In this paper we consider the most common ABox reasoning services for the description logic $\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle(\mathbf{D})$ ($\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$, for short) and prove their decidability via a reduction to the satisfiability problem for the set-theoretic fragment \flqsr. The description logic $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ is very expressive, as it admits various concept and role constructs, and data types, that allow one to represent rule-based languages such as SWRL. Decidability results are achieved by defining a generalization of the conjunctive query answering problem, called HOCQA (Higher Order Conjunctive Query Answering), that can be instantiated to the most wide\-spread ABox reasoning tasks. We also present a \ke\space based procedure for calculating the answer set from $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ knowledge bases and higher order $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ conjunctive queries, thus providing means for reasoning on several well-known ABox reasoning tasks. Our calculus extends a previously introduced \ke\space based decision procedure for the CQA problem.

翻译：摘要：本文考虑了描述逻辑 $\mathcal{DL}\langle \mathsf{4LQS^{R,\!\times}}\rangle(\mathbf{D})$（简记为 $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$）中最常见的ABox推理服务，并通过将其归约到集合论片段 \flqsr 的可满足性问题，证明了这些推理服务的可判定性。描述逻辑 $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ 具有极强的表达能力——它支持多种概念与角色构造子以及数据类型，从而能够表示SWRL等基于规则的语言。通过定义一种称为高阶合取查询回答（HOCQA）的广义合取查询回答问题（该问题可实例化为最广泛的ABox推理任务），我们获得了可判定性结果。此外，本文提出了一种基于 $\ke$ 空间的过程，用于从 $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ 知识库和高阶 $\mathcal{DL}_{\mathbf{D}}^{4,\!\times}$ 合取查询中计算答案集，从而为多种常见ABox推理任务提供推理手段。该演算扩展了先前针对CQA问题提出的基于 $\ke$ 空间的判定过程。