1st Edition

Mathematical Aspects of Logic Programming Semantics

By Pascal Hitzler, Anthony Seda Copyright 2011
    304 Pages 20 B/W Illustrations
    by CRC Press

    304 Pages 20 B/W Illustrations
    by CRC Press

    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 Logic
    Ordered Sets and Fixed-Point Theorems
    First-Order Predicate Logic
    Ordered Spaces of Valuations

    The Semantics of Logic Programs
    Logic Programs and Their Models
    Supported Models
    Stable Models
    Fitting Models
    Perfect Models
    Well-Founded Models

    Topology and Logic Programming
    Convergence 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 Spaces
    Distance Functions in General
    Metrics and Their Generalizations
    Generalized Ultrametrics
    Dislocated Metrics
    Dislocated Generalized Ultrametrics
    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 Semantics
    Two-Valued Supported Models
    Three-Valued Supported Models
    A Hierarchy of Logic Programs
    Consequence Operators and Fitting-Style Operators
    Measurability Considerations

    Stable and Perfect Model Semantics
    The Fixpoint Completion
    Stable Model Semantics
    Perfect Model Semantics

    Logic Programming and Artificial Neural Networks
    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 Thoughts
    Foundations of Programming Semantics
    Quantitative Domain Theory
    Fixed-Point Theorems for Generalized Metric Spaces
    The Foundations of Knowledge Representation and Reasoning
    Clarifying 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
    Separation Properties and Compactness
    Subspaces and Products
    The Scott Topology




    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.

    … Much of the material has been generated by [the authors’] own collaboration over the past decade, but they also integrate research results by others. A major feature is that they significantly transcend the tools and methods from the order theory traditionally used in this context, to include non-traditional methods from mathematical analysis depending on topology, generalized distance functions, and their associated fixed-point theory. …
    SciTech Book News, February 2011