The primary aim of this monograph is to provide a formal framework for the representation and management of uncertainty and vagueness in the field of artificial intelligence. Particular emphasis is put on a thorough analysis of these phenomena and on the development of sound mathematical modeling approaches. The scope of the book also includes implementational aspects and a valuation of existing models and systems. The fundamental claim of the book is that vagueness and uncertainty can be handled adequately by using measure-theoretic methods. The presentation of applicable knowledge…mehr
The primary aim of this monograph is to provide a formal framework for the representation and management of uncertainty and vagueness in the field of artificial intelligence. Particular emphasis is put on a thorough analysis of these phenomena and on the development of sound mathematical modeling approaches. The scope of the book also includes implementational aspects and a valuation of existing models and systems. The fundamental claim of the book is that vagueness and uncertainty can be handled adequately by using measure-theoretic methods. The presentation of applicable knowledge representation formalisms and reasoning algorithms shows that efficiency requirements do not necessarily require renunciation of an uncompromising mathematical modeling approach. The results are used to evaluate systems based on probabilistic methods as well as on non-standard concepts such as certainty factors, fuzzy sets, and belief functions. The book is self-contained and addresses researchers and practitioners in the field of knowledge based systems and decision support systems. It is suitable as a textbook for graduate-level students in AI, operations research, and applied probability. Diese Monographie vermittelt die mathematischen Hilfsmittel für die formale Darstellung und Bewältigung von Unsicherheit und Ungenauigkeit in der Künstlichen Intelligenz. Im Vordergrund steht eine gründliche Analyse dieser Phänomene und die Entwicklung mathematischer Modelle auf der Grundlage maßtheoretischer Methoden. Das Buch richtet sich an Forscher und Entwickler im Bereich wissensbasierter Systeme und ist auch als Lehrbuch für fortgeschrittene Studenten der Informatik und der Mathematik mit Interesse an Künstlicher Intelligenz geeignet.
1. General Considerations of Uncertainty and Vagueness.- 1.1 Artificial Intelligence.- 1.2 Modeling Ignorance.- 1.3 The Scope of the Book.- 2. Introduction.- 2.1 Basic Notations.- 2.2 A Simple Example.- 2.3 Vagueness and Uncertainty.- 3. Vague Data.- 3.1 Basic Concepts.- 3.2 On the Origin of Vague Data.- 3.3 Uncertainty Handling by Means of Layered Contexts.- 3.4 The General Case.- 3.5 Concluding Remarks.- 4. Probability Theory.- 4.1 Basic Concepts.- 4.2 Probabilities on Different Sample Spaces.- 4.3 Bayesian Inference.- 4.4 Classes of Probabilities.- 4.5 Decision Making Aspects.- 4.6 Aggregating Probability Distributions.- 4.7 Concluding Remarks.- 5. Random Sets.- 5.1 Random Variables.- 5.2 The Notion of a Random Set.- 5.3 Decision Making in the Context of Vague Data.- 5.4 The Notion of an Information Source.- 5.5 Concluding Remarks.- 6. Mass Distributions.- 6.1 Basic Concepts.- 6.2 Different Frames of Discernment.- 6.3 Measures for Possibility/Necessity.- 6.4 Generalized Mass Distributions.- 6.5 Decision Making with Mass Distributions.- 6.6 Knowledge Representation with Mass Distributions.- 6.7 Simplifying Assumptions.- 6.8 Concluding Remarks.- 7. On Graphical Representations.- 7.1 Graphs and Trees.- 7.2 Hypergraphs and Hypertrees.- 7.3 Analysis of Simple Hypertrees.- 7.4 Dependency Networks.- 7.5 Triangulated Graphs.- 7.6 Directed Acyclic Graphs.- 7.7 Concluding Remarks.- 8. Modeling Aspects.- 8.1 Rule Based Approaches.- 8.2 Model Based Representations.- 8.3 Dependency Network Based Systems.- 9. Heuristic Models.- 9.1 MYCIN - The Certainty Factor Approach.- 9.2 RUM - Triangular Norms and Conorms.- 9.3 INFERNO - A Bounds Propagation Architecture.- 9.4 Other Heuristic Models.- 10. Fuzzy Set Based Models.- 10.1 Fuzzy Sets.- 10.2 Possibility Distributions.- 10.3Approximate Reasoning.- 10.4 Reasoning with Fuzzy Truth Value.- 10.5 Conclusions.- 11. Reasoning with L-Sets.- 11.1 Knowledge Representation with L-Sets.- 11.2 On the Interpretation of Vague Rules.- 11.3 L-Sets on Product Spaces.- 11.4 Local Computation of Marginal ¿-Sets.- 11.5 The Propagation Algorithm.- 11.6 Aspects of Implementation.- 12. Probability Based Models.- 12.1 The Interpretation of Rules.- 12.2 The Straightforward Use of Probabilities.- 12.3 PROSPECTOR - Inference Networks.- 12.4 Decomposable Graphical Models.- 12.5 Propagation Based on Dependency Networks.- 12.6 Concluding Remarks.- 13. Models Based on the Dempster-Shafer Theory of Evidence.- 13.1 The Mathematical Theory of Evidence.- 13.2 Knowledge Representation Aspects.- 13.3 The Straightforward Use of Belief Functions.- 13.4 Belief Functions in Hierarchical Hypothesis Spaces.- 13.5 MacEvidence - Belief Propagation in Markov Trees.- 13.6 Conclusions.- 14. Reasoning with Mass Distributions.- 14.1 Matrix Notation for Specializations.- 14.2 Specializations in Product Spaces.- 14.3 Knowledge Representation with Mass Distributions.- 14.4 Local Computations with Mass Distributions.- 14.5 The Propagation Algorithm.- 14.6 Aspects of Implementation.- 15. Related Research.- 15.1 Nonstandard Logics.- 15.2 Integrating Uncertainty Calculi and Logics.- 15.3 Symbolic Methods.- 15.4 Conclusions.- References.
