Mathematical Aspects of Logic Programming Semantics: 1st Edition (Paperback) book cover

Mathematical Aspects of Logic Programming Semantics

1st Edition

By Pascal Hitzler, Anthony Seda

CRC Press

304 pages | 20 B/W Illus.

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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.


… 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

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


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



About the Authors

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.

About the Series

Chapman & Hall/CRC Studies in Informatics Series

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Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Information Theory
COMPUTERS / Machine Theory
COMPUTERS / Software Development & Engineering / Systems Analysis & Design