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El. knyga: Generic Inference: A Unifying Theory for Automated Reasoning

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  • Formatas: PDF+DRM
  • Išleidimo metai: 09-May-2011
  • Leidėjas: John Wiley & Sons Inc
  • Kalba: eng
  • ISBN-13: 9781118010846
Kitos knygos pagal šią temą:
Generic Inference: A Unifying Theory for Automated ReasoningDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Išleidimo metai: 09-May-2011
  • Leidėjas: John Wiley & Sons Inc
  • Kalba: eng
  • ISBN-13: 9781118010846
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"The book provides a rigorous algebraic study of the most popular inference formalisms with a special focus on their wide application area and shows that all these tasks can be performed by a single generic inference algorithm. It will include 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)"--

"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"--

Provided by publisher.

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.
List of Instances and Applications
xiii
List of Figures and Tables
xvii
Acknowledgments xxi
Introduction xxiii
PART I LOCAL COMPUTATION
1 Valuation Algebras
3(14)
1.1 Operations and Axioms
4(2)
1.2 First Examples
6(11)
1.3 Conclusion
17(1)
Appendix: Generalizations of the Valuation Algebra Framework
17(84)
A.1 Ordered Sets and Lattices
17(3)
A.1.1 Partitions and Partition Lattices
19(1)
A.2 Valuation Algebras on General Lattices
20(4)
A.3 Valuation Algebras with Partial Projection
24(5)
Problem Sets and Exercises
26(3)
2 Inference Problems
29(18)
2.1 Graphs, Trees and Hypergraphs
30(2)
2.2 Knowledgebases and their Representation
32(1)
2.3 The Inference Problem
33(11)
2.4 Conclusion
44(3)
Problem Sets and Exercises
44(3)
3 Computing Single Queries
47(54)
3.1 Valuation Algebras with Variable Elimination
49(1)
3.2 Fusion and Bucket Elimination
50(19)
3.2.1 The Fusion Algorithm
50(5)
3.2.2 Join Trees
55(2)
3.2.3 The Bucket Elimination Algorithm
57(4)
3.2.4 First Complexity Considerations
61(5)
3.2.5 Some Generalizing Complexity Comments
66(2)
3.2.6 Limitations of Fusion and Bucket Elimination
68(1)
3.3 Valuation Algebras with Neutral Elements
69(4)
3.3.1 Stable Valuation Algebras
70(3)
3.4 Valuation Algebras with Null Elements
73(2)
3.5 Local Computation as Message-Passing Scheme
75(3)
3.5.1 The Complexity of Fusion as Message-Passing Scheme
77(1)
3.6 Covering Join Trees
78(4)
3.7 Join Tree Construction
82(5)
3.7.1 Join Tree Construction by Triangulation
83(4)
3.8 The Collect Algorithm
87(4)
3.8.1 The Complexity of the Collect Algorithm
90(1)
3.8.2 Limitations of the Collect Algorithm
91(1)
3.9 Adjoining an Identity Element
91(1)
3.10 The Generalized Collect Algorithm
92(6)
3.10.1 Discussion of the Generalized Collect Algorithm
96(1)
3.10.2 The Complexity of the Generalized Collect Algorithm
97(1)
3.11 An Application: The Fast Fourier Transform
98(2)
3.12 Conclusion
100(1)
Appendix: Proof of the Generalized Collect Algorithm
101(52)
Problem Sets and Exercises
103(6)
4 Computing Multiple Queries
109(44)
4.1 The Shenoy-Shafer Architecture
111(10)
4.1.1 Collect & Distribute Phase
114(2)
4.1.2 The Binary Shenoy-Shafer Architecture
116(1)
4.1.3 Performance Gains due to the Identity Element
117(1)
4.1.4 Complexity of the Shenoy-Shafer Architecture
118(2)
4.1.5 Discussion of the Shenoy-Shafer Architecture
120(1)
4.1.6 The Super-Cluster Architecture
120(1)
4.2 Valuation Algebras with Inverse Elements
121(2)
4.2.1 Idempotent Valuation Algebras
122(1)
4.3 The Lauritzen-Spiegelhalter Architecture
123(5)
4.3.1 Complexity of the Lauritzen-Spiegelhalter Architecture
127(1)
4.4 The HUGIN Architecture
128(3)
4.4.1 Complexity of the HUGIN Architecture
131(1)
4.5 The Idempotent Architecture
131(3)
4.5.1 Complexity of the Idempotent Architecture
134(1)
4.6 Answering Uncovered Queries
134(9)
4.6.1 The Complexity of Answering Uncovered Queries
141(2)
4.7 Scaling and Normalization
143(4)
4.8 Local Computation with Scaling
147(5)
4.8.1 The Scaled Shenoy-Shafer Architecture
147(3)
4.8.2 The Scaled Lauritzen-Spiegelhalter Architecture
150(1)
4.8.3 The Scaled HUGIN Architecture
151(1)
4.9 Conclusion
152(1)
Appendix: Valuation Algebras with Division
153(28)
D.1 Properties for the Introduction of Division
154(13)
D.1.1 Separative Valuation Algebras
155(9)
D.1.2 Regular Valuation Algebras
164(3)
D.1.3 Idempotent Valuation Algebras
167(1)
