Mathematical aspects of logic programming semantics

282 Pages·2011·3.841 MB·xxx, 274 p. : ill. ; 25 cm\282

by  Pascal Hitzler, Anthony Seda

MATHEMATICAL
ASPECTS OF LOGIC
PROGRAMMING
SEMANTICS
Chapman & Hall/CRC
Studies in Informatics Series
SERIES EDITOR
G. Q. Zhang
Department of EECS
Case Western Reserve University
Cleveland, Ohio, U.S.A.
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Mathematical Aspects of Logic Programming Semantics
Pascal Hitzler and Anthony Seda
Chapman & Hall/CRC
Studies in Informatics Series
MATHEMATICAL
ASPECTS OF LOGIC
PROGRAMMING
SEMANTICS
P H
ascal itzler
Kno.e.sis Center at
Wright State University
Dayton, Ohio, USA
a s
ntHony eda
University College Cork
Ireland
CRC Press
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Dedication
To Anne, to Martine, and to the memory of Barbara and Ellen Lucille
Contents
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
vii
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 ix
Appendix: Transfinite Induction and General Topology 229
A.1 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
List of Figures
1.1 Hasse diagrams for THREE and FOUR. . . . . . . . . . . . 16
2.1 Dependency graph for P . . . . . . . . . . . . . . . . . . . . . 48
1
2.2 Dependency graph for P . . . . . . . . . . . . . . . . . . . . . 48
2
4.1 Dependencies between single-valued fixed-point theorems. . . 113
5.1 The main classes of programs discussed in this book. . . . . . 160
7.1 The neural-symbolic cycle. . . . . . . . . . . . . . . . . . . . . 187
7.2 Unit N in a connectionist network. . . . . . . . . . . . . . . 188
k
7.3 Sketch of a 3-layer recurrent network. . . . . . . . . . . . . . 191
7.4 Two 3-layer feedforward networks of binary threshold units. . 194
7.5 Transforming T into f . . . . . . . . . . . . . . . . . . . . . 198
P P
7.6 The embedding of the T -operator for Program 7.5.1. . . . . 199
P
7.7 The networks from Example 7.5.7. . . . . . . . . . . . . . . . 202
7.8 The embedding and approximation of a T -operator. . . . . . 202
P
7.9 An approximating sigmoidal network for Program 7.5.1. . . . 204
7.10 An approximation using the raised cosine function. . . . . . . 205
7.11 An RBF network approximating a T -operator. . . . . . . . . 206
P
7.12 A two-dimensional version of the Cantor set. . . . . . . . . . 207
7.13 A construction of the two-dimensional Cantor set. . . . . . . 207
7.14 A vector-based approximating network. . . . . . . . . . . . . 208
7.15 A conjunction unit for FOUR. . . . . . . . . . . . . . . . . . 216

xi
List of Tables
1.1 Belnap’s four-valued logic. . . . . . . . . . . . . . . . . . . . . 11
4.1 Generalized metrics: Definition 4.2.1. . . . . . . . . . . . . . . 92
4.2 (Dislocated) generalized ultrametrics: Definition 4.3.1. . . . . 98
4.3 Summary of single-valued fixed-point theorems. . . . . . . . . 113
5.1 Several truth tables for three-valued logics. . . . . . . . . . . 153

xiii