A Simple Algorithm for Consistent Query Answering under Primary Keys

TL;DR

This paper introduces a simple polynomial-time fixpoint algorithm for consistent query answering under primary key constraints, effectively identifying certain queries in specific cases and providing an under-approximation for general cases.

Contribution

The paper presents a new inflationary fixpoint algorithm that efficiently computes certain answers for a class of queries under primary key constraints, extending previous results.

Findings

Algorithm correctly identifies certain queries in known polynomial cases.

The algorithm provides an under-approximation for arbitrary Boolean conjunctive queries.

It runs in polynomial time and is simple to implement.

Abstract

We consider the dichotomy conjecture for consistent query answering under primary key constraints. It states that, for every fixed Boolean conjunctive query q, testing whether q is certain (i.e. whether it evaluates to true over all repairs of a given inconsistent database) is either polynomial time or coNP-complete. This conjecture has been verified for self-join-free and path queries. We propose a simple inflationary fixpoint algorithm for consistent query answering which, for a given database, naively computes a set $\Delta$ of subsets of facts of the database of size at most k, where k is the size of the query q. The algorithm runs in polynomial time and can be formally defined as: (1) Initialize $\Delta$ with all sets $S$ of at most $k$ facts such that $S\models q$. (2) Add any set $S$ of at most k facts to $\Delta$ if there exists a block $B$ (i.e., a maximal set of facts sharing the same key) such that for every fact $a \in B$ there is a set $S' \subseteq S \cup \{a\}$ such that $S'\in \Delta$. For an input database $D$, the algorithm answers "q is certain" iff $\Delta$ eventually contains the empty set. The algorithm correctly computes certainty when the query q falls in the polynomial time cases of the known dichotomies for self-join-free queries and path queries. For arbitrary Boolean conjunctive queries, the algorithm is an under-approximation: the query is guaranteed to be certain if the algorithm claims so. However, there are polynomial time certain queries (with self-joins) which are not identified as such by the algorithm.