Possibility Theory: An Approach to Computerized Processing of Uncertainty

Possibility Theory: An Approach to Computerized Processing of Uncertainty

by Didier Dubois, Henri Prade

Paperback(Softcover reprint of the original 1st ed. 1988)

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In the evolution of scientific theories, concern with uncertainty is almost invariably a concomitant of maturation. This is certainly true of the evolution· of physics, economics, operations research, communication sciences, and a host of other fields. And it is true of what has been happening more recently in the area of artificial intelligence, most notably in the development of theories relating to the management of uncertainty in knowledge-based systems. In science, it is traditional to deal with uncertainty through the use of probability theory. In recent years, however, it has become increasingly clear that there are some important facets of uncertainty which do not lend themselves to analysis by classical probability-based methods. One such facet is that of lexical elasticity, which relates to the fuzziness of words in natural languages. As a case in point, even a simple relation X, Y, and Z, expressed as if X is small and Y is very large then between Z is not very small, does not lend itself to a simple interpretation within the framework of probability theory by reason of the lexical elasticity of the predicates small and large.

Product Details

ISBN-13: 9781468452891
Publisher: Springer US
Publication date: 03/14/2012
Edition description: Softcover reprint of the original 1st ed. 1988
Pages: 280
Product dimensions: 7.01(w) x 10.00(h) x 0.02(d)

Table of Contents

1. Measures of Possibility and Fuzzy Sets.- 1.1. Imprecision and Uncertainty.- 1.2. Traditional Models of Imprecision and Uncertainty.- 1.3. Confidence Measures.- 1.3.1. Measures of Possibility and of Necessity.- 1.3.2. Possibility and Probability.- 1.4. Fuzzy Sets.- 1.5. Elementary Fuzzy Set Operations.- 1.6. Practical Methods for Determining Membership Functions.- 1.6.1. Vague Categories as Perceived by an Individual.- 1.6.2. Fuzzy Sets Constructed from Statistical Data.- 1.6.3. Remarks on the Set of Degrees of Membership.- 1.7. Confidence Measures for a Fuzzy Event.- 1.8. Fuzzy Relations and Cartesian Products of Fuzzy Sets.- References.- 2. The Calculus of Fuzzy Quantities.- 2.1. Definitions and a Fundamental Principle.- 2.1.1. Fuzzy Quantities, Fuzzy Intervals, Fuzzy Numbers.- 2.1.2. The Extension Principle.- 2.2. Calculus of Fuzzy Quantities with Noninteractive Variables.- 2.2.1. Fundamental Result.- 2.2.2. Relation to Interval Analysis.- 2.2.3. Application to Standard Operations.- 2.2.4. The Problem of Equivalent Representations of a Function.- 2.3. Practical Calculation with Fuzzy Intervals.- 2.3.1. Parametric Representation of a Fuzzy Interval.- 2.3.2. Exact Practical Calculation with the Four Arithmetic Operations.- 2.3.3. Approximate Calculation of Functions of Fuzzy Intervals.- 2.4. Further Calculi of Fuzzy Quantities.- 2.4.1. “Pessimistic” Calculus of Fuzzy Quantities with Interactive Variables.- 2.4.2. “Optimistic” Calculus of Fuzzy Quantities with Noninteractive Variables.- 2.5. Illustrative Examples.- 2.5.1. Estimation of Resources in a Budget.- 2.5.2. Calculation of a PERT Analysis with Fuzzy Duration Estimates.- 2.5.3. A Problem in the Control of a Machine Tool.- Appendix: Computer Programs.- References.- 3. The Use of Fuzzy Sets for the Evaluation and Ranking of Objects.- 3.1. A Quantitative Approach to Multiaspect Choice.- 3.1.1. Basic Principles of the Approach.- 3.1.2. Fuzzy Set-Theoretic Operations.- 3.1.3. Application to the Combination of Criteria.- 3.1.4. Identification of Operators.- 3.1.5. Example.- 3.2. Comparison of Imprecise Evaluations.- 3.2.1. Comparison of a Real Number and a Fuzzy Interval.- 3.2.2. Comparison of Two Fuzzy Intervals.- 3.2.3. Ordering of n Fuzzy Intervals.- 3.2.4. Computer Implementation.- 3.2.5. Example.- Appendix: Computer Programs.- References.- 4. Models for Approximate Reasoning in Expert Systems.- 4.1. Remarks on Modeling Imprecision and Uncertainty.- 4.1.1. Credibility and Plausibility.- 4.1.2. Decomposable Measures.- 4.1.3. Vague Propositions.- 4.1.4. Evaluating the Truth Value of a Proposition.- 4.2. Reasoning from Uncertain Premises.- 4.2.1. Deductive Inference with Uncertain Premises.- 4.2.2. Complex Premises.- 4.2.3. Combining Degrees of Uncertainty Relative to the Same Proposition.- 4.3. Inference from Vague or Fuzzy Premises.- 4.3.1. Representation of the Rule “if X is A, then Y is B”.- 4.3.2. “Generalized” Modus Ponens.- 4.3.3. Complex Premises.- 4.3.4. Combining Possibility Distributions.- 4.4. Brief Summary of Current Work and Systems.- 4.5. Example.- Appendix A..- Appendix B: Computer Programs.- References.- 5. Heuristic Search in an Imprecise Environment, and Fuzzy Programming.- 5.1. Heuristic Search in an Imprecise Environment.- 5.1.1. A and A* Algorithms.- 5.1.2. The Classical Traveling Salesman Problem (Reminder).- 5.1.3. Heuristic Search with Imprecise Evaluations.- 5.1.4. Heuristic Search with Fuzzy Values.- 5.2. An Example of Fuzzy Programming: Tracing the Execution of an Itinerary Specified in Imprecise Terms.- 5.2.1. Execution and Chaining of Instructions.- 5.2.2. Illustrative Example.- 5.2.3. Problems Arising in Fuzzy Programming.- 5.2.4. Concluding Remarks.- Appendix: Computer Programs.- A.1. Selection of “the Smallest” of N Fuzzy Numbers.- A.2. Tracing Imprecisely Specified Itineraries.- References.- 6. Handling of Incomplete or Uncertain Data and Vague Queries in Database Applications.- 6.1. Representation of Incomplete or Uncertain Data.- 6.1.1. Representing Data by Means of Possibility Distributions.- 6.1.2. Differences and Similarities with Other Fuzzy Approaches.- 6.1.3. Dependencies and Possibilistic Information.- 6.2. The Extended Relational Algebra and the Corresponding Query Language.- 6.2.1. Generalization of ?-Selection.- 6.2.2. Cartesian Product, ?-Join, and Projection.- 6.2.3. Union and Intersection—Redundancy.- 6.2.4. Queries Employing Other Operations.- 6.3. Example.- 6.3.1. Representation of Data.- 6.3.2. Examples of Queries.- 6.4. Conclusion.- Appendix: Computer Program.- A.1. Data Structures.- A.2. Representation of Queries.- A.3. Description of Implemeted Procedures.- References.

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