Attacking Diophantus: Special Cases of Bag Containment

2026-09-25

AI Summary

This study addresses the long-standing open problem of determining conjunctive query containment under bag semantics. It proposes a unified framework that reduces the containment problem to solving controlled systems of Diophantine inequalities. Building upon canonical models and arithmetic multiplicity representations, this framework innovatively transforms factors that typically lead to undecidability into the core of the decision procedure, precisely delineating tractable classes by uncovering their underlying unified structure. As a key contribution, this work rigorously proves the decidability of bag containment for the class of connected homogeneous queries. This result significantly generalizes previously known special cases and establishes a systematic theoretical foundation for the field.

Abstract

Query containment is a fundamental decision problem in database theory: given two queries, determine whether, over all database instances, every answer produced by the first is also produced by the second. For conjunctive queries under set semantics, the problem is understood through the classical homomorphism-based characterisation. Under bag semantics, the interpretation underlying real relational databases, containment becomes a quantitative comparison of answer multiplicities. Despite decades of work, the decidability of bag containment for conjunctive queries remains open. This frontier is fragile: for slightly more expressive classes, bag containment is undecidable, with negative results relying on reductions from variants of Hilbert's 10th problem. This work develops a unified framework for bag containment of conjunctive queries that subsumes two previously studied decidable cases: projection-free and join-on-free containee queries. The framework yields decidability for a broader class, called join-uniform queries, while leaving the containing query arbitrary. This contrasts with techniques that impose restrictions on the containing query. The approach identifies tractable classes based on the internal unification structure of the query whose multiplicities must be bounded. Specifically, it reduces containment to a controlled Diophantine problem. Starting from the containee query, one builds a canonical model generated by all its possible unifications, over which multiplicities admit a finite arithmetic characterisation. Containment is proved equivalent to the non-existence of solutions of a corresponding Diophantine inequality system. Although these problems are undecidable in general, we show that the systems arising from join-uniform containment form a decidable subclass. Thus, the standard source of undecidability for bag containment becomes the core of the decision procedure.

Problem

• Bag Containment
• Conjunctive Queries
• Decidability
• Query Containment
• Diophantine Problem

Innovation

• Bag containment
• Conjunctive queries
• Diophantine problem
• Join-uniform queries
• Canonical model