Restricted Chase Termination: You Want More than Fairness

2025-05-22

AI Summary

This paper investigates the universal termination problem of the restricted chase under existential rules. Addressing a long-standing open question regarding its computational complexity, we establish that the problem is complete for the analytical hierarchy level $Pi^1_2$, thereby identifying fairness conditions as the fundamental source of its high complexity. To mitigate this, we propose weakened application conditions that relax fairness requirements while preserving semantic soundness, reducing the complexity to the computable level $Pi^0_2$-completeness. Methodologically, we integrate formal logic, database theory, and computability analysis to construct a rigorous modeling framework grounded in the analytical hierarchy. Our results yield a novel termination guarantee paradigm for rule-based data reasoning—one that balances theoretical depth with practical decidability.

Technology Category

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

Application Category

• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking

Abstract

The chase is a fundamental algorithm with ubiquitous uses in database theory. Given a database and a set of existential rules (aka tuple-generating dependencies), it iteratively extends the database to ensure that the rules are satisfied in a most general way. This process may not terminate, and a major problem is to decide whether it does. This problem has been studied for a large number of chase variants, which differ by the conditions under which a rule is applied to extend the database. Surprisingly, the complexity of the universal termination of the restricted (aka standard) chase is not fully understood. We close this gap by placing universal restricted chase termination in the analytical hierarchy. This higher hardness is due to the fairness condition, and we propose an alternative condition to reduce the hardness of universal termination.

Problem

• Determining universal termination of restricted chase
• Analyzing complexity due to fairness condition
• Proposing alternative condition to reduce hardness

Innovation

• Analyzes restricted chase termination complexity
• Introduces alternative to fairness condition
• Places termination in analytical hierarchy