viii
Contents
4.4 Dislocated Metrics . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5 Dislocated Generalized Ultrametrics . . . . . . . . . . . . . . 104
4.6 Quasimetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.7 A Hierarchy of Fixed-Point Theorems . . . . . . . . . . . . . 112
4.8 Relationships Between the Various Spaces . . . . . . . . . . . 114
4.9 Fixed-Point Theory for Multivalued Mappings . . . . . . . . 125
4.10 Partial Orders and Multivalued Mappings . . . . . . . . . . . 127
4.11 Metrics and Multivalued Mappings . . . . . . . . . . . . . . 129
4.12 Generalized Ultrametrics and Multivalued Mappings . . . . . 129
4.13 Quasimetrics and Multivalued Mappings . . . . . . . . . . . 132
4.14 An Alternative to Multivalued Mappings . . . . . . . . . . . 136
5 Supported Model Semantics
139
5.1 Two-Valued Supported Models . . . . . . . . . . . . . . . . . 140
5.2 Three-Valued Supported Models . . . . . . . . . . . . . . . . 151
5.3 A Hierarchy of Logic Programs . . . . . . . . . . . . . . . . . 159
5.4 Consequence Operators and Fitting-Style Operators . . . . . 161
5.5 Measurability Considerations . . . . . . . . . . . . . . . . . . 166
6 Stable and Perfect Model Semantics
169
6.1 The Fixpoint Completion . . . . . . . . . . . . . . . . . . . . 169
6.2 Stable Model Semantics . . . . . . . . . . . . . . . . . . . . . 171
6.3 Perfect Model Semantics . . . . . . . . . . . . . . . . . . . . 175
7 Logic Programming and Artificial Neural Networks
185
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.2 Basics of Artificial Neural Networks . . . . . . . . . . . . . . 188
7.3 The Core Method as a General Approach to Integration . . . 191
7.4 Propositional Programs . . . . . . . . . . . . . . . . . . . . . 192
7.5 First-Order Programs . . . . . . . . . . . . . . . . . . . . . . 196
7.6 Some Extensions – The Propositional Case . . . . . . . . . . 212
7.7 Some Extensions – The First-Order Case . . . . . . . . . . . 218
8 Final Thoughts
221
8.1 Foundations of Programming Semantics . . . . . . . . . . . . 221
8.2 Quantitative Domain Theory . . . . . . . . . . . . . . . . . . 222
8.3 Fixed-Point Theorems for Generalized Metric Spaces . . . . 223
8.4 The Foundations of Knowledge Representation and Reasoning 223
8.5 Clarifying Logic Programming Semantics . . . . . . . . . . . 224
8.6 Symbolic and Subsymbolic Representations . . . . . . . . . . 225
8.7 Neural-Symbolic Integration . . . . . . . . . . . . . . . . . . 225
8.8 Topology, Programming, and Artificial Intelligence . . . . . . 226
Contents
4.4 Dislocated Metrics . . . . . . . . . . . . . . . . . . . . . . . . 102
4.5 Dislocated Generalized Ultrametrics . . . . . . . . . . . . . . 104
4.6 Quasimetrics . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.7 A Hierarchy of Fixed-Point Theorems . . . . . . . . . . . . . 112
4.8 Relationships Between the Various Spaces . . . . . . . . . . . 114
4.9 Fixed-Point Theory for Multivalued Mappings . . . . . . . . 125
4.10 Partial Orders and Multivalued Mappings . . . . . . . . . . . 127
4.11 Metrics and Multivalued Mappings . . . . . . . . . . . . . . 129
4.12 Generalized Ultrametrics and Multivalued Mappings . . . . . 129
4.13 Quasimetrics and Multivalued Mappings . . . . . . . . . . . 132
4.14 An Alternative to Multivalued Mappings . . . . . . . . . . . 136
5 Supported Model Semantics
139
5.1 Two-Valued Supported Models . . . . . . . . . . . . . . . . . 140
5.2 Three-Valued Supported Models . . . . . . . . . . . . . . . . 151
5.3 A Hierarchy of Logic Programs . . . . . . . . . . . . . . . . . 159
5.4 Consequence Operators and Fitting-Style Operators . . . . . 161
5.5 Measurability Considerations . . . . . . . . . . . . . . . . . . 166
6 Stable and Perfect Model Semantics
169
6.1 The Fixpoint Completion . . . . . . . . . . . . . . . . . . . . 169
6.2 Stable Model Semantics . . . . . . . . . . . . . . . . . . . . . 171
6.3 Perfect Model Semantics . . . . . . . . . . . . . . . . . . . . 175
7 Logic Programming and Artificial Neural Networks
185
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
7.2 Basics of Artificial Neural Networks . . . . . . . . . . . . . . 188
7.3 The Core Method as a General Approach to Integration . . . 191
7.4 Propositional Programs . . . . . . . . . . . . . . . . . . . . . 192
7.5 First-Order Programs . . . . . . . . . . . . . . . . . . . . . . 196
7.6 Some Extensions – The Propositional Case . . . . . . . . . . 212
7.7 Some Extensions – The First-Order Case . . . . . . . . . . . 218
8 Final Thoughts
221
8.1 Foundations of Programming Semantics . . . . . . . . . . . . 221
8.2 Quantitative Domain Theory . . . . . . . . . . . . . . . . . . 222
8.3 Fixed-Point Theorems for Generalized Metric Spaces . . . . 223
8.4 The Foundations of Knowledge Representation and Reasoning 223
8.5 Clarifying Logic Programming Semantics . . . . . . . . . . . 224
8.6 Symbolic and Subsymbolic Representations . . . . . . . . . . 225
8.7 Neural-Symbolic Integration . . . . . . . . . . . . . . . . . . 225
8.8 Topology, Programming, and Artificial Intelligence . . . . . . 226