ISBN: 3790814148
TITLE: Propositional, Probabilistic and Evidential Reasoning
AUTHOR: Liu
TOC:

Foreword vii
Preface ix
1 Introduction
1.1 Classical and Non-monotonic Reasoning 1
1.2 Classifications of Non-monotonic Systems 3
1.2.1 Purely Symbolic Non-monotonic Systems 3
1.2.2 Purely Numerical Non-monotonic Systems 4
1.2.3 Hybrid Non-monotonic Systems 4
1.3 Hybrid Systems: Some Examples 6
1.3.1 Extending Classical Logic 7
1.3.2 Associating Numerical Values with Non-monotonic Rules 10
1.3.3 Extending the ATMS 11
1.3.4 Possibilistic Logic 13
1.3.5 Argumentation 16
1.4 Qualitative Probabilistic Reasoning 17
1.4.1 Purely Qualitative Probabilistic Networks 17
1.4.2 Semi-qualitative Probabilistic Networks 19
1.4.3 Incorporating Qualitative Reasoning into Quantitative Reasoning 21
1.5 Rough Sets Theory 22
1.6 Incidence Calculus 23
1.7 Structure of the Book 24
1.8 Remarks 26
1.9 Summary 27
2 Incidence Calculus
2.1 Incidence Calculus Theories 29
2.1.1 Basic Definitions 29
2.1.2 Incidence Calculus Theories 33
2.2 The Legal Assignment Finder 37
2.2.1 Assignments and Inference Rules 37
2.2.2 Constraint Sets 39
2.2.3 Termination Decisions of the Inference Procedure 40
2.3 Examples of Using the Legal Assignment Finder 41
2.4 Assigning Incidences to Formulae 48
2.5 Summary 53
3 Generalizing Incidence Calculus
3.1 Generalized Incidence Calculus Theories 55
3.2 Basic Incidence Assignment 63
3.3 An Incidence Function Has a Unique Basic Incidence Assignment 69
3.4 A Basic Incidence Assignment Maps to a Family of Incidence Assignments 72
3.5 Summary 77
4 From Numerical to Symbolic Assignments
4.1 An Algorithm for Assigning Incidences 80
4.2 Unique Output of the Algorithm 82
4.3 Similarity of Separate Incidence Assignments 88
4.4 Fundamental Nature of Basic Incidence Assignments 94
4.5 Summary 98
5 Combining Multiple Pieces of Evidence
5.1 Effects of New Information 102
5.2 Combination Rule in Generalized Incidence Calculus 104
5.3 DS-independent Information 109
5.4 Examples 112
5.5 Summary 117
6 The Dempster-Shafer Theory of Evidence
6.1 Basic Concepts in the Dempster-Shafer Theory of Evidence 121
6.2 Probability Background of Mass Functions 123
6.2.1 Dempster's Probability Prototype of Mass Functions 124
6.2.2 Deriving Mass Functions from Probability Spaces 126
6.3 Problems with Dempster's Combination Rule 127
6.3.1 Dempster's Combination Framework 128
6.3.2 The Condition for Using Dempster's Combination Rule 130
6.3.3 Examples 132
6.4 Computational Complexity Problem in DS Theory 142
6.4.1 Linear Algorithms of Dempster's Combination Rule 142
6.4.2 Algorithms Based on Markov Trees 144
6.4.3 Parallel Techniques for Managing Complexity 146
6.5 Heuristic Knowledge Representation in DS Theory 150
6.5.1 Yen's Probabilistic Mapping 151
6.5.2 Evidential Mapping 152
6.6 The Open World Assumption 156
6.7 Summary 157
7 A Comprehensive Comparison of Generalized Incidence Calculus and Dempster-Shafer Theory
7.1 Comparison 1: Representing Evidence 160
7.2 Comparison 11: Combining DS-Independent Evidence 165
7.3 Comparison 111: Combining Dependent Evidence 170
7.4 Comparison IV: Some Other Aspects of the Two Theories 175
7.4.1 Recovering Mass Functions 175
7.4.2 Recovering Probability Spaces 178
7.5 Summary 180
8 Assumption-Based Truth Maintenance Systems
8.1 Reasoning Mechanism in the ATMS 183
8.1.1 Structure of Nodes 183
8.1.2 Types of Nodes 185
8.2 Non-Redundant Justification Sets and Environments 187
8.3 Probabilistic Assumption Sets 189
8.4 Conclusion 194
9 Relations Between Extended Incidence Calculus and Assumption-Based Truth Maintenance System
9.1 Review of Generalized Incidence Calculus 198
9.1.1 Essential Semantic Implication Sets in Incidence Calculus l98
9.1.2 Similarities of Reasoning Models in Generalized Incidence Calculus and the ATMS 202
9.2 Constructing Labels and Calculating Beliefs in Nodes Using Generalized Incidence Calculus 203
9.2.1 An Example 203
9.2.2 Algorithm of Equivalent Transformation from an ATMS to Generalized Incidence Calculus 208
9.2.3 Formal Proof 210
9.2.4 Comparison with Laskey and Lehner's Work 215
9.3 Generalized Incidence Calculus Can Provide Justifications for the ATMS 219
9.4 Conclusion 220
10 Conclusion
10.1 Rough Sets and Incidence Calculus 223
10.1.1 Basics of Rough Sets 224
10.1.2 Set Semantics of Propositional Logic 227
10.1.3 Relationship Between Rough Sets and Incidence Calculus 236
10.2 Related Works 238
10.2.1 Interval Structures 238
10.2.2 Bacchus's Propositional Probability Structure 240
10.2.3 Multiple-valued Logics 241
10.3 Summary 242
10.3.1 Coupling Incidence Calculus with other Theories in Practice 242
10.3.2 Where Incidences Come From? 243
10.3.3 Significance of Numerical-symbolic Reasoning 244
Bibliography 245
Index 265
Mathematical Notation 269
List of Figures 271
List of Tables 273
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