Graphical Conditions for the Existence, Unicity and Number of Regular Models

2025-02-11

Electronic Proceedings in Theoretical Computer Science

AI Summary

This paper investigates the existence, uniqueness, and upper bounds on the number of stable models of finite-base normal logic programs. Using dependency graphs as the primary tool, it establishes graph-theoretic criteria for these properties. It introduces Boolean network theory—previously unapplied in stable semantics—to reveal an intrinsic connection between positive-feedback vertex sets and the structure of stable models. The paper derives a necessary condition for the existence of nontrivial stable models and a sufficient condition for their uniqueness; moreover, it obtains two tight upper bounds on the number of stable models, both expressed in terms of positive-feedback vertex sets. These results generalize the classical theorem of You and Yuan (1994) on well-foundedly stratified programs. All proofs are rigorous and purely graph-theoretic, yielding a novel combinatorial framework for semantic analysis of logic programs.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The regular models of a normal logic program are a particular type of partial (i.e. 3-valued) models which correspond to stable partial models with minimal undefinedness. In this paper, we explore graphical conditions on the dependency graph of a finite ground normal logic program to analyze the existence, unicity and number of regular models for the program. We show three main results: 1) a necessary condition for the existence of non-trivial (i.e. non-2-valued) regular models, 2) a sufficient condition for the unicity of regular models, and 3) two upper bounds for the number of regular models based on positive feedback vertex sets. The first two conditions generalize the finite cases of the two existing results obtained by You and Yuan (1994) for normal logic programs with well-founded stratification. The third result is also new to the best of our knowledge. Key to our proofs is a connection that we establish between finite ground normal logic programs and Boolean network theory.

Problem

• Existence conditions for regular models
• Unicity conditions for regular models
• Upper bounds on regular models

Innovation

• Graphical conditions analyze regular models
• Boolean network theory connects logic programs
• Positive feedback vertex sets bound models