Reasoning in Fuzzy Description Logics with General Concept Inclusion Axioms

描述逻辑（dl）是一组基于逻辑的知识表示语言，通过允许知识工程师在此逻辑中定义应用程序领域的相关概念，然后使用终止推理算法对这些定义进行推理，这些语言专门用于表示术语知识。为了处理概念的成员和非成员之间的界限模糊的应用程序（例如，“高个子”，“高血压”或“繁重的网络负载”），dl与模糊逻辑相结合，产生了模糊描述逻辑（模糊dl）。考虑到过去20年关于模糊描述逻辑的文献，人们可能会得到这样的印象，从算法的角度来看，模糊人工智能的行为与它们的清晰对立物非常相似：对于基于简单t-norm的模糊dl（如g¨model），可以使用调用相应清晰dl推理器的黑盒过程，而基于更复杂t-norm（如product和Lukasiewicz）的模糊dl可以通过适当修改基于表的清晰dl推理器来处理。然而，最近证明，在所谓的一般概念包含公理（gci）存在的情况下，已发布的基于表的推理器到模糊dl的扩展不能正确工作。事实上，我们能够证明gci可以导致基于产品t-norm的某些模糊dl的不可判定性。然而，对于大多数模糊dl来说，推理的可判定性仍然是开放的。本课题的目的是研究具有gci的模糊dl的可判定性和不可判定性之间的边界。一方面，我们将尝试为特定的模糊dl显示更多的不可判定结果，然后尝试从这些结果中推导出暗示不可判定的一般标准。另一方面，我们将尝试确定可确定的特殊情况，将基于表和自动化的决策过程扩展到模糊情况，并研究用于非表达性dl的其他推理方法。

Description logics (DLs) are a family of logic-based knowledge representation languages that are tailored towards representing terminological knowledge, by allowing the knowledge engineer to define the relevant concepts of an application domain within this logic and then reason about these definitions using terminating inference algorithms. In order to deal with applications where the boundaries between members and non-members of concepts (e.g., “tall man,” “high blood pressure,” or “heavy network load”) are blurred, DLs have been combined with fuzzy logics, resulting in fuzzy description logics (fuzzy DLs). Considering the literature on fuzzy description logics of the last 20 years, one could get the impression that, from an algorithmic point of view, fuzzy DLs behave very similarly to their crisp counterparts: for fuzzy DLs based on simple t-norms such as G¨odel, black-box procedures that call reasoners for the corresponding crisp DLs can be used, whereas fuzzy DLs based on more complicated t-norms (such as product and Lukasiewicz) can be dealt with by appropriately modifying the tableau-based reasoners for the crisp DLs. However, it has recently turned out that, in the presence of so-called general concept inclusion axioms (GCIs), the published extensions of tableaubased reasoners to fuzzy DLs do not work correctly. In fact, we were able to show that GCIs can cause undecidability for certain fuzzy DLs based on product t-norm. However, for most fuzzy DLs, the decidability status of reasoning w.r.t. GCIs is still open. The purpose of this project is to investigate the border between decidability and undecidability for fuzzy DLs with GCIs. On the one hand, we will try to show more undecidability results for specific fuzzy DLs, and then attempt to derive from these results general criteria that imply undecidability. On the other hand, we will try to determine decidable special cases, by extending tableau- and automatabased decision procedures for DLs to the fuzzy case, and also looking at other reasoning approaches for inexpressive DLs.