Mathematical Aspects of Logic Programming Semantics

Mathematical Aspects of Logic Programming Semantics

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Features

  • Provides comprehensive coverage of the mathematics and foundations of logic programs
  • Presents a comparative treatment of all major semantics in logic programming
  • Discusses topological and metrical bases
  • Includes a self-contained, detailed account of fixed-point theory relevant to logic programming and the semantics of computing
  • Illustrates the integration of logic programs and artificial neural networks
  • Explores promising applications and future directions for research

Summary

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.

Table of Contents

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
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 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
Introduction
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
Convergence
Separation Properties and Compactness
Subspaces and Products
The Scott Topology

Bibliography

Index

Author Bio(s)

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.

Editorial Reviews

… 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