[Paper Review] A Trichotomy in the Data Complexity of Certain Query Answering for Conjunctive Queries

Paraschos Koutris, Jef Wijsen|arXiv (Cornell University)|Jan 30, 2015

Data Management and Algorithms18 references3 citations

TL;DR

This paper establishes an effective complexity trichotomy for certain query answering (CERTAINTY(q)) over self-join-free Boolean conjunctive queries in uncertain databases. By analyzing the attack graph of a query, the authors present a decidable procedure that determines whether CERTAINTY(q) is in FO, P∖FO, or coNP-complete, resolving a decade-old open problem in data complexity.

ABSTRACT

A relational database is said to be uncertain if primary key constraints can possibly be violated. A repair (or possible world) of an uncertain database is obtained by selecting a maximal number of tuples without ever selecting two distinct tuples with the same primary key value. For any Boolean query q, CERTAINTY(q) is the problem that takes an uncertain database db on input, and asks whether q is true in every repair of db. The complexity of this problem has been particularly studied for q ranging over the class of self-join-free Boolean conjunctive queries. A research challenge is to determine, given q, whether CERTAINTY(q) belongs to complexity classes FO, P, or coNP-complete. In this paper, we combine existing techniques for studying the above complexity classification task. We show that for any self-join-free Boolean conjunctive query q, it can be decided whether or not CERTAINTY(q) is in FO. Further, for any self-join-free Boolean conjunctive query q, CERTAINTY(q) is either in P or coNP-complete, and the complexity dichotomy is effective. This settles a research question that has been open for ten years, since [9].

Motivation & Objective

• To resolve the long-standing open problem of classifying the data complexity of CERTAINTY(q) for self-join-free Boolean conjunctive queries.
• To determine whether CERTAINTY(q) is first-order expressible (FO), tractable (P∖FO), or intractable (coNP-complete).
• To provide an effective decision procedure based on query structure that distinguishes these three complexity classes.
• To extend prior results that were limited to acyclic queries, simple primary keys, or queries with exactly two atoms.

Proposed method

• The authors introduce a novel complexity-theoretic approach to first-order inexpressibility, replacing Hanf locality arguments with a more robust framework.
• They define a reduction from an uncertain database to a transformed database using a 'dissolve' operation on query atoms, preserving truth of the query across repairs.
• The method relies on constructing a database db' such that repairs of db satisfy q if and only if corresponding repairs of db' satisfy the transformed query q*.
• The attack graph of the query is used as a structural basis for determining complexity, with cycles and reachability in the graph informing the classification.
• The reduction ensures that the number of relevant cycles is polynomially bounded, enabling polynomial-time computation of the transformed database.
• The proof uses a bijective correspondence between repairs of the original and transformed databases, preserving query satisfaction under specific valuations.

Experimental results

Research questions

• RQ1 Can we decide, for any self-join-free Boolean conjunctive query q, whether CERTAINTY(q) is in FO?
• RQ2 Is there an effective dichotomy that classifies CERTAINTY(q) as either in P or coNP-complete for all such queries?
• RQ3 Can the complexity classification be extended beyond acyclic queries or simple primary keys?
• RQ4 Does a structural characterization of CERTAINTY(q) complexity exist that is both decidable and effective?

Key findings

• For any self-join-free Boolean conjunctive query q, it is decidable whether CERTAINTY(q) is in FO, resolving a limitation of prior work that required acyclicity assumptions.
• If CERTAINTY(q) is not in FO, then it is L-hard, establishing a stronger inexpressibility result using complexity-theoretic techniques rather than Hanf locality.
• For every self-join-free Boolean conjunctive query, CERTAINTY(q) is either in P or coNP-complete, and this dichotomy is effective, extending prior results that required simple primary keys.
• The complexity trichotomy is effective: a procedure based on the attack graph of q can determine the exact complexity class of CERTAINTY(q).
• The reduction from db to db' preserves query satisfaction across repairs, enabling a polynomial-time reduction that maintains the logical equivalence of query answers across repairs.