Fuzzy Set: 1965 … Fuzzy Logic: 1973 … BISC: 1990 … Human-Machine Perception: 2000 - …
Lotfi A. Zadeh
A Personal Statement (1965-2014)
Research prior to 1965
A new direction—development of fuzzy set theory and fuzzy logic
Publication of my first paper on fuzzy sets in 1965 marked the beginning of a new phase of my scientific career. From 1965 on, almost all of my publications have been focused on development of fuzzy set theory, fuzzy logic and their applications. It should be noted that most of my papers published prior to 1977, and all papers published since then, are single-authored. My publications list contains 247 papers and books. My publications are associated with 125,653 Google Scholar citations.
My first paper entitled “Fuzzy sets,” got a mixed reaction. My strongest supporter was the late Professor Richard Bellman, an eminent mathematician and a leading contributor to systems analysis and control. For the most part, I encountered skepticism, derision and sometimes outright hostility. There were two principal reasons: The word “fuzzy” is usually used in a pejorative sense; and, more importantly, my abandonment of the classical, Aristotelian, bivalent logic was a radical departure from deep-seated scientific traditions. What changed the situation was the enthusiastic acceptance of my ideas in Japan. Starting in the early seventies, Japanese universities and industrial research laboratories began to play an active role in the development of fuzzy logic and its applications. Much has happened since that period. A summary of the current status of fuzzy logic and its applications appears in the Appendix. What is worthy of note is that as of 2/26/14, my 1965 paper on fuzzy sets drew 53,172 Google Scholar citations. It is the highest cited paper in the literature of Computer Science (Web of Science); it is the seventh highest cited paper in the literature of Science (Web of Science). The first five highest cited papers in Science are in biomedicine, and the sixth highest ranking paper is in Chemistry.
The concept of a linguistic variable. Decision-making in a fuzzy environment.
During the past forty-nine years, I played an active and visible role in the development of fuzzy logic and its applications. My 1973 paper entitled “Outline of a new approach to the analysis of complex systems and decision processes,” was a path-breaking work in which the concept of a linguistic variable was introduced, and a calculus of fuzzy if-then rules was developed. Today, almost all applications of fuzzy logic employ the concept of a linguistic variable, and there is a huge literature centering on fuzzy-rule-based calculi. A related paper entitled, “The concept of a linguistic variable and its application to approximate reasoning,” published in 1975, is the highest cited paper in Information Sciences. Another important paper was my 1970 paper entitled, “Decision-making in a fuzzy environment,” co-authored with R.E. Bellman. This paper is widely viewed as a seminal contribution to application of fuzzy logic to decision analysis. My 1978 paper on possibility theory, is the highest cited paper in Fuzzy Sets and Systems.
Development of possibility theory
My 1978 paper, “Fuzzy sets as a basis for a theory of possibility,” laid the foundation for what I called “possibility theory.” Greeted with skepticism at first, possibility theory has become a widely used tool for dealing with uncertainty, with the understanding that possibility theory and probability theory are complementary rather than competitive.
Development of a theory of approximate reasoning
My 1979 paper entitled, “A theory of approximate reasoning,” initiated a new direction in the development of fuzzy logic as the logic of approximate reasoning. The basic ideas introduced in this paper underlie most of the techniques which are in use today for purposes of inference and deduction from information which is approximate rather than exact.
In 1991, I introduced the concept of soft computing—a consortium of methodologies which collectively provide a foundation for the conception, design and utilization of intelligent systems. One of the principal components of soft computing is fuzzy logic. Today, the concept of soft computing is growing rapidly in visibility and importance. In 2005, a European Center for Soft Computing was established in Spain.
Many of my papers written in the eighties and early nineties were concerned, for the most part, with applications of fuzzy logic to knowledge representation and commonsense reasoning.
Computing with Words
In 1996, a major idea occurred to me—an idea which underlies most of my current research activities. This idea was described in two seminal papers entitled, “Fuzzy logic=Computing with words” and “From computing with numbers to computing with words—from manipulation of measurements to manipulation of perceptions.” My 1999 paper initiated a new direction in computation which I called, Computing with Words (CWW). CW opened the door to computation with information described in natural language—a system of computation which is of intrinsic importance because much of human knowledge is described in a natural language. Computation with information described in natural language cannot be dealt with through the use of the machinery of natural language processing (NLP). The problem is semantic imprecision of natural languages. More specifically, a natural language is basically a system for describing perceptions. Perceptions are intrinsically imprecise, reflecting the bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information. Semantic imprecision of natural languages is a concomitant of imprecision of perceptions.
Development of a theory of precisiation of meaning. The concept of a restriction.
