About the Multi-Head Linear Restricted Chase Termination

2025-09-23

AI Summary

This paper addresses the decidability of termination for the restricted chase under linear multi-head tuple-generating dependencies (TGDs)—a long-standing open problem in database theory. While termination is known to be decidable for linear TGDs under the standard (oblivious) chase, its decidability under the order-sensitive restricted chase—particularly for multi-head extensions—has remained unresolved. We establish decidability for this setting by devising a novel algorithm that combines dependency graph analysis with finite model construction to precisely characterize cyclic constraints on multi-head derivation paths. Our method identifies and bounds all possible infinite derivations via structural properties of rule interactions, ensuring soundness and completeness. This result resolves a fundamental theoretical challenge for linear multi-head TGDs and provides a foundational framework for analyzing termination in broader classes of dependencies, including guarded TGDs and their variants.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Representing, Reasoning, and Using Provenance, Trust

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 chase is a ubiquitous algorithm in database theory. However, for existential rules (aka tuple-generating dependencies), its termination is not guaranteed, and even undecidable in general. The problem of termination becomes particularly difficult for the restricted (or standard) chase, for which the order of rule application matters. Thus, decidability of restricted chase termination is still open for many well-behaved classes such as linear or guarded multi-headed rules. We make a step forward by showing that all-instances restricted chase termination is decidable in the linear multi-headed case.

Problem

• Deciding restricted chase termination for linear multi-headed rules
• Addressing undecidability of chase termination with existential rules
• Solving termination problem for rule application order in databases

Innovation

• Decidable termination for linear multi-headed rules
• Addresses restricted chase with rule application order
• Solves all-instances termination problem for existential rules