Lotfi A. Zadeh

Fuzzy logic = computing with words

As its name suggests, computing with words (CW) is a methodology in which words are used in place of numbers for computing and reasoning. The point of this note is that fuzzy logic plays a pivotal role in CW and vice-versa. Thus, as an approximation, fuzzy logic may be equated to CW. There are two major imperatives for computing with words. First, computing with words is a necessity when the available information is too imprecise to justify the use of numbers, and second, when there is a tolerance for imprecision which can be exploited to achieve tractability, robustness, low solution cost, and better rapport with reality. Exploitation of the tolerance for imprecision is an issue of central importance in CW. In CW, a word is viewed as a label of a granule; that is, a fuzzy set of points drawn together by similarity, with the fuzzy set playing the role of a fuzzy constraint on a variable. The premises are assumed to be expressed as propositions in a natural language. In coming years, computing with words is likely to evolve into a basic methodology in its own right with wide-ranging ramifications on both basic and applied levels.

Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic

There are three basic concepts that underlie human cognition: granulation, organization and causation. Informally, granulation involves decomposition of whole into parts; organization involves integration of parts into whole; and causation involves association of causes with effects. Granulation of an object A leads to a collection of granules of A, with a granule being a clump of points (objects) drawn together by indistinguishability, similarity, proximity or functionality. For example, the granules of a human head are the forehead, nose, cheeks, ears, eyes, etc. In general, granulation is hierarchical in nature. A familiar example is the granulation of time into years, months, days, hours, minutes, etc. Modes of information granulation (IG) in which the granules are crisp (c-granular) play important roles in a wide variety of methods, approaches and techniques. Crisp IG, however, does not reflect the fact that in almost all of human reasoning and concept formation the granules are fuzzy (f-granular). The granules of a human head, for example, are fuzzy in the sense that the boundaries between cheeks, nose, forehead, ears, etc. are not sharply defined. Furthermore, the attributes of fuzzy granules, e.g., length of nose, are fuzzy, as are their values: long, short, very long, etc. The fuzziness of granules, their attributes and their values is characteristic of ways in which humans granulate and manipulate information.

Probability measures of Fuzzy events

In probability theory [I], an event, A, is a member of a a-field, CY, of subsets of a sample space ~2. A probability measure, P, is a normed measure over a measurable space (Q, GY); that is, P is a real-valued function which assigns to every A in Gk’ a probability, P(A), such that (a) P(A) > 0 for all A E a, (b) P(Q) = 1; and (c) P is countably additive, i.e., if {Ai} is any collection of disjoint events, then

A computational approach to fuzzy quantifiers in natural languages

The generic term fuzzy quantifier is employed in this paper to denote the collection of quantifiers in natural languages whose representative elements are: several, most, much, not many, very many, not very many, few, quite a few, large number, small number, close to five, approximately ten, frequently, etc. In our approach, such quantifiers are treated as fuzzy numbers which may be manipulated through the use of fuzzy arithmetic and, more generally, fuzzy logic. A concept which plays an essential role in the treatment of fuzzy quantifiers is that of the cardinality of a fuzzy set. Through the use of this concept, the meaning of a proposition containing one or more fuzzy quantifiers may be represented as a system of elastic constraints whose domain is a collection of fuzzy relations in a relational database. This representation, then, provides a basis for inference from premises which contain fuzzy quantifiers. For example, from the propositions “Most Us are As” and “Most As are Bs,” it follows that “Most2 Us are Bs,” where most2 is the fuzzy product of the fuzzy proportion most with itself. The computational approach to fuzzy quantifiers which is described in this paper may be viewed as a derivative of fuzzy logic and test-score semantics. In this semantics, the meaning of a semantic entity is represented as a procedure which tests, scores and aggregates the elastic constraints which are induced by the entity in question.

