We clarify the complexity of answering unions of conjunctive queries over knowledge bases formulated in the description logic $\mathcal S$, the extension of $\mathcal{ALC}$ with transitive roles. Contrary to what existing partial results suggested, we show that the problem is in fact 2ExpTime-complete; hardness already holds in the presence of two transitive roles and for Boolean conjunctive queries. We complement this result by showing that the problem remains in coNExpTime when the input query is rooted or is restricted to use at most one transitive role (but may use arbitrarily many non-transitive roles).

翻译：我们阐明了在描述逻辑 $\mathcal S$（即 $\mathcal{ALC}$ 的传递角色扩展）中，基于知识库的合取查询并集回答问题的复杂性。与现有部分结果所暗示的相反，我们证明该问题实际上是 2ExpTime-完全的；即使在仅有两个传递角色且查询为布尔合取查询的情形下，难度已经成立。我们进一步补充了该结果：当输入查询是根查询或仅限于使用至多一个传递角色（但可使用任意多个非传递角色）时，问题仍保持在 coNExpTime 内。