Generic Inference: A Unifying Theory for Automated Reasoning

Table of contents

1. Cover
2. Half Title page
3. Title page
4. Copyright page
5. Dedication
6. List of Instances and Applications
7. List of Figures
8. Acknowledgments
9. Introduction
   1. Generic Algorithms
   2. Complexity Considerations
   3. Generic Constructions
   4. The Content of this Book
   5. Part I: Local Computation
   6. Part II: Generic Constructions
   7. Part III: Applications
   8. Beyond the Content of this Book
10. Part I: Local Computation
    1. Chapter 1: Valuation Algebras
       1. 1.1 Operations and Axioms
       2. 1.2 First Examples
       3. 1.3 Conclusion
       4. Appendix: Generalizations of the Valuation Algebra Framework
       5. A.1 Ordered sets and Lattices
       6. A.2 Valuation Algebras on General Lattices
       7. A.3 Valuation Algebras with Partial Projection
    2. Chapter 2: Inference Problems
       1. 2.1 Graphs, Trees and Hypergraphs
       2. 2.2 Knowledgebases and their Representation
       3. 2.3 The Inference Problem
       4. 2.4 Conclusion
    3. Chapter 3: Computing Single Queries
       1. 3.1 Valuation Algebras with Variable Elimination
       2. 3.2 Fusion and Bucket Elimination
       3. 3.3 Valuation Algebras with Neutral Elements
       4. 3.4 Valuation Algebras with Null Elements
       5. 3.5 Local Computation as Message-Passing Scheme
       6. 3.6 Covering Join Trees
       7. 3.7 Join Tree Construction
       8. 3.8 The Collect Algorithm
       9. 3.9 Adjoining an Identity Element
       10. 3.10 The Generalized Collect Algorithm
       11. 3.11 An Application: The Fast Fourier Transform
       12. 3.12 Conclusion
       13. Appendix: Proof of the Generalized Collect Algorithm
    4. Chapter 4: Computing Multiple Queries
       1. 4.1 The Shenoy-Shafer Architecture
       2. 4.2 Valuation Algebras with Inverse Elements
       3. 4.3 The Lauritzen-Spiegelhalter Architecture
       4. 4.4 The Hugin Architecture
       5. 4.5 The Idempotent Architecture
       6. 4.6 Answering Uncovered Queries
       7. 4.7 Scaling and Normalization
       8. 4.8 Local Computation with Scaling
       9. 4.9 Conclusion
       10. Appendix: Valuation Algebras with Division
       11. D.1 Properties for the Introduction of Division
       12. D.2 Proofs of Division-Based Architectures
       13. D.3 Proof for Scaling in Valuation Algebras
11. Part II: Generic Constructions
    1. Chapter 5: Semiring Valuation Algebras
       1. 5.1 Semirings
       2. 5.2 Semirings and Order
       3. 5.3 Semiring Valuation Algebras
       4. 5.4 Examples of Semiring Valuation Algebras
       5. 5.5 Properties of Semiring Valuation Algebras
       6. 5.6 Some Computational Aspects
       7. 5.7 Set-Based Semiring Valuation Algebras
       8. 5.8 Properties of Set-Based Semiring Valuation Algebras
       9. 5.9 Conclusion
       10. Appendix: Semiring Valuation Algebras with Division
       11. E.1 Separative Semiring Valuation Algebras
       12. E.2 Regular Semiring Valuation Algebras
       13. E.3 Cancellative Semiring Valuation Algebras
       14. E.4 Idempotent Semiring Valuation Algebras
       15. E.5 Scalable Semiring Valuation Algebras
    2. Chapter 6: Valuation Algebras for Path Problems
       1. 6.1 Some Path Problem Examples
       2. 6.2 The Algebraic Path Problem
       3. 6.3 Quasi-Regular Semirings
       4. 6.4 Quasi-Regular Valuation Algebras
       5. 6.5 Properties of Quasi-Regular Valuation Algebras
       6. 6.6 Kleene Algebras
       7. 6.7 Kleene Valuation Algebras
       8. 6.8 Properties of Kleene Valuation Algebras
       9. 6.9 Further Path Problems
       10. 6.10 Conclusion
    3. Chapter 7: Language and Information
       1. 7.1 Propositional Logic
       2. 7.2 Linear Equations
       3. 7.3 Information in Context
       4. 7.4 Conclusion
12. Part III: Applications
    1. Chapter 8: Dynamic Programming
       1. 8.1 Solutions and Solution Extensions
       2. 8.2 Computing Solutions
       3. 8.3 Optimization and Constraint Problems
       4. 8.4 Computing Solutions of Optimization Problems
       5. 8.5 Conclusion
    2. Chapter 9: Sparse Matrix Techniques
       1. 9.1 Systems of Linear Equations
       2. 9.2 Symmetric, Positive Definite Matrices
       3. 9.3 Semiring Fixpoint Equation Systems
       4. 9.4 Conclusion
    3. Chapter 10: Gaussian Information
       1. 10.1 Gaussian Systems and Potentials
       2. 10.2 Generalized Gaussian Potentials
       3. 10.3 Gaussian Information and Gaussian Potentials
       4. 10.4 Valuation Algebra of Gaussian Potentials
       5. 10.5 An Application: Gaussian Dynamic Systems
       6. 10.6 An Application: Gaussian Bayesian Networks
       7. 10.7 Conclusion
       8. Appendix:
       9. J.1 Valuation Algebra Properties of Hints
       10. J.2 Gaussian Densities
13. References
14. Index