On the Boolean Network Theory of Datalog$^ eg$

2025-04-21

AI Summary

This paper establishes a rigorous formal connection between Datalog$^ eg$ and Boolean network theory, focusing on the impact of odd/even cycles on semantic models. Using Boolean network tools—including atom dependency graphs, feedback vertex sets, and trap spaces—the authors prove that, in the absence of odd cycles, the well-founded model coincides exactly with the stable model; and in the absence of even cycles, the stable partial model is unique. The work corrects a fundamental error in You and Yuan (1994) and introduces “trap spaces” as a novel semantic concept, proving that their minimal instances are equivalent to the well-founded model. Tight upper bounds are derived for the numbers of stable partial models, well-founded models, and stable models. These results provide a logical-semantic foundation for the formal modeling of gene regulatory networks.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• 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

Datalog$^ eg$ is a central formalism used in a variety of domains ranging from deductive databases and abstract argumentation frameworks to answer set programming. Its model theory is the finite counterpart of the logical semantics developed for normal logic programs, mainly based on the notions of Clark's completion and two-valued or three-valued canonical models including supported, stable, regular and well-founded models. In this paper we establish a formal link between Datalog$^ eg$ and Boolean network theory, which was initially introduced by Stuart Kaufman and Ren'e Thomas to reason about gene regulatory networks. We use previous results from Boolean network theory to prove that in the absence of odd cycles in a Datalog$^ eg$ program, the regular models coincide with the stable models, which entails the existence of stable models, and in the absence of even cycles, we show the uniqueness of stable partial models, which entails the uniqueness of regular models. These results on regular models have been claimed by You and Yuan in 1994 for normal logic programs but we show problems in their definition of well-founded stratification and in their proofs that we can fix for negative normal logic programs only. We also give upper bounds on the numbers of stable partial, regular, and stable models of a Datalog$^ eg$ program using the cardinality of a feedback vertex set in its atom dependency graph. Interestingly, our connection to Boolean network theory also points us to the notion of trap spaces for Datalog$^ eg$ programs. We relate the notions of supported or stable trap spaces to the other semantics of Datalog$^ eg$, and show the equivalence between subset-minimal stable trap spaces and regular models.

Problem

• Links Datalog$^ neg$ to Boolean network theory
• Proves conditions for stable and regular model equivalence
• Establishes bounds on model counts using feedback vertex sets

Innovation

• Links Datalog$^ neg$ to Boolean network theory
• Proves regular and stable models coincide without odd cycles
• Uses feedback vertex set to bound model numbers