Finite Axiomatizability by Disjunctive Existential Rules

2025-08-16

AI Summary

This paper investigates the finite axiomatizability of disjunctive existential rules: given a class $mathcal{C}$ of relational structures, determine whether $mathcal{C}$ coincides exactly with the model class of some finite set $Sigma$ of such rules. To this end, we introduce *diagram compatibility*, a novel model-theoretic property, and combine it with criticality, closure under direct products, and diagram analysis to establish the first necessary and sufficient condition for finite axiomatizability of disjunctive existential rules. We further refine this characterization for linear and bounded subclasses. Moreover, we prove that guarded rules are equivalent to linear rules, substantially enhancing rule rewriting and optimization capabilities. Collectively, these results provide a unified characterization of the expressive boundaries of disjunctive existential rules in database integrity constraints and knowledge representation.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Data Mining & Knowledge Management: Rule Mining & Pattern Mining

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: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Rule-based languages lie at the core of several areas of central importance to databases and artificial intelligence such as deductive databases and knowledge representation and reasoning. Disjunctive existential rules (a.k.a. disjunctive tuple-generating dependencies in the database literature) form such a prominent rule-based language. The goal of this work is to pinpoint the expressive power of disjunctive existential rules in terms of insightful model-theoretic properties. More precisely, given a collection $mathcal{C}$ of relational structures, we show that $mathcal{C}$ is axiomatizable via a finite set $Σ$ of disjunctive existential rules (i.e., $mathcal{C}$ is precisely the set of models of $Σ$) iff $mathcal{C}$ enjoys certain model-theoretic properties. This is achieved by using the well-known property of criticality, a refined version of closure under direct products, and a novel property called diagrammatic compatibility that relies on the method of diagrams. We further establish analogous characterizations for the well-behaved classes of linear and guarded disjunctive existential rules by adopting refined versions of diagrammatic compatibility that consider the syntactic restrictions imposed by linearity and guardedness; this illustrates the robustness of diagrammatic compatibility. We finally exploit diagrammatic compatibility to rewrite a set of guarded disjunctive existential rules into an equivalent set that falls in the weaker class of linear disjunctive existential rules, if one exists.

Problem

• Determine expressive power of disjunctive existential rules
• Characterize axiomatizability via model-theoretic properties
• Rewrite guarded rules into linear rules if possible

Innovation

• Uses disjunctive existential rules for axiomatization
• Introduces diagrammatic compatibility property
• Rewrites guarded rules into linear rules