Henri Prade

Fuzzy real algebra: Some results

Abstract In this paper, the possibility to perform easily most of the extended n-ary operations on fuzzy subsets of the real line is shown. A general algorithm is given. These results are particularized for usual operations such as addition, subtraction, multiplication, division, ‘max’ and ‘min’ operations for normalized convex fuzzy subsets of the real line, i.e. fuzzy numbers. A three parameters representation for fuzzy numbers is shown to be very convenient to perform usual operations. Lastly, interpretative comments about fuzzy real algebra are given and possible applications pointed out.

Fuzzy Sets in Approximate Reasoning and Information Systems

Series Foreword. Contributing Authors. Introduction. Part I: Reasoning. 1. Fuzzy Sets and Possibility Theory in Approximate and Plausible Reasoning B. Bouchon-Meunier, et al. 2. Weighted Inference Systems V. Novak. 3. Closure Operators in Fuzzy set Theory L. Biacino, G. Gerla. Part II: Learning and Fusion. 4. Learning Fuzzy Decision Rules B. Bouchon-Meunier, C. Marsala. 5. Neuro-Fuzzy Methods in Fuzzy Rule Generation D. Nauck, R. Kruse. 6. Merging Fuzzy Information D. Dubois, et al. Part III: Fuzzy Information Systems. 7. Fuzzy Databases P. Bosc, et al. 8. Fuzzy Set Techniques in Information Retrieval D.H. Kraft, et al. Summary. References

A theorem on implication functions defined from triangular norms.

Several transformation which enable implication functions in multivalued logics to be generated from conjunctions have been proposed in the literature. It is proved that for a rather general class of conjunctions modeled by triangular norms, the generation process is closed, thus shedding some light on the relationships between seemingly independent classes of implication functions.

Belief Change Rules in Ordinal and Numerical Uncertainty Theories

The situation of belief change in numerical uncertainty frameworks differs from the situation in classical logic in two respects: on the one hand, uncertainty theories are tailored to representing epistemic states involving shades of belief and are more expressive than classical logic in that respect. Indeed, considering a propositional belief set (a set of accepted beliefs) from the standpoint of uncertainty, a proposition is only surely true (it belongs to the belief set), surely false (its negation belongs to the belief set) or unknown (neither the proposition nor its negation belong to the belief set). Uncertainty theories express the extent to which an ultimately true or false proposition is believed. On the other hand, many logical-oriented belief revision theories are syntax-dependent (e.g., [Nebel, 1992], and in this book)—although they are semantically meaningful—while uncertainty theories adopt a semantic representation of epistemic states. In syntax-dependent theories of revision two logically equivalent, syntactically distinct, belief bases are generally not revised in the same way. The syntactic dimension of logical approaches introduces a level of increased complexity and expressiveness that cannot be grasped in the usual uncertainty theories. So, uncertainty theories are altogether more refined and less expressive than logical syntax-dependent approaches for the purpose of belief change.

Soft Methods for Handling Variability and Imprecision

Probability theory has been the only well-founded theory of uncertainty for a long time. It was viewed either as a powerful tool for modelling random phenomena, or as a rational approach to the notion of degree of belief. During the last thirty years, in areas centered around decision theory, artificial intelligence and information processing, numerous approaches extending or orthogonal to the existing theory of probability and mathematical statistics have come to the front. The common feature of those attempts is to allow for softer or wider frameworks for taking into account the incompleteness or imprecision of information. Many of these approaches come down to blending interval or fuzzy interval analysis with probabilistic methods. This book gathers contributions to the 4th International Conference on Soft methods in Probability and Statistics. Its aim is to present recent results illustrating such new trends that enlarge the statistical and uncertainty modeling traditions, towards the handling of incomplete or subjective information. It covers a broad scope ranging from philosophical and mathematical underpinnings of new uncertainty theories, with a stress on their impact in the area of statistics and data analysis, to numerical methods and applications to environmental risk analysis and mechanical engineering. A unique feature of this collection is to establish a dialogue between fuzzy random variables and imprecise probability theories.

