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