intelligent decisions, and learn from experience.

In the nearly four decades which have passed since the launching of the Sputnik, great progress has been achieved in our understanding of how to model, identify and control complex systems. However, to be able to design systems having high MIQ (Machine Intelligence Quotient), a profound change in the orientation of control theory may be required. More specifically, what may be needed is the employment of soft computing—rather than hard computing—in systems analysis and design. Soft computing—unlike hard computing—is tolerant of imprecision, uncertainty and partial truth.

At this juncture, the principal constituents of soft computing are fuzzy logic, neurocomputing and probabilistic reasoning. In this paper, the focus is on the role of fuzzy logic. The basic ideas underlying fuzzy logic and its applications to modeling, identification and control are described and illustrated by examples. The role model for fuzzy logic is the human mind.

Soft computing is a collection of methodologies that aim to exploit the tolerance for imprecision and uncertainty to achieve tractahility, robustness, and low solution cost. Its principal constituents are fuzzy logic, neurocomputing, and probabilistic reasoning. Soft computing is likely to play an increasingly important role in many application areas, including software engineering. The role model for soft computing is the human mind.

The relationship between probability theory and fuzzy logic has long been an object of discussion and some controversy. The position articulated in this article is that probability theory by itself is not sufficient for dealing with uncertainty and imprecision in real-world settings. To enhance its effectiveness, probability theory needs an infusion of concepts and techniques drawn from fuzzy logic—especially the concept of a linguistic variable and the calculus of fuzzy if-then rules. In the final analysis, probability theory and fuzzy logic are complementary rather than competitive.

Text of a lecture presented on the occasion of the award of Doctorate Honoris Causa, University Of Oviedo, Spain.

Right Honourable and most distinguished Rector, most worthy officers of the University, dear professors, researchers, and students, Ladies and Gentlemen,

It is a great honor for me to stand here today as a recipient of the Doctorate Honoris Causa. The award of the Doctorate Honoris Causa has a special significance for me. First, because Spain is a country in which the theory of fuzzy sets and its concomitants have been embraced by many prominent mathematicians, scientists and engineers; and second, because Spain and Spanish people have always occupied a very warm spot in my heart. One cannot but admire the richness of Spanish culture and its intellectual traditions. Spain has produced and it continuing to produce men and women who have contributed and are contributing so much to arts, music, literature, and science. But what touches me most is the warmth and generosity of the Spanish people. In today's world of turbulence and conflict, these are qualities that are in short supply.

The following sections are included:

• SUBJECT INDEX