Uncertainty and Vagueness in Knowledge Based Systems Numerical Methods

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. It puts particular emphasis on a thorough analysis of these phenomena and on the development of sound mathematical modeling approaches. Beyond this theoretical basis the scope of the book includes also implementational aspects and a valuation of existing models and systems. The fundamental ambition of this book is to show that vagueness and un­ certainty can be handled adequately by using measure-theoretic methods. The presentation of applicable knowledge representation formalisms and reasoning algorithms substantiates the claim that efficiency requirements do not necessar­ ily require renunciation of an uncompromising mathematical modeling. These results are used to evaluate systems based on probabilistic methods as well as on non-standard concepts such as certainty factors, fuzzy sets or belief functions. The book is intended to be self-contained and addresses researchers and practioneers in the field of knowledge based systems. It is in particular suit­ able as a textbook for graduate-level students in AI, operations research and applied probability. A solid mathematical background is necessary for reading this book. Essential parts of the material have been the subject of courses given by the first author for students of computer science and mathematics held since 1984 at the University in Braunschweig.

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.