Uncertainty and Vagueness in Knowledge Based Systems

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.- 2.3.1 Modeling Vague Data.- 2.3.2 Modeling Partial Belief.- 3. Vague Data.- 3.1 Basic Concepts.- 3.2 On the Origin of Vague Data.- 3.3 Uncertainty Handling by Means of Layered Contexts.- 3.3.1 Possibility and Necessity.- 3.3.2 Operations with Vague Data.- 3.3.3 On the Interpretation of Vague Data.- 3.4 The General Case.- 3.5 Concluding Remarks.- 4. Probability Theory.- 4.1 Basic Concepts.- 4.1.1 Axiomatic Probability Theory.- 4.1.2 On the Interpretation of a Probability.- 4.1.3 Practical Aspects.- 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.- 5.2.1 Weighted Sets versus Random Sets.- 5.2.2 On the Updating of Random Sets.- 5.3 Decision Making in the Context of Vague Data.- 5.4 The Notion of an Information Source.- 5.4.1 Updating Information Sources.- 5.4.2 The Combination of Information Sources.- 5.5 Concluding Remarks.- 6. Mass Distributions.- 6.1 Basic Concepts.- 6.1.1 Condensed Representations of Random Sets.- 6.1.2 Belief Functions.- 6.2 Different Frames of Discernment.- 6.2.1 Specializations.- 6.2.2 Strict Specializations.- 6.2.3 Orthogonal Extensions.- 6.2.4 Conjunctive and Disjunctive Extensions.- 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.- 6.6.1 Encoding Knowledge by Mass Distributions.- 6.6.2 Integration of Different Pieces of Knowledge.- 6.7 Simplifying Assumptions.- 6.8 Concluding Remarks.- 7. On Graphical Representations.- 7.1 Graphs and Trees.- 7.1.1 Undirected Graphs.- 7.1.2 Trees.- 7.2 Hypergraphs and Hypertrees.- 7.2.1 Hypertrees.- 7.2.2 Simple Hypertrees.- 7.3 Analysis of Simple Hypertrees.- 7.3.1 Markov Trees.- 7.3.2 Knowledge Representation with Hypergraphs.- 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.- 8.2.1 Requirements on Models.- 8.2.2 On the Structure of Models.- 8.2.3 On the Choice of Mathematical Models.- 8.2.4 Selected Problems with Mathematical Models.- 8.3 Dependency Network Based Systems.- 9. Heuristic Models.- 9.1 MYCIN - The Certainty Factor Approach.- 9.1.1 The Mathematical Model.- 9.1.2 Uncertainty Representation in MYCIN.- 9.1.3 Related Models and Proposals.- 9.1.4 Conclusions.- 9.2 RUM - Triangular Norms and Conorms.- 9.2.1 Families of Uncertainty Calculi - Triangular Norms and Conorms.- 9.2.2 RUM.- 9.2.3 Final Remarks.- 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.- 10.3 Approximate 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.- 12.2.1 The Model of Ishizuka et al.- 12.2.2 The Model of Adams.- 12.2.3 Discussions.- 12.3 PROSPECTOR - Inference Networks.- 12.3.1 The Inference Network Model.- 12.3.2 PROSPECTOR.- 12.3.3 Discussion and Related Work.- 12.4 Decomposable Graphical Models.- 12.4.1 The Model of Pearl.- 12.4.2 MUNIN - An Application.- 12.4.3 HUGIN - A Professional Tool.- 12.5 Propagation Based on Dependency Networks.- 12.5.1 Knowledge Representation.- 12.5.2 Graph Structure and Conditional Independence.- 12.5.3 Local Computation of Marginal Probability Distributions...- 12.5.4 The Propagation Algorithm.- 12.5.5 Aspects of Implementation.- 12.5.6 Numerical Example.- 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.- 13.2.1 Representing Pieces of Knowledge.- 13.2.2 Integration of Pieces of Evidence.- 13.3 The Straightforward Use of Belief Functions.- 13.3.1 The Model of Ishizuka et al.- 13.3.2 The Model of Ginsberg.- 13.3.3 Discussion, Related Work.- 13.4 Belief Functions in Hierarchical Hypothesis Spaces.- 13.4.1 Gordon and Shortliffe's Extension to MYCIN.- 13.4.2 The Model of Yen - A Quasi-Probabilistic Approach.- 13.5 MacEvidence - Belief Propagation in Markov Trees.- 13.5.1 Belief Propagation in Markov Trees.- 13.5.2 MacEvidence.- 13.5.3 Discussion.- 13.6 Conclusions.- 14. Reasoning with Mass Distributions.- 14.1 Matrix Notation for Specializations.- 14.1.1 Specialization Matrices.- 14.1.2 Composition of Specialization Matrices.- 14.1.3 Properties of Specialization Matrices.- 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.