We consider the issue of answering unions of conjunctive queries (UCQs) with disjunctive existential rules and mappings. While this issue has already been well studied from a chase perspective, query rewriting within UCQs has hardly been addressed yet. We first propose a sound and complete query rewriting operator, which has the advantage of establishing a tight relationship between a chase step and a rewriting step. The associated breadth-first query rewriting algorithm outputs a minimal UCQ-rewriting when one exists. Second, we show that for any ``truly disjunctive'' nonrecursive rule, there exists a conjunctive query that has no UCQ-rewriting. It follows that the notion of finite unification sets (fus), which denotes sets of existential rules such that any UCQ admits a UCQ-rewriting, seems to have little relevance in this setting. Finally, turning our attention to mappings, we show that the problem of determining whether a UCQ admits a UCQ-rewriting through a disjunctive mapping is undecidable. We conclude with a number of open problems.

翻译：我们考虑使用带析取存在规则和映射来回答合取查询的并集（UCQs）的问题。虽然从追逐（chase）角度该问题已被充分研究，但UCQs内的查询重写却鲜有涉及。本文首先提出一个完备且正确的查询重写算子，其优势在于建立了追逐步骤与重写步骤之间的紧密关联。该关联的广度优先查询重写算法能在存在解时输出最小UCQ重写。其次，我们证明对于任意"真正析取"的非递归规则，存在一个没有UCQ重写的合取查询。由此可知，有限统一集（fus）——即任意UCQ均存在UCQ重写的一组存在规则——在此场景下似乎意义不大。最后，将关注点转向映射时，我们证明判断UCQ是否通过析取映射存在UCQ重写的问题是不可判定的。我们以若干开放问题作为结语。