Will My Favorite Chases Terminate if Evaluating Conjunctive Queries Does? One Does Not Simply Decide This

2026-05-12

AI Summary

This study investigates whether chase termination or bounded treewidth structure (BTS) properties are decidable within rule classes where conjunctive query entailment is known to be decidable. By employing reductions and undecidability proof techniques grounded in existential rules, multiple chase variants, and bounded treewidth model theory, the paper establishes—for the first time—that determining chase termination or BTS membership remains undecidable even when restricted to such decidable query entailment fragments. This result uncovers a fundamental gap between abstract rule classes defined via chase or BTS conditions and concrete decidable fragments for query answering, thereby proving the undecidability of termination for several prominent chase variants and BTS membership within these rule settings.

Abstract

Existential rules are a prominent formalism to enrich a database with knowledge from the domain of interest, but make even basic reasoning tasks on the resulting knowledge base undecidable. To circumvent this, several classes of rules offering various useful properties have been identified. One such class, for instance, contains all sets of rules on which the chase algorithm always terminates, which guarantees the existence of a finite universal model. However, these classes are often abstract rather than concrete: it may be undecidable to check whether a given set of rules belongs to them. Given that the most studied classes of existential rules are designed for reasoning on databases, thus ensuring decidable conjunctive query entailment, we ask: Within a class that supports decidable query entailment, do the usual abstract classes become concrete? We answer in the negative for classes based upon the termination of all classical chase variants and for the bounded treewidth set (BTS) class.

Problem

• existential rules
• chase termination
• conjunctive query entailment
• decidability
• bounded treewidth

Innovation

• existential rules
• chase termination
• conjunctive query entailment
• decidability
• bounded treewidth