We investigate semiring provenance--a successful framework originally defined in the relational database setting--for description logics. In this context, the ontology axioms are annotated with elements of a commutative semiring and these annotations are propagated to the ontology consequences in a way that reflects how they are derived. We define a provenance semantics for a language that encompasses several lightweight description logics and show its relationships with semantics that have been defined for ontologies annotated with a specific kind of annotation (such as fuzzy degrees). We show that under some restrictions on the semiring, the semantics satisfies desirable properties (such as extending the semiring provenance defined for databases). We then focus on the well-known why-provenance, for which we study the complexity of problems related to the provenance of an assertion or a conjunctive query answer. Finally, we consider two more restricted cases which correspond to the so-called positive Boolean provenance and lineage in the database setting. For these cases, we exhibit relationships with well-known notions related to explanations in description logics and complete our complexity analysis. As a side contribution, we provide conditions on an $\mathcal{ELHI}_\bot$ ontology that guarantee tractable reasoning.

翻译：我们研究半环溯源——一个最初在关系数据库领域成功定义的框架——在描述逻辑中的应用。在此背景下，本体公理被赋予交换半环的注释元素，这些注释以反映其推导方式的形式传播至本体结论。我们为一个涵盖多种轻量级描述逻辑的语言定义了一种溯源语义，并展示了其与已为特定注释类型（如模糊度）标注的本体所定义的语义之间的关系。我们证明，在对半环施加某些限制时，该语义满足若干理想性质（例如扩展为数据库定义的半环溯源）。随后，我们聚焦于著名的何故溯源，研究与断言或合取查询答案溯源相关的问题复杂度。最后，我们考虑两种更具限制性的情况，对应数据库中的正布尔溯源与谱系。针对这些情况，我们揭示了其与描述逻辑中关于解释的熟知概念之间的关系，并完成了复杂度分析。作为附加贡献，我们给出了确保一个$\mathcal{ELHI}_\bot$本体具备可处理推理的条件。