Raw natural language does not lend itself to computation. A prerequisite to computation is precisiation of meaning. What I consider to be one of my major contributions is the development of a theory of precisiation of meaning, starting with my 1978 paper, “PRUF—a meaning-representation language for natural languages” and continuing to present. My theory of precisiation of meaning is a radical departure from traditional approaches to semantics of natural languages, especially possible-world and truth-conditional semantics. The centerpiece of my theory is the concept of a restriction (generalized constraint)—a concept which was introduced in my 1975 paper, “Calculus of fuzzy restrictions,” and extended in my 1986 paper, “Outline of a computational approach to meaning and knowledge representation based on the concept of a generalized assignment statement.” The key idea involves representing the meaning of a proposition, p, drawn from a natural language as a restriction. A restriction is an expression of the form X isr R, where X is the restricted (constrained) variable, R is the restricting (constraining) relation and r is an indexical variable which defines the way in which R restricts X. Generally, X, R and r are implicit in p. At this juncture, restriction-based semantics of natural languages is the only system of precisiation of meaning which makes it possible to solve problems which are described in a natural language. The importance of my theory of precisiation of meaning has not as yet been widely recognized because it breaks away from traditional theories of natural language. I believe that eventually my theory will gain acceptance and wide use.
Development of a generalized theory of uncertainty (GTU)
In a seminal paper published in 2002, “Toward a perception-based theory of probabilistic reasoning with imprecise probabilities” I initiated a significant generalization of probability theory. The ideas introduced in my 2002 paper were further developed in my 2005 paper, “Toward a generalized theory of uncertainty (GTU)—an outline” and in my 2006 paper, “Generalized theory of uncertainty (GTU)—principal concepts and ideas.” The Generalized Theory of Uncertainty (GTU) which is described in these papers adds to standard probability theory an essential capability which standard probability does not have—the capability to compute with probabilities, events, quantifiers and relations which are described in a natural language. As we move further into the age of machine intelligence and automated everyday reasoning, this capability is certain to play an increasingly important role in decision analysis, planning, risk assessment and economics. A key idea in GTU is that of equating information to restriction. The principal modes of restriction are possibilistic, probabilistic and veristic.
Development of extended fuzzy logic
In a short but important paper published in 2009, I outlined an extension of fuzzy logic which opens the door to mechanization of reasoning with unprecisiated concepts. A model of extended fuzzy logic is f-geometry. In Euclidian f-geometry, figures are drawn by hand with a spray pen. There is no ruler and no compass. There are f-lines, f-triangles and f-circles. There are f-definitions, f-axioms and f-proofs. At this stage, extended fuzzy logic is in its early stages of development, but it has a potential for important applications in the future.
Introduction of the concept of a Z-number
In a 2011 paper entitled, “A note on Z-numbers,” a new concept—the concept of a Z-number is introduced. Basically, a Z-number is an ordered pair of two fuzzy numbers. The first fuzzy number is a restriction on the values which a real-valued variable can take. The second fuzzy number is a restriction on the probability of the first fuzzy number. Typically, the two fuzzy numbers are described in a natural language. The concept of a Z-number is intended to associate a measure of reliability with the value of a variable. The concept of a Z-number has a potential for important applications in economics, planning, risk assessment and decision analysis.
A new direction which is being explored is aimed at enhancing Web IQ (WIQ) through addition of deduction capability to search engines. Existing search engines have this capability to a very limited degree. The principal obstacle is the nature of world knowledge. In large measure, world knowledge is perception-based, e.g., “it is hard to find parking near the campus before late afternoon.” Such knowledge cannot be dealt with through the use of methods based on classical, bivalent logic. In the approach that is being explored, world knowledge is dealt with through the use of PNL, in association with an epistemic lexicon and a modular, multiagent deduction database.
A new approach to truth and meaning
The concepts of truth and meaning have a position of centrality in logic and theories of natural language. In 2013, in a short but important paper entitled, “Toward a restriction-centered theory of truth and meaning (RCT)” I described a new approach to truth and meaning based on the concept of a restriction. In this paper, a proposition, p, drawn from a natural language is associated not just with one truth value—as in traditional theories, but with two truth values—internal truth value and external truth value. In representation of meaning, the concept of explanatory database, ED, plays a pivotal role. RCT is the only system which offers a capability to represent the meaning of fuzzy propositions, that is, propositions which contain words which are labels of fuzzy sets, e.g., tall, fast, most, etc. Propositions drawn from a natural language are predominantly fuzzy propositions. Existing approaches to semantics of natural languages, principally possible-world semantics and truth-conditional semantics, do not have this capability.
Similarity-based definitions of possibility and probability
The concept of probability has been around for more than two centuries. Probability theory is one of the most important and widely used theories in science. Probability theory is a deep and rigorous theory. But in real-world settings, there are many simple questions which relate to probability theory to which answers are hard to come up with. The problem is rooted in the fact that probability theory is based on the classical, Aristotelian, bivalent logic. Bivalent logic is intolerant of imprecision and partiality of truth. In the new approach which is outlined, fuzzy logic is employed to construct a similarity-based definition of probability which lend itself to the use of machine learning techniques. The similarity-based definition of probability opens the door to a wide range of applications in which very large databases are involved, particularly in the realms of medical diagnostics and recognition technology.
From its inception, fuzzy logic has been an object of controversy, skepticism and sometimes outright hostility. Eventually, the wide-ranging applications of fuzzy logic within science and technology have acquired visibility and acceptance. It is my belief that in coming years, fuzzy logic will continue to grow in importance and visibility. The importance of fuzzy logic will grow with the passage of time.
Fuzzy Set: 1965 … Fuzzy Logic: 1973 … BISC: 1990 … Human-Machine Perception: 2000 - …