The role of fuzzy logic in the management of uncertainty in expert systems

Management of uncertainty is an intrinsically important issue in the design of expert systems because much of the information in the knowledge base of a typical expert system is imprecise, incomplete or not totally reliable. In the existing expert systems, uncertainty is dealt with through a combination of predicate logic and probability-based methods. A serious shortcoming of these methods is that they are not capable of coming to grips with the pervasive fuzziness of information in the knowledge base, and, as a result, are mostly ad hoc in nature. An alternative approach to the management of uncertainty which is suggested in this paper is based on the use of fuzzy logic, which is the logic underlying approximate or, equivalently, fuzzy reasoning. A feature of fuzzy logic which is of particular importance to the management of uncertainty in expert systems is that it provides a systematic framework for dealing with fuzzy quantifiers, e.g., most, many, few, not very many, almost all, infrequently, about 0.8, etc. In this way, fuzzy logic subsumes both predicate logic and probability theory, and makes it possible to deal with different types of uncertainty within a single conceptual framework. In fuzzy logic, the deduction of a conclusion from a set of premises is reduced, in general, to the solution of a nonlinear program through the application of projection and extension principles. This approach to deduction leads to various basic syllogisms which may be used as rules of combination of evidence in expert systems. Among syllogisms of this type which are discussed in this paper are the intersection/product syllogism, the generalized modus ponens, the consequent conjunction syllogism, and the major-premise reversibility rule.

Toward a generalized theory of uncertainty (GTU): an outline

It is a deep-seated tradition in science to view uncertainty as a province of probability theory. The generalized theory of uncertainty (GTU) which is outlined in this paper breaks with this tradition and views uncertainty in a much broader perspective.Uncertainty is an attribute of information. A fundamental premise of GTU is that information, whatever its form, may be represented as what is called a generalized constraint. The concept of a generalized constraint is the centerpiece of GTU. In GTU, a probabilistic constraint is viewed as a special-albeit important-instance of a generalized constraint.A generalized constraint is a constraint of the form X isr R, where X is the constrained variable, R is a constraining relation, generally non-bivalent, and r is an indexing variable which identifies the modality of the constraint, that is, its semantics. The principal constraints are: possibilistic (r=blank); probabilistic (r=p); veristic (r=v); usuality (r=u); random set (r=rs); fuzzy graph (r=fg); bimodal (r=bm); and group (r=g). Generalized constraints may be qualified, combined and propagated. The set of all generalized constraints together with rules governing qualification, combination and propagation constitutes the generalized constraint language (GCL).The generalized constraint language plays a key role in GTU by serving as a precisiation language for propositions, commands and questions expressed in a natural language. Thus, in GTU the meaning of a proposition drawn from a natural language is expressed as a generalized constraint. Furthermore, a proposition plays the role of a carrier of information. This is the basis for equating information to a generalized constraint.In GTU, reasoning under uncertainty is treated as propagation of generalized constraints, in the sense that rules of deduction are equated to rules which govern propagation of generalized constraints. A concept which plays a key role in deduction is that of a protoform (abbreviation of prototypical form). Basically, a protoform is an abstracted summary-a summary which serves to identify the deep semantic structure of the object to which it applies. A deduction rule has two parts: symbolic-expressed in terms of protoforms-and computational.GTU represents a significant change both in perspective and direction in dealing with uncertainty and information. The concepts and techniques introduced in this paper are illustrated by a number of examples.

A Fuzzy-Set-Theoretic Interpretation of Linguistic Hedges

Abstract A basic idea suggested in this paper is that a linguistic hedge such as very, more or less, much, essentially. slightly, etc. may be viewed as an operator which acts on the fuzzy set representing the meaning of its operand. For example, in the case of the composite term very tall man, the operator very acts on the fuzzy meaning of the term tall man. To represent a hedge as an operator, it is convenient to define several elementary operations on fuzzy sets from which more complicated operations may be built up by combination or composition. In this way, an approximate representation for a hedge can be expressed in terms of such operations as complementation, intersection, concentration, dilation, contrast intensification, fuzzification, accentuation, etc. Two categories of hedges are considered. In the case of hedges of Type I, e.g., very, much, more or less, slightly, etc., the hedge can be approximated by an operator acting on a single fuzzy set. In the case of hedges of Type II, e.g., technical...

