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.

翻译：本研究探讨了在标注有自然有序正半环值的数据库上查询的一致性答案。在此设定下，查询的一致性答案被定义为该查询在所有不一致数据库的修复上取得的半环值的最小值。研究主要聚焦于无自连接合取查询与键约束，这是标准数据库上一致性查询应答研究最为深入的场景。我们引入了一种具有受限否定形式的一阶逻辑变体，定义了合适的半环语义，进而确立了本文的主要结果：在键约束下，一个无自连接合取查询的一致性查询答案在该逻辑中可重写，当且仅当该查询的攻击图中不包含环。这一结果推广了Koutris和Wijsen针对普通数据库的类似结论，同时也为包括包半环、热带半环和模糊半环在内的多种半环带来了新的发现。此外，对于包半环，我们证明了计算任何攻击图包含强环的无自连接合取查询的一致性答案不仅是NP难的，而且即使以恒定相对近似保证来近似这些一致性答案也是NP难的。