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. PUBLISHED TITLES Stochastic Relations: Foundations for Markov Transition Systems Ernst-Erich Doberkat Conceptual Structures in Practice Pascal Hitzler and Henrik Schärfe Context-Aware Computing and Self-Managing Systems Waltenegus Dargie Introduction to Mathematics of Satisfiability Victor W. Marek Ubiquitous Multimedia Computing Qing Li and Timothy K. Shih Mathematical Aspects of Logic Programming Semantics Pascal Hitzler and Anthony Seda ChapmanChapman & Hall/CRC & Hall/CRC StudiesStudies in Informatics in Informatics Series Series MATHEMATICALMATHEMATICAL ASPECTS ASPECTS OF OF LOGIC LOGIC PROGRAMMINGPROGRAMMING SEMANTICSSEMANTICS PASCALPASCAL HITZLER HITZLER Kno.e.sisKno.e.sis Center Center at at WrightWright State UniversityState University Dayton,Dayton, Ohio, Ohio,USA USA ANTHONYANTHONY SEDA SEDA UniversityUniversity College College Cork Cork IrelandIreland CRC Press Taylor & Francis Group CRC6000 PressBroken Sound Parkway NW, Suite 300 TaylorBoca Raton, & Francis FL 33487-2742 Group 6000 Broken Sound Parkway NW, Suite 300 Boca© 2011 Raton, by Taylor FL 33487-2742 and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business © 2011 by Taylor and Francis Group, LLC CRCNo claim Press to is original an imprint U.S. of Government Taylor & Francis works Group, an Informa business NoPrinted claim in to the original United U.S. States Government of America works on acid-free paper 10 9 8 7 6 5 4 3 2 1 Printed in the United States of America on acid-free paper 10International 9 8 7 6 5 4 Standard3 2 1 Book Number: 978-1-4398-2961-5 (Hardback) InternationalThis book contains Standard information Book Number: obtained 978-1-4398-2961-5 from authentic (Hardback) and highly regarded sources. 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Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are Visit the Taylor & Francis Web site at used only for identification and explanation without intent to infringe. http://www.taylorandfrancis.com Visit the Taylor & Francis Web site at http://www.taylorandfrancis.comand the CRC Press Web site at http://www.crcpress.com and the CRC Press Web site at http://www.crcpress.com 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 T HREE and FOUR. 16 2.1 Dependency graph for P1..................... 48 2.2 Dependency graph for P2..................... 48 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 Nk in a connectionist network. 188 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 TP into fP . 198 7.6 The embedding of the TP -operator for Program 7.5.1. 199 7.7 The networks from Example 7.5.7. 202 7.8 The embedding and approximation of a TP -operator. 202 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 TP -operator. 206 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 Preface This book presents a rigorous, comprehensive, modern, and detailed account of the mathematical methods and tools required for the semantic analysis of logic programs. It is, in part, the outcome of a fruitful research collabora- tion between the authors over the last decade or so and contains many of the results we obtained during that period.