Fundamentals of fuzzy sets

Foreword L.A. Zadeh. Preface. Series Foreword. Contributing Authors. General Introduction D. Dubois, H. Prade. Part I: Fuzzy Sets. 1. Fuzzy Sets: History and Basic Notions D. Dubois, et al. 2. Fuzzy Set-Theoretic Operators and Quantifiers J. Fodor, R.R. Yager. 3. Measurement of Membership Functions: Theoretical and Empirical Work T. Bilgic, I.B. Turksen. Part II: Fuzzy Relations. 4. An Introduction to Fuzzy Relations S. Ovchinnikov. 5. Fuzzy Equivalence Relations: Advanced Material D. Boixader, et al. 6. Analytical Solution Methods for Fuzzy Relational Equations B. De Baets. Part III: Uncertainty. 7. Possibility Theory, Probability and Fuzzy Sets: Misunderstandings, Bridges and Gaps D. Dubois, et al. 8. Measures of Uncertainty and Information G.J. Klir. 9. Quantifying Different Facets of Fuzzy Uncertainty N.R. Pal, J.C. Bezdek. Part IV: Fuzzy Sets on the Real Line. 10. Fuzzy Interval Analysis D. Dubois, et al. 11. Metric Topology of Fuzzy Numbers and Fuzzy Analysis P. Diamond, P. Kloeden. Index.

PRUF—a meaning representation language for natural languages

PRUF—an acronym for Possibilistic Relational Universal Fuzzy—is a meaning representation language for natural languages which departs from the conventional approaches to the theory of meaning in several important respects. First, a basic assumption underlying PRUF is that the imprecision that is intrinsic in natural languages is, for the most part, possibilistic rather than probabilistic in nature. Thus, a proposition such as “Richard is tall” translates in PRUF into a possibility distribution of the variable Height (Richard), which associates with each value of the variable a number in the interval [0,11 representing the possibility that Height (Richard) could assume the value in question. More generally, a proposition, p , translates into a procedure, P, which returns a possibility distribution, Π p , with P and Π p representing, respectively, the meaning of p and the information conveyed by p . In this sense, the concept of a possibility distribution replaces that of truth as a foundation for the representation of meaning in natural languages. Second, the logic underlying PRUF is not a two-valued or multivalued logic, but a fuzzy logic, FL, in which the truth-values are linguistic, that is, are of the form true, not true, very true, more or less true, not very true , etc., with each such truth-value representing a fuzzy subset of the unit interval. The truth-value of a proposition is defined as its compatibility with a reference proposition, so that given two propositions p and r, one can compute the truth of p relative to r . Third, the quantifiers in PRUF—like the truth-values—are allowed to be linguistic, i.e. may be expressed as most, many, few, some, not very many, almost all, etc. Based on the concept of the cardinality of a fuzzy set, such quantifiers are given a concrete interpretation which makes it possible to translate into PRUF propositions exemplified by “Many tall men are much taller than most men,” “All tall women are blonde is not very true,” etc. The translation rules in PRUF are of four basic types: Type I—pertaining to modification; Type II—pertaining to composition; Type III—pertaining to quantification; and Type IV—pertaining to qualification and, in particular, to truth qualification, probability qualification and possibility qualification. The concepts of semantic equivalence and semantic entailment in PRUF provide a basis for question-answering and inference from fuzzy premises. In addition to serving as a foundation for approximate reasoning, PRUF may be employed as a language for the representation of imprecise knowledge and as a means of precisiation of fuzzy propositions expressed in a natural language.

A Fuzzy-Algorithmic Approach to the Definition of Complex or Imprecise Concepts

It may be argued, rather persuasively, that most of the concepts encountered in various domains of human knowledge are, in reality, much too complex to admit of simple or precise definition. This is true, for example, of the concepts of recession and utility in economics; schizophrenia and arthritis in medicine; stability and adaptivity in system theory; sparseness and stiffness in numerical analysis; grammaticality and meaning in linguistics; performance measurement and correctness in computer science; truth and causality in philosophy; intelligence and creativity in psychology; and obscenity and insanity in law.