On the Trap Space Semantics of Normal Logic Programs

2026-01-07

Electronic Proceedings in Theoretical Computer Science

AI Summary

This work addresses the lack of a unified perspective that integrates dynamic and model-theoretic aspects in the semantics of traditional normal logic programs—such as stable models, regular models, and well-founded models. It introduces a novel semantic framework based on the notion of trap spaces, originally from Boolean networks, generalized here to arbitrary normal logic programs. The proposed trap space semantics uniquely combines model-theoretic and dynamical systems characteristics. By leveraging update operators, state transition graphs, and three-valued/two-valued semantic analyses, the paper establishes a rigorous theoretical foundation for this semantics and formally proves its precise correspondences with classical semantics, including stable models, regular models, L-stable models, and dynamic supportedness classes. This provides a unified framework and a fresh perspective for understanding and reasoning about the behavior of logic programs.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The logical semantics of normal logic programs has traditionally been based on the notions of Clark's completion and two-valued or three-valued canonical models, including supported, stable, regular, and well-founded models. Two-valued interpretations can also be seen as states evolving under a program's update operator, producing a transition graph whose fixed points and cycles capture stable and oscillatory behaviors, respectively. We refer to this view as dynamical semantics since it characterizes the program's meaning in terms of state-space trajectories, as first introduced in the stable (supported) class semantics. Recently, we have established a formal connection between Datalog^\neg programs (i.e., normal logic programs without function symbols) and Boolean networks, leading to the introduction of the trap space concept for Datalog^\neg programs. In this paper, we generalize the trap space concept to arbitrary normal logic programs, introducing trap space semantics as a new approach to their interpretation. This new semantics admits both model-theoretic and dynamical characterizations, providing a comprehensive approach to understanding program behavior. We establish the foundational properties of the trap space semantics and systematically relate it to the established model-theoretic semantics, including the stable (supported), stable (supported) partial, regular, and L-stable model semantics, as well as to the dynamical stable (supported) class semantics. Our results demonstrate that the trap space semantics offers a unified and precise framework for proving the existence of supported classes, strict stable (supported) classes, and regular models, in addition to uncovering and formalizing deeper relationships among the existing semantics of normal logic programs.

Problem

• normal logic programs
• trap space semantics
• model-theoretic semantics
• dynamical semantics
• stable models

Innovation

• trap space semantics
• normal logic programs
• dynamical semantics
• Boolean networks
• model-theoretic semantics