 | Hitzler P. Mathematical aspects of logic programming semantics / P.Hitzler, A.Seda. - Boca Raton: CRC Press, 2011. - xxx, 274 p.: ill. - (Chapman & Hall/CRC studies in informatics series). - Abridgement of: Encyclopedia of microbiology. - 3rd ed. / editor-in-chief, M.Schaechter. 2009. - Bibliogr.: p.243-264. - Ind.: p.265-274. - ISBN 978-1-4398-2961-5
|
List of Figures ................................................ xi
List of Tables ............................................... xiii
Preface ........................................................ xv
Introduction .................................................. xix
About the Authors ............................................ xxix
1 Order and Logic .............................................. 1
1.1 Ordered Sets and Fixed-Point Theorems ................... 1
1.2 First-Order Predicate Logic ............................. 7
1.3 Ordered Spaces of Valuations ........................... 12
2 The Semantics of Logic Programs ............................. 23
2.1 Logic Programs and Their Models ........................ 23
2.2 Supported Models ....................................... 28
2.3 Stable Models .......................................... 32
2.4 Fitting Models ......................................... 37
2.5 Perfect Models ......................................... 43
2.6 Well-Founded Models .................................... 56
3 Topology and Logic Programming .............................. 65
3.1 Convergence Spaces and Convergence Classes ............. 66
3.2 The Scott Topology on Spaces of Valuations ............. 69
3.3 The Cantor Topology on Spaces of Valuations ............ 76
3.4 Operators on Spaces of Valuations Revisited ............ 83
4 Fixed-Point Theory for Generalized Metric Spaces ............ 87
4.1 Distance Functions in General .......................... 88
4.2 Metrics and Their Generalizations ...................... 91
4.3 Generalized Ultrametrics ............................... 97
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
Appendix: Transfinite Induction and General Topology .......... 229
A.l The Principle of Transfinite Induction ................ 229
A.2 Basic Concepts from General Topology .................. 234
A.3 Convergence ........................................... 237
A.4 Separation Properties and Compactness ................. 238
A.5 Subspaces and Products ................................ 239
A.6 The Scott Topology .................................... 240
Bibliography .................................................. 243
Index ......................................................... 265
|
|
