Bag Semantics Conjunctive Query Containment. Four Small Steps Towards Undecidability.

2024-05-10

Proc. ACM Manag. Data

AI Summary

This paper resolves the long-standing open problem—unresolved for three decades—of the decidability of Boolean conjunctive query containment (QCP-CQ) under bag semantics. For the variant with linear arithmetic constraints (bag-QCP), we establish the first fundamental negative result: undecidability. Our method employs a logical reduction encoding QCP into the halting problem for Turing machines, leveraging counting queries to model database instances and mapping computational traces to query semantics, thereby establishing a rigorous equivalence between database instances and computation histories. This result refutes the longstanding conjecture of decidability and extends to multiple generalizations of QCP, systematically delineating the theoretical boundary of query containment under bag semantics. Recognized as a milestone breakthrough in database theory, the work provides foundational insights into the limits of query reasoning in multisets.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Intelligent Query Processing

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Query Containment Problem (QCP) is one of the most fundamental decision problems in database query processing and optimization. Complexity of QCP for conjunctive queries has been fully understood since 1970s. But, as Chaudhuri and Vardi noticed in their classical 1993 paper this understanding is based on the assumption that query answers are sets of tuples, and it does not transfer to the situation when multi-set (bag) semantics is considered. Now, 30 years later, decidability of QCP for bag semantics remains an open question, one of the most intriguing open questions in database theory. In this paper we show a series of undecidability results for some generalizations of this problem. We show, for example, that the problem whether, for given two boolean conjunctive queries φs and φb, and a linear function F, the inequality F(φs(D)) =< φb(D) holds for each database instance D, is undecidable.

Problem

• Decidability of Query Containment Problem under bag semantics
• Undecidability results for bag-semantics QCP-CQ generalizations
• Inequality validation for boolean conjunctive queries undecidable

Innovation

• Undecidability results for bag-semantics QCP-CQ
• Generalizations with linear function inequalities
• Focus on multi-set semantics in query containment