Rewriting Consistent Answers on Annotated Data

TL;DR

This paper studies how to compute consistent answers to queries over annotated databases using semiring semantics, providing a logical characterization and complexity results for various semirings.

Contribution

It introduces a logical framework for consistent query answering over semiring-annotated data and characterizes when answers are rewritable based on attack graph cycles.

Findings

Rewritability of consistent answers is characterized by the absence of cycles in the attack graph.

For the bag semiring, computing consistent answers with cycles is NP-hard and hard to approximate.

The results generalize previous work and apply to multiple semirings including tropical and fuzzy.

Abstract

We embark on a study of the consistent answers of queries over databases annotated with values from a naturally ordered positive semiring. In this setting, the consistent answers of a query are defined as the minimum of the semiring values that the query takes over all repairs of an inconsistent database. The main focus is on self-join free conjunctive queries and key constraints, which is the most extensively studied case of consistent query answering over standard databases. We introduce a variant of first-order logic with a limited form of negation, define suitable semiring semantics, and then establish the main result of the paper: the consistent query answers of a self-join free conjunctive query under key constraints are rewritable in this logic if and only if the attack graph of the query contains no cycles. This result generalizes an analogous result of Koutris and Wijsen for ordinary databases, but also yields new results for a multitude of semirings, including the bag semiring, the tropical semiring, and the fuzzy semiring. Further, for the bag semiring, we show that computing the consistent answers of any self-join free conjunctive query whose attack graph has a strong cycle is not only NP-hard but also it is NP-hard to even approximate the consistent answers with a constant relative approximation guarantee.