D.2 Proofs of Division-Based Architectures
167(2)
D.2.1 Proof of the Lauritzen-Spiegelhalter Architecture
167(1)
D.2.2 Proof of the HUGIN Architecture
168(1)
D.3 Proof for Scaling in Valuation Algebras
169(12)
Problem Sets and Exercises
173(8)
PART II GENERIC CONSTRUCTIONS
5 Semiring Valuation Algebras
181(24)
5.1 Semirings
182(4)
5.1.1 Multidimensional Semirings
185(1)
5.1.2 Semiring Matrices
185(1)
5.2 Semirings and Order
186(4)
5.3 Semiring Valuation Algebras
190(2)
5.4 Examples of Semiring Valuation Algebras
192(3)
5.5 Properties of Semiring Valuation Algebras
195(3)
5.5.1 Semiring Valuation Algebras with Neutral Elements
196(1)
5.5.2 Stable Semiring Valuation Algebras
196(1)
5.5.3 Semiring Valuation Algebras with Null Elements
197(1)
5.6 Some Computational Aspects
198(2)
5.7 Set-Based Semiring Valuation Algebras
200(3)
5.8 Properties of Set-Based Semiring Valuation Algebras
203(1)
5.8.1 Neutral and Stable Set-Based Semiring Valuations
203(1)
5.8.2 Null Set-Based Semiring Valuations
204(1)
5.9 Conclusion
204(1)
Appendix: Semiring Valuation Algebras with Division
205(88)
E.1 Separative Semiring Valuation Algebras
205(4)
E.2 Regular Semiring Valuation Algebras
209(3)
E.3 Cancellative Semiring Valuation Algebras
212(1)
E.4 Idempotent Semiring Valuation Algebras
213(2)
E.5 Scalable Semiring Valuation Algebras
215(4)
Problem Sets and Exercises
217(2)
6 Valuation Algebras for Path Problems
219(46)
6.1 Some Path Problem Examples
220(3)
6.2 The Algebraic Path Problem
223(9)
6.3 Quasi-Regular Semirings
232(5)
6.4 Quasi-Regular Valuation Algebras
237(8)
6.5 Properties of Quasi-Regular Valuation Algebras
245(1)
6.6 Kleene Algebras
246(7)
6.6.1 Matrices over Kleene Algebras
249(4)
6.7 Kleene Valuation Algebras
253(3)
6.8 Properties of Kleene Valuation Algebras
256(1)
6.9 Further Path Problems
256(6)
6.10 Conclusion
262(3)
Problem Sets and Exercises
262(3)
7 Language and Information
265(28)
7.1 Propositional Logic
266(8)
7.1.1 Language and Semantics
266(3)
7.1.2 Propositional Information
269(2)
7.1.3 Some Computational Aspects
271(3)
7.2 Linear Equations
274(6)
7.2.1 Equations and Solution Spaces
275(4)
7.2.2 Algebra of Affine Spaces
279(1)
7.3 Information in Context
280(9)
7.3.1 Contexts: Models and Language
281(5)
7.3.2 Tuple Model Structures
286(3)
7.4 Conclusion
289(4)
Problem Sets and Exercises
290(3)
PART III APPLICATIONS
8 Dynamic Programming
293(28)
8.1 Solutions and Solution Extensions
294(3)
8.2 Computing Solutions
297(7)
8.2.1 Computing all Solutions with Distribute
297(2)
8.2.2 Computing some Solutions without Distribute
299(4)
8.2.3 Computing all Solutions without Distribute
303(1)
8.3 Optimization and Constraint Problems
304(7)
8.3.1 Totally Ordered Semirings
304(3)
8.3.2 Optimization Problems
307(4)
8.4 Computing Solutions of Optimization Problems
311(6)
8.5 Conclusion
317(4)
Problem Sets and Exercises
318(3)
9 Sparse Matrix Techniques
321(64)
9.1 Systems of Linear Equations
322(16)
9.1.1 Gaussian Variable Elimination
323(2)
9.1.2 Fill-ins and Local Computation
325(3)
9.1.3 Regular Systems
328(7)
9.1.4 LDR-Decomposition
335(3)
9.2 Symmetric, Positive Definite Matrices
338(26)
9.2.1 Valuation Algebra of Symmetric Systems
339(4)
9.2.2 Solving Symmetric Systems
343(5)
9.2.3 Symmetric Gaussian Elimination
348(6)
9.2.4 Symmetric Decompositions and Elimination Sequences
354(3)
9.2.5 An Application: The Least Squares Method
357(7)
9.3 Semiring Fixpoint Equation Systems
364(18)
9.3.1 Computing Quasi-Inverse Matrices
365(1)
9.3.2 Local Computation with Quasi-Regular Valuations
366(5)
9.3.3 Local Computation with Kleene Valuations
371(11)
9.4 Conclusion
382(3)
Problem Sets and Exercises
382(3)
10 Gaussian Information
385(43)
10.1 Gaussian Systems and Potentials
386(9)
10.2 Generalized Gaussian Potentials
395(5)
10.3 Gaussian Information and Gaussian Potentials
400(14)
10.3.1 Assumption-Based Reasoning
400(3)
10.3.2 General Gaussian Information
403(4)
10.3.3 Combination of Gaussian Information
407(1)
10.3.4 Projection of Gaussian Information
408(6)
10.4 Valuation Algebra of Gaussian Potentials
414(3)
10.5 An Application: Gaussian Dynamic Systems
417(6)
10.6 An Application: Gaussian Bayesian Networks
423(5)
10.7 Conclusion
428(1)
Appendix
428(9)
J.1 Valuation Algebra Properties of Hints
428(5)
J.2 Gaussian Densities
433(4)
Problem Sets and Exercises
435(2)
References 437(10)
Index 447
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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