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Generic Inference: A Unifying Theory for Automated Reasoning

ISBN: 978-1-118-01086-0
250 pages
January 2012
Generic Inference: A Unifying Theory for Automated Reasoning (1118010868) cover image


This book provides a rigorous algebraic study of the most popular inference formalisms with a special focus on their wide application area, showing that all these tasks can be performed by a single generic inference algorithm. Written by the leading international authority on the topic, it includes an algebraic perspective (study of the valuation algebra framework), an algorithmic perspective (study of the generic inference schemes) and a "practical" perspective (formalisms and applications). Researchers in a number of fields including artificial intelligence, operational research, databases and other areas of computer science; graduate students; and professional programmers of inference methods will benefit from this work.
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Table of Contents

List of Instances and Applications.

List of Figures and Tables.



Part I Local Computation.

1 Valuation Algebras.

1.1 Operations and Axioms.

1.2 First Examples.

1.3 Conclusion.

2 Inference Problems.

2.1 Graphs, Trees and Hypergraphs.

2.2 Knowledgebases and their Representation.

2.3 The Inference Probloem.

2.4 Conclusion.

3 Computing Single Queries.

3.1 Valuation Algebras with Variable Elimination.

3.2 Fusion and Bucker 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.

4 Computing Multiple Queries.

4.1 The Shenoy Shafer.

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.

Part II Generic Constructions.

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.

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.

7 Language and Information.

7.1 Propositional Logic.

7.2 Linear Equations.

7.3 Information in Context.

7.4 Conclusion.

Part III Applications.

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.

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.

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.




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Author Information

Marc Pouly, PhD, received the Award for Outstanding PhD Thesis in Computer Science at the University of Fribourg (Switzerland), in 2008. He was visiting researcher at the Cork Constraint Computation Centre in Ireland and, since 2010, he is researcher at the Interdisciplinary Centre for Security, Reliability and Trust of the University of Luxembourg.

Jürg Kohlas, PhD, is Professor of Theoretical Computer Science in the Department of Informatics at the University of Fribourg (Switzerland). His research interests include algebraic theory of information and probabilistic argumentation.

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