Generalized Theory Of Uncertainty Research Articles

A Guide to Help Prepare Readers for J M Keynes's A Treatise on Probability (1921)

A Guide to Help Prepare Readers for J M Keynes's A Treatise on Probability (1921)

Fuzzy logic—a personal perspective

This paper marks the 50th anniversary of the publication of my first paper on fuzzy sets, “Fuzzy sets,” Information and Control, 1965. What is of historical interest is that initially—and for some time thereafter—my paper was an object of indifference, skepticism and derision. A prominent school of thought claimed that fuzzy set theory is probability theory in disguise. Positive comments were few and far between. In contrast, my ideas were welcomed with open arms in Japan. In the seventies and eighties of last century, fuzzy set theory and fuzzy logic began to gain acceptance in Europe and, more particularly, in Eastern Europe and the Soviet Union. In part, many negative reactions to my papers reflected the fact that the word “fuzzy” has pejorative connotations. In large measure, science is based on the classical, Aristotelian, bivalent logic. Binarization—drawing a sharply defined boundary between two classes—is a deeply entrenched Cartesian tradition. What is not widely recognized is that this tradition has outlived its usefulness. One of the principal contributions of fuzzy logic is providing a basis for a progression from binarization to graduation, from binarism to pluralism, from black and white to shades of gray. Graduation involves association of a class which has unsharp (fuzzy) boundaries with degrees/grades of membership. Classes with unsharp boundaries are pervasive in human cognition. Most words in natural language are labels of such classes. This paper is a concise exposition of what I consider to be my principal contributions to the development of fuzzy set theory and fuzzy logic. Among the contributions which are discussed are: introduction of the concept of a fuzzy set, FL-generalization, the concept of a linguistic variable, information granulation, precisiation of meaning, generalized theory of uncertainty (GTU), the concept of a restriction, restriction-centered theory of truth and meaning, the information principle, and similarity-based definitions of possibility and probability.

IMPLICATIVE FILTERS OF R0-ALGEBRAS BASED ON FUZZY POINTS

Abstract. As a generalization of the concept of a fuzzy implicative fil-ter which is introduced by Liu and Li [3], the notion of (∈,∈∨q k )-fuzzyimplicative filters is introduced, and related properties are investigated.The relationship between (∈, ∈∨q k )-fuzzy filters and (∈, ∈∨q k )-fuzzyimplicative filters is established. Conditions for an (∈,∈∨q k )-fuzzy filterto be an (∈, ∈∨q k )-fuzzy implicative filter are considered. Character-izations of an (∈, ∈∨q k )-fuzzy implicative filter are provided, and theimplication-based fuzzy implicative filters of an R 0 -algebra is discussed. 1. IntroductionOne important task of artificial intelligence is to make the computers sim-ulate beings in dealing with certainty and uncertainty in information. Logicappears in a “sacred” (respectively, a “profane”) form which is dominant inproof theory (respectively, model theory). The role of logic in mathematicsand computer science is twofold – as a tool for applications in both areas, anda technique for laying the foundations. Non-classical logic including many-valued logic, fuzzy logic, etc., takes the advantage of classical logic to handleinformation with various facets of uncertainty (see [11] for generalized theory ofuncertainty), such as fuzziness, randomness etc. Non-classical logic has becomea formal and useful tool for computer science to deal with fuzzy informationand uncertain information. Among all kinds of uncertainties, incomparabil-ity is an important one which can be encountered in our life. The conceptof R

Some Approaches to Combining Probabilistic and Fuzzy Uncertainties

Different types of uncertainty are widely spread in all areas of human activity. Probabilistic uncertainties are related to the chances of occurrence of random events. To deal with this kind of uncertainty, statistics and probability theory are successfully employed. Another kind of uncertainty, fuzzy uncertainties refer to imprecision and fuzziness of different kinds of measurements. To cope with this kind of uncertainty, the fuzzy set theory is used. This paper addresses widespread approaches to combining probabilistic and fuzzy uncertainties. The theoretical fundamentals of the approaches are considered within the context of the generalized theory of uncertainty (GTU).

Evolution of fuzzy logic: from intelligent systems and computation to human mind

Inspired by human's remarkable capability to perform a wide variety of physical and mental tasks without any measurements and computations and dissatisfied with classical logic as a tool for modeling human reasoning in an imprecise environment, Lotfi A. Zadeh developed the theory and foundation of fuzzy logic with his 1965 paper "Fuzzy sets" (Zadeh in Inf Control 8:378---53, 1965) and extended his work with his 2005 paper "Toward a generalized theory of uncertainty (GTU)--an outline" (Zadeh in Inf Control, 2005). Fuzzy logic has at least two main sources over the past century. The first of these sources was initiated by Peirce in the form what he called a logic of vagueness in 1900s, and the second source is Lotfi's A. Zadeh work, fuzzy sets and fuzzy Logic in the 1960s and 1970s.

Generalized theory of uncertainty (GTU)—principal concepts and ideas

Uncertainty is an attribute of information. The path-breaking work of Shannon has led to a universal acceptance of the thesis that information is statistical in nature. Concomitantly, existing theories of uncertainty are based on probability theory. The generalized theory of uncertainty (GTU) departs from existing theories in essential ways. First, the thesis that information is statistical in nature is replaced by a much more general thesis that information is a generalized constraint, with statistical uncertainty being a special, albeit important case. Equating information to a generalized constraint is the fundamental thesis of GTU. Second, bivalence is abandoned throughout GTU, and the foundation of GTU is shifted from bivalent logic to fuzzy logic. As a consequence, in GTU everything is or is allowed to be a matter of degree or, equivalently, fuzzy. Concomitantly, all variables are, or are allowed to be granular, with a granule being a clump of values drawn together by a generalized constraint. And third, one of the principal objectives of GTU is achievement of NL-capability, that is, the capability to operate on information described in natural language. NL-capability has high importance because much of human knowledge, including knowledge about probabilities, is described in natural language. NL-capability is the focus of attention in the present paper. The centerpiece of GTU is the concept of a generalized constraint. The concept of a generalized constraint is motivated by the fact that most real-world constraints are elastic rather than rigid, and have a complex structure even when simple in appearance. The paper concludes with examples of computation with uncertain information described in natural language.

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 general theory of uncertainty based on t-conorms

We first extend a t-conorm C to C(f) for certain f:Rn→R, sup ∥f∥<+∞ are used to define a generalized fuzzy measure on a class of subsets of Rn. The generalized fuzzy measure is employed to develop a general theory of uncertain variables including: (1) uncertainty distributions (u.d.'s) of functions of uncertain variables; (2) expected value and variance; (3) marginal and joint u.d.'s; and (4) conditional u.d.'s. Numerous examples are presented showing the theory contains both discrete and continuous possibility theory and probability theory.