Mathematical Aspects of Logic Programming Semantics

Covering the authors’ own state-of-the-art research results, Mathematical Aspects of Logic Programming Semantics presents a rigorous, modern account of the mathematical methods and tools required for the semantic analysis of logic programs. It significantly extends the tools and methods from traditional order theory to include nonconventional methods from mathematical analysis that depend on topology, domain theory, generalized distance functions, and associated fixed-point theory. The book covers topics spanning the period from the early days of logic programming to current times. It discusses applications to computational logic and potential applications to the integration of models of computation, knowledge representation and reasoning, and the Semantic Web. The authors develop well-known and important semantics in logic programming from a unified point of view using both order theory and new, nontraditional methods. They closely examine the interrelationships between various semantics as well as the integration of logic programming and connectionist systems/neural networks. For readers interested in the interface between mathematics and computer science, this book offers a detailed development of the mathematical techniques necessary for studying the semantics of logic programs. It illustrates the main semantics of logic programs and applies the methods in the context of neural-symbolic integration.

Order and LogicOrdered Sets and Fixed-Point Theorems First-Order Predicate Logic Ordered Spaces of Valuations The Semantics of Logic ProgramsLogic Programs and Their Models Supported Models Stable Models Fitting Models Perfect Models Well-Founded Models Topology and Logic ProgrammingConvergence Spaces and Convergence Classes The Scott Topology on Spaces of Valuations The Cantor Topology on Spaces of Valuations Operators on Spaces of Valuations Revisited Fixed-Point Theory for Generalized Metric SpacesDistance Functions in General Metrics and Their Generalizations Generalized UltrametricsDislocated Metrics Dislocated Generalized Ultrametrics Quasimetrics A Hierarchy of Fixed-Point Theorems Relationships between the Various Spaces Fixed-Point Theory for Multivalued Mappings Partial Orders and Multivalued Mappings Metrics and Multivalued Mappings Generalized Ultrametrics and Multivalued Mappings Quasimetrics and Multivalued Mappings An Alternative to Multivalued Mappings Supported Model SemanticsTwo-Valued Supported Models Three-Valued Supported Models A Hierarchy of Logic Programs Consequence Operators and Fitting-Style Operators Measurability Considerations Stable and Perfect Model SemanticsThe Fixpoint Completion Stable Model Semantics Perfect Model Semantics Logic Programming and Artificial Neural NetworksIntroduction Basics of Artificial Neural Networks The Core Method as a General Approach to Integration Propositional Programs First-Order Programs Some Extensions — The Propositional Case Some Extensions — The First-Order Case Final ThoughtsFoundations of Programming Semantics Quantitative Domain Theory Fixed-Point Theorems for Generalized Metric Spaces The Foundations of Knowledge Representation and ReasoningClarifying Logic Programming Semantics Symbolic and Subsymbolic Representations Neural-Symbolic Integration Topology, Programming, and Artificial Intelligence Appendix: Transfinite Induction and General Topology The Principle of Transfinite Induction Basic Concepts from General Topology Convergence Separation Properties and Compactness Subspaces and Products The Scott Topology Bibliography Index

Pascal Hitzler is an assistant professor in the Kno.e.sis Center for Knowledge-Enabled Computing, which is an Ohio Center of Excellence at Wright State University. Dr. Hitzler is editor-in-chief of the journal Semantic Web — Interoperability, Usability, Applicability and co-author of the textbook Foundations of Semantic Web Technologies (CRC Press, August 2009). His research interests encompass the Semantic Web, neural-symbolic integration, knowledge representation and reasoning, denotational semantics, and set-theoretic topology. Anthony Seda is a senior lecturer in the Department of Mathematics and co-founder of the Boole Centre for Research in Informatics at University College Cork. Dr. Seda is an editorial board member of Information and the International Journal of Advanced Intelligence. His research interests include measure theory, functional analysis, topology, fixed-point theory, denotational semantics, and the semantics of logic programs.