1. General Considerations of Uncertainty and Vagueness.- 1.1 Artificial Intelligence.- 1.2 Modeling Ignorance.- 1.3 The Scope of the Book.- 2. Introduction.- 2.1 Basic Notations.- 2.2 A Simple Example.- 2.3 Vagueness and Uncertainty.- 3. Vague Data.- 3.1 Basic Concepts.- 3.2 On the Origin of Vague Data.- 3.3 Uncertainty Handling by Means of Layered Contexts.- 3.4 The General Case.- 3.5 Concluding Remarks.- 4. Probability Theory.- 4.1 Basic Concepts.- 4.2 Probabilities on Different Sample Spaces.- 4.3 Bayesian Inference.- 4.4 Classes of Probabilities.- 4.5 Decision Making Aspects.- 4.6 Aggregating Probability Distributions.- 4.7 Concluding Remarks.- 5. Random Sets.- 5.1 Random Variables.- 5.2 The Notion of a Random Set.- 5.3 Decision Making in the Context of Vague Data.- 5.4 The Notion of an Information Source.- 5.5 Concluding Remarks.- 6. Mass Distributions.- 6.1 Basic Concepts.- 6.2 Different Frames of Discernment.- 6.3 Measures for Possibility/Necessity.- 6.4 Generalized Mass Distributions.- 6.5 Decision Making with Mass Distributions.- 6.6 Knowledge Representation with Mass Distributions.- 6.7 Simplifying Assumptions.- 6.8 Concluding Remarks.- 7. On Graphical Representations.- 7.1 Graphs and Trees.- 7.2 Hypergraphs and Hypertrees.- 7.3 Analysis of Simple Hypertrees.- 7.4 Dependency Networks.- 7.5 Triangulated Graphs.- 7.6 Directed Acyclic Graphs.- 7.7 Concluding Remarks.- 8. Modeling Aspects.- 8.1 Rule Based Approaches.- 8.2 Model Based Representations.- 8.3 Dependency Network Based Systems.- 9. Heuristic Models.- 9.1 MYCIN - The Certainty Factor Approach.- 9.2 RUM - Triangular Norms and Conorms.- 9.3 INFERNO - A Bounds Propagation Architecture.- 9.4 Other Heuristic Models.- 10. Fuzzy Set Based Models.- 10.1 Fuzzy Sets.- 10.2 Possibility Distributions.- 10.3Approximate Reasoning.- 10.4 Reasoning with Fuzzy Truth Value.- 10.5 Conclusions.- 11. Reasoning with L-Sets.- 11.1 Knowledge Representation with L-Sets.- 11.2 On the Interpretation of Vague Rules.- 11.3 L-Sets on Product Spaces.- 11.4 Local Computation of Marginal ¿-Sets.- 11.5 The Propagation Algorithm.- 11.6 Aspects of Implementation.- 12. Probability Based Models.- 12.1 The Interpretation of Rules.- 12.2 The Straightforward Use of Probabilities.- 12.3 PROSPECTOR - Inference Networks.- 12.4 Decomposable Graphical Models.- 12.5 Propagation Based on Dependency Networks.- 12.6 Concluding Remarks.- 13. Models Based on the Dempster-Shafer Theory of Evidence.- 13.1 The Mathematical Theory of Evidence.- 13.2 Knowledge Representation Aspects.- 13.3 The Straightforward Use of Belief Functions.- 13.4 Belief Functions in Hierarchical Hypothesis Spaces.- 13.5 MacEvidence - Belief Propagation in Markov Trees.- 13.6 Conclusions.- 14. Reasoning with Mass Distributions.- 14.1 Matrix Notation for Specializations.- 14.2 Specializations in Product Spaces.- 14.3 Knowledge Representation with Mass Distributions.- 14.4 Local Computations with Mass Distributions.- 14.5 The Propagation Algorithm.- 14.6 Aspects of Implementation.- 15. Related Research.- 15.1 Nonstandard Logics.- 15.2 Integrating Uncertainty Calculi and Logics.- 15.3 Symbolic Methods.- 15.4 Conclusions.- References.
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