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, which allows to compute the semiring provenance for every additively and multiplicatively idempotent commutative semiring, and for which we study the complexity of problems related to the provenance of an axiom 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 ELHI_bot ontology that guarantee tractable reasoning.

翻译：我们研究了半环溯源——这一最初在关系数据库环境中成功的框架——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元素，这些注释以反映其推导过程的方式传播到本体推论中。我们为涵盖多种轻量级描述逻辑的语言定义了溯源语义，并展示了其与为特定类型注释（如模糊度）标注的本体所定义的语义之间的关系。我们证明，在半环的某些限制下，该语义满足理想性质（例如，扩展了为数据库定义的半环溯源）。随后，我们重点关注著名的为什么溯源（why-provenance），它允许为每个加法幂等和乘法幂等交换半环计算半环溯源，并研究与此类溯源相关的公理或合取查询答案问题的复杂性。最后，我们考虑了两个更受限的情况，对应于数据库中所谓的正布尔溯源（positive Boolean provenance）和谱系（lineage）。针对这些情况，我们展示了与描述逻辑中解释相关的著名概念之间的联系，并完成了复杂性分析。作为附加贡献，我们提供了保证ELHI_bot本体可推理性（tractable reasoning）的条件。