Fuzzy-Set Based Logics — an History-Oriented Presentation of their Main Developments

The historical development of fuzzy logic may look somewhat erratic. The concept of approximate reasoning developed by Zadeh in the seventies in considerable details did not receive great attention at the time, neither from the logical community nor from the engineering community let alone the artificial intelligence community, despite isolated related works in the eighties. Engineers exploited very successful, sometimes ad hoc, numerical techniques borrowing only a small part of fuzzy set concepts. They did not implement the combination projection principle, which is the backbone of approximate reasoning. Finally, there is a long tradition of mutual distrust between artificial intelligence and fuzzy logic, because of the numerical flavor of the latter. A new trend, the fundamental thesis of Zadeh—namely, the “fuzzy logic is a logic of approximate reasoning” is again left on the side of the road. Yet the contention is that, a good approach to ensuring a full revival of fuzzy logic is to demonstrate its capability to reasoning about knowledge and uncertainty.

Introduction: The Real Contribution of Fuzzy Systems

Fuzzy logic (FL) was invented in the sixties by a leading expert in control engineering who realized that control theory had become beautiful enough to carry on its development on its own, but that there was many real problems it could not solve. Most real complex system control problems involve man. Hence applying control theory to complex control problems may require a formal understanding of how a human operator understands his system, what his goals are, and how he proceeds when controlling it. It requires a dedicated tool for representing human-originated information in a flexible way. And this is where fuzzy logic enters the picture (Zadeh [28]). In a paper that can be considered as the origin of the fuzzy rule-based approach, Zadeh [31] claimed that systems analysis and control requires a trade-off between representations that are very precise and accurate, such as numerical ones stemming from numerical functions, differential equations and the like, and representations that are intelligible, meaningful to humans, hence summarized, possibly in linguistic terms. However these two kinds of representations are sort of antagonistic because the more precise and accurate a representation, the less understandable it is, and conversely. This is Zadeh’s principle of incompatibility, and it explains the particular position of fuzzy logic in control and systems engineering, and the misunderstandings and controversies it has created in the control engineering community. The contribution of fuzzy systems may not be at the level where traditional systems engineers expect contributions: where they expect improved performance, they might get improved intelligibility, flexibility and transparency in the representation of dynamic systems and the design of controllers.

Bipolar representations in reasoning, knowledge extraction and decision processes

This paper surveys various areas in information engineering where an explicit handling of positive and negative sides of information is appropriate. Three forms of bipolarity are laid bare. They can be instrumental in logical representations of incompleteness, rule representation and extraction, argumentation, and decision analysis.

A Glance at Non-Standard Models and Logics of Uncertainty and Vagueness

Historically, it is well known that the notion of probability emerged in the 17th century as a dual concept: chance, related to gaming problems, and subjective uncertainty, related to the question of reliability of testimonies. In the works of pioneers of probability theory, such as J. Bernoulli, chance, very soon connected to frequency of occurrence, was an additive notion but subjective probability was not so. However with the development of physical sciences, the non-additive side of probability was forgotten (see Shafer, 1978). So much so as 20th century researchers in decision theory have devoted much effort in the non-frequentist justification of additive probability as a model for subjective uncertainty in rational decision strategies.

Decision, nonmonotonic reasoning and possibilistic logic

The paper survey recent AI-oriented works in qualitative decision developed by the authors in the framework of possibility theory. Lottery-based and act-based axiomatics underlying pessimistic and optimistic criteria for decision under uncertainty are first briefly restated, when uncertainty and preferences are encoded with an ordinal scale. A logical machinery capable of computing optimal decisions in the sense of these criteria is presented. Then an approach to qualitative decision under uncertainty which does not require a commensurateness hypothesis between the uncertainty and the preference scales is proposed; this approach is closely related to nonmonotonic reasoning, but turns our to be ineffective for practical decision. Lastly, the modeling of preference as prioritized sets of goals, as sets of solutions reaching some given level of satisfaction, or in terms of possibilistic constraints is discussed briefly.